HOMOTOPY THEORY OF COMODULES OVER A HOPF
ALGEBROID
MARK HOVEY
Abstract.Given a good homology theory E and a topological space X,
E*X is not just an E*-module but also a comodule over the Hopf algebroid
(E*, E*E). We establish a framework for studying the homological algebra
of comodules over a well-behaved Hopf algebroid (A, ). That is, we con-
struct the derived category Stable( ) of (A, ) as the homotopy category*
* of
a Quillen model structure on Ch( ), the category of unbounded chain com-
plexes of -comodules. This derived category is obtained by inverting t*
*he
homotopy isomorphisms, NOT the homology isomorphisms. We establish the
basic properties of Stable( ), showing that it is a compactly generated *
*tensor
triangulated category.
Introduction
Given a commutative ring k, a Hopf algebroid over k is a cogroupoid object
in the category of commutative k-algebras. That is, a Hopf algebroid is a pair
(A, ) of commutative k-algebras, so that, given a commutative k-algebra R, the
set k-alg(A, R) is naturally the objects of a groupoid with morphisms k-alg( , *
*R).
This gives several structure maps of which we remined the reader below. The rea-
son for our interest in Hopf algebroids is that, if E*(-) is a well-behaved hom*
*ology
theory on topological spaces, then E*X is naturally a comodule over the Hopf al-
gebroid (E*, E*E). In particular, the study of comodules over the Hopf algebroid
BP*BP led to the Landweber exact functor theorem [Lan76], a result of funda-
mental importance in algebraic topology. When (E*, E*E) is a Hopf algebroid, the
E2-term of the Adams spectral sequence based on E is the bigraded Ext in the
category of E*E-comodules.
Thus we would like to understand the homological algebra of comodules over
a Hopf algebroid. The simplest kind of Hopf algebroid is a discrete Hopf alge-
broid (A, A). The associated groupoid has no non-identity maps, and a comodule
over (A, A) is the same thing as an A-module. One of the most useful tools in
studying the homological algebra of A-modules is the unbounded derived category
D(A), obtained by inverting the homology isomorphisms in the category Ch(A) of
unbounded chain complexes of A-modules. The goal of this paper is to construct
D(A, ), the derived category of a Hopf algebroid (A, ). We stress that homolo*
*gy
isomorphisms are NOT the right thing to invert to form D(A, ). This is already
clear in case (A, ) is a Hopf algebra over a field k, such as the Steenrod alg*
*e-
bra. In this case, D(A, ) was constructed by the author in [Hov99 , Section 2.*
*5]
and studied by Palmieri [Pal01]. The idea is that chain complexes of comodules
are like topological spaces; they have homotopy as well as homology, and it is *
*the
____________
Date: May 28, 2003.
1
2 MARK HOVEY
homotopy isomorphisms we should invert, not the homology isomorphisms. For a
discrete Hopf algebroid (A, A), homotopy and homology coincide, but not in gen-
eral. To avoid confusion, we refer to D(A, ) as the stable homotopy category *
*of
-comodules, and denote it by Stable( ).
We want Stable( ) to have all the usual properties of D(A) or the stable ho-
motopy category; it should be a triangulated category with a compatible closed
symmetric monoidal structure, and there should be a good set of generators. In
fact, we want Stable( ) to be a stable homotopy category in the sense of [HPS97*
* ].
We also want
Stable( )(S0A, S0M)* ~=Ext*(A, M)
for a comodule M, where S0N denotes the complex consisting of N in degree 0 and
0 everywhere else. This will guarantee that we recover the E2-term of the Adams
spectral sequence based on E.
The axioms for a stable homotopy category, or, indeed, even for a triangulated
category, are so painful to check that the best way to construct such a categor*
*y is as
the homotopy category of a Quillen model structure [Qui67]. One of the main goa*
*ls
of [Hov99 ] was to enumerate the conditions we need on a model structure so tha*
*t its
homotopy category is a stable homotopy category. We also point out that there a*
*re
many advantages of a model structure over its associated homotopy category; the
model structure allows one to perform constructions, such as homotopy limits and
colimits, that are inaccessible in the homotopy category, and allows one to make
comparisons with other model categories.
Thus, the bulk of this paper is devoted to constructing a model structure on
Ch( ) in which the weak equivalences are the homotopy isomorphisms. We ex-
pect that the associated stable homotopy category Stable( ) will have many good
properties and will provide insight into homotopy theory, as Palmieri's work [P*
*al01,
Pal99] on the Steenrod algebra has done. We have in fact shown that Stable(E*E)
is a Bousfield localization of Stable(BP*BP ) for any Landweber exact commuta-
tive ring spectrum E in [Hov02a ]; this then gives rise to a general change of *
*rings
theorem containing the change of rings theorem of Miller-Ravenel [MR77 ] and the
author and Sadofsky [HS99a ]. The construction of the model structure is compli-
cated enough that we do not discuss such applications in this foundational pape*
*r.
We do establish some beginning properties of Stable( ) in Section 6. In particu*
*lar,
we show that Stable( ) is monogenic (that is, bigraded suspensions of the sphere
weakly generate the category) when = BP*BP or = E*E for E any Landweber
exact homology theory over BP .
In order to establish our model structure, we need to first study the structu*
*re
of the abelian category -comod of -comodules, which we do in Section 1. Most
of the results in this section seem to be new, at least in the generality in wh*
*ich we
give them, and of independent interest. For example, we study duality in -como*
*d,
showing that a comodule is dualizable if and only if it is finitely generated a*
*nd
projective as an A-module.
We follow the usual plan to construct our model structure. That is, we start
by building an auxiliary model structure in Section 2 called the projective mod*
*el
structure. This model structure is easy to construct, but has too few weak equi*
*va-
lences (unless the Hopf algebroid is discrete). So we must localize it by makin*
*g the
homotopy isomorphisms weak equivalences. This first necessitates a study of the
homotopy isomorphisms in Section 3 and a reminder, with a few new results, about
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 3
localization of model categories in 4. We finally construct the desired model s*
*truc-
ture in 5, and study some of the basic properties of Stable( ) in the aforement*
*ioned
Section 6.
We should note that our results do not apply to an arbitrary Hopf algebroid. *
*We
need our Hopf algebroid to be amenable, defined precisely in Definition 2.3.2. *
*All of
the amenable Hopf algebroids we know are in fact Adams Hopf algebroids, defined
in [GH00 ] but implicit in Adams' blue book [Ada74 , Section III.13]. If E is a*
* ring
spectrum that satisfies Adams' condition, which we call topologically flat, tha*
*t E
be a minimal weak colimit of finite spectra Xffsuch that E*Xffis finitely gener*
*ated
and projective over E*, then (E*, E*E) is an Adams Hopf algebroid (Section 1.4).
To make sure our results apply in cases of interest, we must check that interes*
*ting
ring spectra E are topologically flat. We do this in Section 1.4, building on r*
*esults
of Adams and of Hopkins.
Hopf algebroids also arise in algebraic geometry in connection with stacks [F*
*C90 ].
More precisely, a Hopf algebroid (A, ) is the same thing as a representable sh*
*eaf
of groupoids Spec(A, ) with respect to the flat topology on the category of af*
*fine
schemes. A -comodule is equivalent to a quasi-coherent sheaf over Spec(A, ),
as is proved in [Hov02b ]. It is then natural to ask whether the approach we ta*
*ke
in this paper to study the homological algebra of comodules can be generalized *
*to
quasi-coherent sheaves over possibly non-representable sheaves of groupoids, su*
*ch
as stacks. We do not know how to do this. The best we can say is that the Quill*
*en
equivalence class of our model structure on chain complexes of comodules depends
only on the homotopy type of the associated stack of the given Hopf algebroid. *
*The
stacks that arise in algebraic topology, such as the stack of formal groups, ar*
*e the
associated stacks of Hopf algebroids, so this does give us a good construction *
*of the
unbounded derived category for such stacks.
The author has been trying to prove the results in this paper since 1997, when
Doug Ravenel strongly encouraged him to build a stable homotopy category of
BP*BP -comodules. It is a pleasure to acknowledge the author's debt to Neil Str*
*ick-
land, who constructed Stable(BP*BP ) in a fairly ad hoc way, without a model
structure, about 1997. The crucial input that finally enabled the author to bui*
*ld
the model structure came from the paper of Paul Goerss and Mike Hopkins [GH00 ].
1. The abelian category of comodules
We begin with a fairly comprehensive study of the category -comod of comod-
ules over a Hopf algebroid (A, ). Some of these results are well-known, but ot*
*hers
are apparently new.
Before we begin, we establish notation and remind the reader of some of the
basic structure maps of Hopf algebroids. The symbol (A, ) will always denote a
Hopf algebroid [Rav86 , Appendix 1], and the symbol always denotes A , the
tensor product of A-bimodules. Given an A-bimodule M, fM denotes M with the
A-actions reversed.
With these conventions, the structure maps of (A, ) include maps of commuta-
tive k-algebras jL :A -! corepresenting the source of a morphism, jR :A -!
corepresenting the target of a morphism, and ": -!A corepresenting the identi*
*ty
maps of the groupoid. This makes into an A-bimodule, with jL giving the
left A-action and jR giving the right A-action. There are then additional stru*
*c-
ture maps of k-algebras Ø: -! ecorepresenting the inverse of a morphism, and
4 MARK HOVEY
: -! corepresenting the composition of a pair of morphisms. Of course,
these maps must satisfy some relations assuring that we get a groupoid. For exa*
*m-
ple, "jR = "jL = 1A , since the source and target of the identity map at x are *
*both
x. The remaining relations can be found in [Rav86 , Appendix 1].
1.1. Basic structure. Recall that a left -comodule is a left A-module M equipp*
*ed
with a map _ :M -! M satisfying a coassociativity and counit condition. There
is an obvious notion of a map of comodules.
Lemma 1.1.1. Suppose is flat as a right A-module. Then the category -comod
is a cocomplete abelian subcategory of A-mod.
Proof.Since the tensor product commutes with colimits, the A-module colimit of
a diagram of comodules is again a comodule, and is the colimit in -comod. That
-comod is abelian when is flat is proved in [Rav86 , Theorem A1.1.3]; we req*
*uire
flatness in order to conclude that the A-module kernel of a comodule map is aga*
*in
a comodule.
Because of this lemma, we will assume throughout the paper that (A, ) is a
flat Hopf algebroid; that is, that is flat as a right A-module. Note that the
conjugation Ø defines an isomorphism between the left A-module and the right
A-module , so is also flat as a left A-module.
Lemma 1.1.2. The category -comod is a symmetric monoidal category. We
denote the symmetric monoidal product by M ^ N.
The symmetric monoidal product is of course given by the tensor product, but
the author, following Margolis [Mar83 ], thinks it is better to reserve the not*
*ation
M N for the tensor product of A-bimodules.
Proof.We define M ^ N = M N, the tensor product of left A-modules, with
comodule structure given by the composite
M N _-_--!( M) ( N) g-! M N,
where g(x m y n) = xy m n. Note that g involves both multiplication a*
*nd
the twist isomorphism, and we must do both of these together to get the necessa*
*ry
bilinearity. We leave it to the reader to check that this does define a map fro*
*m the
tensor product, and that the composition above is a comodule structure. The unit
of the tensor product is A, with comodule structure given by jL.
We now point out that the category of comodules is natural. Recall that a map
: (A, ) -!(B, ) of Hopf algebroids is a pair of ring homomorphisms 0: A -!B
and 1: -! that corepresents a natural morphism of groupoids. This means
that 0ffl = ffl 1, 1jL = jL 0, 1jR = jR 0, and ( 1 1) = 1.
Lemma 1.1.3. A map : (A, ) -!(B, ) induces a symmetric monoidal functor
*: -comod -! -comod.
Proof.Define *M = B A M. The -comodule structure on B A M is given by
the composite
B M 1-_--!B M -! M ~= B (B M),
where the map B -! takes b x to b 1(x).
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 5
In light of this lemma, the following definition is natural.
Definition 1.1.4. A map : (A, ) -! (B, ) is a weak equivalence if * is an
equivalence of categories.
The notion of weak equivalence is a fundamentally new feature that arises in
studying Hopf algebroids; any weak equivalence of discrete Hopf algebroids is n*
*ec-
essarily an isomorphism, but there are many examples of weak equivalences of
Hopf algebroids that are not isomorphisms given in [Hov02b ] and [HS02 ]. For e*
*x-
ample, if E = v-1nBP and F = E(n), it is proved in [HS02 ] that the evident map
(E*, E*E) -!(F*, F*F ) is a weak equivalence of Hopf algebroids. The author used
a different definition of weak equivalence in [Hov02b ], but the two definition*
*s are
in fact equivalent [HS02 ]. In fact, is a weak equivalence of Hopf algebroids*
* if and
only if it induces a homotopy equivalence of the associated stacks (that is, if*
* and
only if induces a weak equivalence in the Hollander model structure on sheaves
of groupoids [Hol01]).
A particular example of a map of Hopf algebroids is the map ffl: (A, ) -!(A,*
* A)
that is the identity on A and the counit ffl on . Geometrically, this is the i*
*nclusion of
the identity maps of a groupoid into the whole groupoid. The functor ffl* is ju*
*st the
forgetful functor from -comodules to A-modules. As is well known [Rav86 , A1.2*
*.1],
this functor has a right adjoint that takes an A-module M to the -comodule M,
with structure map 1. This is called the extended comodule on M; in case M
is itself a free A-module on the set S, then M is called the cofree comodul*
*e on
M. We have a natural isomorphism
A-mod (M, N) -! -comod(M, N)
for -comodules M and A-modules N. This natural isomorphism takes a map
f :M -!N of A-modules to the map of comodules (1 f)_, and a map of comodules
g :M -! N to the map (ffl 1)g of A-modules.
It is less well-known that the extended comodule functor M 7! M itself
has a right adjoint R: -comod -!A-mod , defined by RN = -comod( , N). The
A-module action on RN is defined by (af)(x) = f(xjR (a)).
Note that, if M is itself a comodule, then we can form the extended comodule
M and the tensor product ^ M. The following lemma is well-known.
Lemma 1.1.5. Suppose (A, ) is a flat Hopf algebroid, M is an A-module, and N
is a -comodule. Then there is a natural isomorphism of comodules
( M) ^ N -! (M N).
In particular, when M = A, we get a natural isomorphism of comodules
^ N -! N.
Proof.We first note that ( M) ^ N is the tensor product of the left A-modules
M and N. There is a natural comodule map
fMN :( M) ^ N -! (M N)
adjoint to " 1 1. For fixed N, this is a natural transformation of right ex*
*act
functors of M that commutes with direct sums. Since every A-module is a quotient
of a map of free A-modules, it suffices to show that fAN is an isomorphism.
In fact, we construct an inverse g = gAN to fAN . We define g to be the compo*
*site
M 1-_--! M (~O(1-Ø))-1------! ^ M,
6 MARK HOVEY
which is, a priori, only a map of A-modules. Note that, though the multiplicati*
*on ~
does not factor through , the composite ~O(1 Ø) does do so, since Ø switches
the left and right units. A diagram chase shows that g and fAN are inverses (a*
*nd
therefore that g is a comodule map).
It is tempting to think that, given an A-module M, one can think of M as a
trivial -comodule, via the map jL 1: M -! M. This is wrong; for example,
v-1nBP*, for n > 0, cannot be given the structure of a BP*BP -comodule [JY80 ,
Proposition 2.9]. The difficulty is that jL is not a map of A-bimodules. Howeve*
*r,
there is a symmetric monoidal trivial comodule functor from the category of abe*
*lian
groups to -comodules that takes the abelian group M to A Z M with the trivial
comodule structure given by jL Z 1. This functor has a right adjoint that takes
the comodule N to the abelian group of primitive elements in N.
1.2. Limits. In general, right adjoints such as limits are difficult to constru*
*ct for
comodules, because the forgetful functor from -comodules to A-modules does not
preserve products, though it does preserve kernels. We give a general method for
constructing right adjoints, involving resolutions by extended comodules. This *
*is
the same method, using right adjoints rather than left adjoints, used in [BW85 *
* ] to
construct colimits in the category of algebras over a triple. It is really an a*
*pplication
of the special adjoint functor theorem.
For a comodule M, the adjoint to the identity map is the map _ :M -! M,
which we now think of as a map of comodules, giving M the extended comodule
structure. The map _ is of course an embedding, since it is split over A by ffl*
* 1. In
particular, if p: M -!N denotes the cokernel of _, which is itself a comodu*
*le,
then we have a natural diagram
(1.2.1) M _-! M _p--! N
expressing M as the kernel of a map of extended comodules.
Now, if R is a right adjoint, then R will have to preserve kernels, so R is c*
*om-
pletely determined by its restriction to the full subcategory of extended comod*
*ules.
We first use this idea to show that -comod is complete.
Proposition 1.2.2. Suppose (A, ) is a flat Hopf algebroid. Then -comod has
products, and so is complete.
Q
Proof.Let us denote the comodule product we are trying to construct by iMi.
Adjointness shows that, if {Mi} is a set of A-modules, then
Y Y
( Mi) ~= Mi.
i i
Now suppose we have a set ofQcomodule maps fi: Mi -! Ni. We need to
define the comodule product ifi. We define it to be the composite
Y 1 Y 1 ff Y
Mi---! Mi---! ( Mi)
-1-Q-fi--! Y ( N 1 Q (ffl 1) Y
i) -------! Ni,
where ff is the evident natural transformation. The reader can then check that *
*this
is a good definition of theQproduct on the fullQsubcategory of extended comodul*
*es,
and, in particular, that i(1 gi) = 1 gi for A-module maps gi.
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 7
The definition of the product for a family of general comodules Miis now forc*
*ed
on us, as explained in the paragraph preceding this proposition. To wit, given *
*a set
of comodules Mi, we have left exact sequences
0 -!Mi-_! Mi-_pi-! Ni,
Q
and so we define iMi by the left exact sequence
Y Y Q (_p ) Y
0 -! Mi-! Mi- i---i-! Ni.
i i i
Q
We leave to the reader the proof that iMiis indeed the product in -comod.
Remark. An alternative approach to the category of -comodules that is some-
times used (e.g., by Boardman [Boa82]) is to establish an equivalence of catego*
*ries
between -comodules and a subcategory of *-modules. Here * = Hom rA( , A),
the A-bimodule of right A-module maps from to A. It turns out that * is a
(noncommutative) algebra over k and there is a map of algebras A -! *. There is
a map M -ff!HomA( *, M). Using this map, any -comodule becomes a *-
module, and this clearly defines a faithful functor from -comodules to *-modu*
*les.
However, this functor will not in general be full, because the map ff need not *
*be
injective. If is projective over A, then ff is injective, but is not projec*
*tive over
A for many of the Hopf algebroids we are interested in. If is projective over*
* A,
one can establish an equivalence between -comodules and a full coreflective su*
*b-
category of *-modules. This means that the inclusion functor from -comodules
to *-modules has a right adjoint R. Indeed, if M is a *-module, let us denote*
* by
~*: M -!Hom A ( *, M) the adjoint to the structure map of M. Then
RM = {x 2 M|~*(x) = ff(y) for somey}.
We refer to RM as the largest subcomodule of M. One can then define the product
of a set of comodules {Mi} to be the largest subcomodule of the A-module produc*
*t.
Building on the remark above, note that, for a set of comodules {Mi}, there is
the natural commutative diagram of A-modules below.
Q Q Q
i Mi ----! ? iMi ----! ?iNi
ff?y ff?y
Q Q Q
iMi ----!Qi_ i( Mi)----!Qi_pii( Ni).
Q Q
This means that there is a natural induced map of A-modules iMi -! iMi.
This map is injective when ff is injective, which is certainly true if is pro*
*jective
over A. It is an isomorphism when ff is so, which is true if is finitely gene*
*rated
and projective over A.
Since the product is right adjoint to the exact diagonal functor, the product*
* is
left exact. But it need not be exact in general. Indeed, let A = Q and = A[x],
thought of as a primitively generated Hopf algebra over A. Let Xn = =(xn) for
n 1, and let Yn = A. There is a surjection Xn -! YnQthat sendsQxn-1 to 1 and
every other power of x toQ0. But one can check that Yn ~= n Yn, andQthat
there is no element of Xn thatQhits (1, 1, . .,.1, . .).. Indeed, Xn cons*
*ists of
those elements (f1, f2, . .).of Xn such that the degrees of fi are bounded.
8 MARK HOVEY
We can also use this technique of constructing right adjoints to prove the fo*
*llow-
ing proposition.
Proposition 1.2.3. Suppose : (A, ) -! (B, ) is a map of Hopf algebroids.
Then the functor *: -comod -! -comod has a right adjoint *.
Proof.An adjointness argument shows that we must define
*( B N) = N
when N is a B-module. Given a comodule map f : B N -! B N0, we define
*(f): N -! N0 as the following composite.
N --1-! N 1-ff--! B N 1-f--! B N0 1-ffl-1--! N0.
Here the map ff is defined by ff(x n) = 1(x) n. We leave it to the reader
to check that this definition is functorial, so that we have defined * on the *
*full
subcategory of extended comodules.
As usual, given an arbitrary -comodule N, we write N as the kernel of
_p: B N -! B N0,
where p: B N -! N0 is the cokernel of _. We then define *(N) as the kernel
of *
N --(_p)--! N0.
We leave to the reader the check that * is right adjoint to *.
In fact, as pointed out to the author by Mark Behrens, *(N) ~=( A B) N,
where the symbol denotes the cotensor product. This construction is studied
briefly in [Rav86 , Lemma A1.1.8], though the adjointness is not proved there.
1.3. Duality and finite presentation. We now show that -comod is in fact a
closed symmetric monoidal category, and we characterize the dualizable comodule*
*s.
Theorem 1.3.1. If (A, ) is a flat Hopf algebroid, then the category -comod is
closed symmetric monoidal. Furthermore, the closed structure F (M, N) is left e*
*xact
in N and right exact in M.
Proof.An adjointness argument shows that we must define
F (M, N) = Hom A(M, N).
Suppose we have a map f : N -f! N0 of extended comodules. We need to
define the map F (M, f): Hom A(M, N) -! Hom A(M, N0). This map will
be adjoint to a map
Hom A(M, N) -!Hom A (M, N0)
of A-modules, which will in turn be adjoint to the composite
Hom A(M, N) M 1-Ev--! N f-! N0 ffl-1-!N0.
We leave to the reader the check that this definition is functorial, so that we*
* have
defined F (M, -) on the full subcategory of extended comodules. We also leave to
the reader the check that there is a natural isomorphism
-comod(L, F (M, N)) ~= -comod(L ^ M, N),
where naturality refers to an arbitrary map of extended comodules N -! N0.
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 9
We then have no choice but to define F (M, N) as the kernel of
Hom (M, N) F(M,_p)------! Hom (M, N0)
where p: N -!N0 is the cokernel of _. The necessary adjunction isomorphism
follows immediately.
Since F (M, -) is right adjoint to the right exact functor M -, it is left *
*exact.
Now suppose we have a right exact sequence
M0 -!M -!M00-!0.
Then we have a right exact sequence
L ^ M0 -!L ^ M -!L ^ M00-!0,
and so a left exact sequence of abelian groups
0 -! -comod(L ^ M00, N) -! -comod(L ^ M, N) -! -comod(L ^ M0, N).
Applying adjointness, we find that F (M00, N) has the universal property charac*
*ter-
izing the kernel of F (M, N) -!F (M0, N).
It would be nice to have a better understanding of F (M, N). The following
proposition is helpful.
Proposition 1.3.2. Suppose (A, ) is a flat Hopf algebroid, and M and N are
-comodules.
(a) There is a natural map F (M, N) øMN---!HomA(M, N) of A-modules.
(b) If M is finitely generated over A, then øMN is injective.
(c) If M is finitely presented over A, then øMN is an isomorphism.
Proof.Consider the natural diagram below.
F (M, N) ----! Hom A(M, N) ----! Hom A(M, N0)
?? ?
y ?y
Hom A(M, N) ----! Hom A(M, N) ----! Hom A(M, N0)
The vertical arrows take x f to the map that takes m to x f(m). It is not
obvious that this diagram is commutative, but a careful diagram chase shows that
it is. The rows both express their left-hand entry as a kernel, the first row *
*by
definition, and the second row by applying Hom A(M, -) to diagram 1.2.1. Thus,
there is a natural induced map F (M, N) -!Hom A (M, N), proving part (a).
Parts (b) and (c) will follow if we can show that the vertical maps are injec*
*tions
when M is finitely generated and isomorphisms when M is finitely presented. Sin*
*ce
is flat as a right A-module, we can write = colimCi, where the Ci are finit*
*ely
generated projective A-modules. The natural map
Ci Hom A(M, N) -!Hom A (M, Ci N)
is therefore an isomorphism. Hence, the map
Hom A(M, N) ~=colimiCi Hom A(M, N) -!colimiHom A(M, Ci N)
is also an isomorphism. When M is finitely generated over A, the map
colimiHomA (M, Ci N) -!Hom A (M, N)
is injective; when M is finitely presented over A, it is an isomorphism. Parts *
*(b)
and (c) follow.
10 MARK HOVEY
We now recall that an object M in a cocomplete category C is called ~-present*
*ed,
for a regular cardinal ~, if C(M, -) commutes with ~-filtered colimits (See [Bo*
*r94,
Section 6.4]). When ~ = !, we get the usual notion of a finitely presented obje*
*ct.
An A-module M is ~-presented if and only if it is a quotient of a map of free
modules, each of which has rank < ~.
Proposition 1.3.3. Suppose (A, ) is a flat Hopf algebroid, M is a -comodule,
and ~ is a regular cardinal. Consider the following three statements.
(a) M is ~-presented as a -comodule.
(b) M is ~-presented as an A-module.
(c) The functor F (M, -) commutes with ~-filtered colimits.
Then (a) and (b) are equivalent, and (c) implies (a). In particular, M is ~-pre*
*sented
for some ~.
Proof.We first show that (a) implies (b). So suppose that M is ~-presented as a
-comodule, and we have a ~-filtered diagram of A-modules Ni. Then
colimHom A(M, Ni) ~=colim -comod(M, Ni)
~= -comod(M, colim( Ni)) ~= -comod(M, colimNi)
~=Hom A(M, colimNi).
We now show that (b) implies (a). Suppose that M is ~-presented as an A-
module, and we have a ~-filtered diagram of comodules Ni. We must show that
the map
colim -comod(M, Ni) -! -comod(M, colimNi)
is an isomorphism. It is obviously injective, since the forgetful functor to A-*
*modules
is faithful, and M is ~-presented as an A-module. On the other hand, suppose we
have a map f :M -! colimNi of comodules. As a map of A-modules f factors
through some map g :M -! Ni for some i. The difficulty is that g may not be a
map of comodules, since _g may not be equal to (1 g)_. But they are equal as
maps to
colimNj ~=colim( Nj),
so they must be equal in some Nj. It follows that the composite
M g-!Ni-! Nj
is the desired factorization of f.
We now show that (c) implies (a). So suppose that F (M, -) commutes with
~-filtered colimits, and we have a ~-filtered diagram of comodules Ni. Then
colim -comod(M, Ni) ~=colim -comod(A, F (M, Ni))
~= -comod(A, colimF (M, Ni)) ~= -comod(A, F (M, colimNi))
~= -comod(M, colimNi),
where the second isomorphism holds because A is finitely presented.
In any closed symmetric monoidal category with unit A, we define DM =
F (M, A). There is always a natural map DM ^ N -! F (M, N). When this map
is an isomorphism for all N, M is called strongly dualizable, which we generally
abbreviate to dualizable. The author does not know to whom this concept is due;
perhaps Puppe [Pup79 ]. An excellent reference is [LMSM86 , Chapter III], and *
*the
basic properties of dualizable objects are summarized in [HPS97 , Theorem A.2.5*
*].
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 11
Proposition 1.3.4. Suppose (A, ) is a flat Hopf algebroid. Then a -comodule
M is dualizable in -comod if and only if M is finitely generated and projectiv*
*e as
an A-module.
Proof.First suppose that M is finitely generated and projective as an A-module.
We need to check that the map F (M, A) ^ N -! F (M, N) is an isomorphism.
But F (M, A) ~=Hom A(M, A) and F (M, N) ~=Hom A(M, N) by Proposition 1.3.2.
It is well known and easy to check that the natural map Hom A(M, A) N -!
Hom A(M, N) is an isomorphism when M is a finitely generated projective.
Now suppose that M is dualizable. Then the functor F (M, -) commutes with
colimits, since it is isomorphic to F (M, A) ^ (-). Proposition 1.3.3 then imp*
*lies
that M is finitely presented as an A-module. We now show that M must in fact be
projective over A. Indeed, the functor F (M, -) is always left exact, and becau*
*se
M is dualizable, F (M, -) ~=F (M, A) ^ (-) is also right exact. Hence F (M, -) *
*is
an exact functor on the category of -comodules. But Proposition 1.3.2 tells us
that F (M, -) ~=Hom A(M, -) since M is finitely presented over A. Now suppose
E is an exact sequence of A-modules. Then Hom A(M, E) is again exact. But,
since M is finitely presented,
Hom A(M, E) ~= Hom A(M, E)
by the argument of Proposition 1.3.2. Since is faithfully flat over A, we con*
*clude
that Hom A(M, E) is exact, so M is projective over A.
1.4. Generators and Adams Hopf algebroids. We have just seen that the
category of -comodules has many good properties when (A, ) is a flat Hopf al-
gebroid. But those properties are still not enough for us, because we need a go*
*od
set of generators for the category of -comodules. Recall that a set of objects*
* G in
an abelian category C is said to generate C when C(P, f) = 0 for all P 2 G impl*
*ies
that f = 0.
For much of the sequel, we will require that the dualizable comodules generate
the category of -comodules. The main advantage of this hypothesis is the follo*
*wing
proposition.
Proposition 1.4.1. Suppose (A, ) is a flat Hopf algebroid for which the dualiz*
*able
comodules generate the category of -comodules. Then the category of -comodules
is a locally finitely presentable Grothendieck category. Furthermore:
(a) Every comodule is a quotient of a direct sum of dualizable comodules.
(b) If M is a comodule and x 2 M, then there is a dualizable comodule P and
a map P -! M of comodules whose image contains x.
(c) Every comodule that is finitely generated over A is a quotient of a dual*
*izable
comodule.
(d) Every comodule is the union of its subcomodules that are finitely genera*
*ted
over A.
(e) Every comodule is a filtered colimit of finitely presented comodules.
Most of these facts are true in a general locally finitely presentable Grothe*
*ndieck
category; see [Ste75] for details. We therefore give only a sketch of the proof.
Proof.There is only a set of isomorphism classes of dualizable comodules, and
dualizable comodules are finitely presented. Hence -comod is a locally finite*
*ly
presentable Grothendieck category. For part (a), let G denote a set containing *
*one
12 MARK HOVEY
element from each isomorphism class of dualizable comodules, and let T be the s*
*et
of all maps f with dom f 2 G and codom f = M. Consider the map
M
ff: dom f -!M,
f2T
and let fi denote the cokernel of this map. Then -comod(P, fi) = 0 for all dua*
*lizable
P , so fi = 0. Hence ff is surjective.
Part (b) is an immediate corollary of part (a), and part (c) and part (d) fol*
*low
easily from part (b). For part (e), we choose a small skeleton F of the categor*
*y of
finitely presented comodules and consider the category F=M consisting of all ma*
*ps
from an element of F to M. There is an obvious map
colimf2F=Mdom f -!M.
One can readily verify that F is filtered, and that this map is a monomorphism,
for any flat Hopf algebroid (A, ). If the dualizable comodules generate, then *
*it is
an epimorphism by part (b).
We also have the following corollary, which answers the question left open by
Proposition 1.3.3.
Corollary 1.4.2. Suppose (A, ) is a flat Hopf algebroid for which the dualizab*
*le
comodules generate the category of -comodules, M is a -comodule, and ~ is a
regular cardinal. If M is ~-presented, then F (M, -) commutes with ~-filtered c*
*ol-
imits.
Proof.Suppose we have a ~-filtered system of comodules Ni. We need to show that
colimF (M, Ni) ff-!F (M, colimNi)
is an isomorphism. Because the dualizable comodules generate, it suffices to sh*
*ow
that -comod(P, f) is an isomorphism for all dualizable comodules P . Since dua*
*l-
izable comodules are in particular finitely presented, we have
-comod(P, colimF (M, Ni)) ~=colim -comod(P, F (M, Ni))
~=colim -comod(P ^ M, Ni) ~=colim -comod(M, DP ^ Ni)
~= -comod(M, colim(DP ^ Ni)) ~= -comod(M, DP ^ colimNi)
~= -comod(M ^ P, colimNi) ~= -comod(P, F (M, colimNi)).
We now owe the reader some examples of flat Hopf algebroids for which the du-
alizable comodules generate the category of -comodules. We learned the followi*
*ng
definition from [GH00 ], but it is implicit in [Ada74 , Section III.13].
Definition 1.4.3. A Hopf algebroid (A, ) is said to be an Adams Hopf algebroid
when is the colimit of a filtered system of comodules i, where i is finitely
generated and projective over A.
In particular, any Adams Hopf algebroid is flat, since the colimit of project*
*ive
modules is flat. Thus the i are dualizable comodules. The following proposition
is a restatement of [GH00 , Lemma 3.4].
Proposition 1.4.4. If (A, ) is an Adams Hopf algebroid, then the dualizable co-
modules generate the category of -comodules.
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 13
Proof.Suppose (A, ) is Adams, and M is a -comodule. Then we have:
M ~=Hom A(A, M) ~= -comod(A, M)
~= -comod(A, ^ M) ~= -comod(A, colim i^ M)
~=colim -comod(A, i^ M) ~=colim -comod(D i, M).
The result follows.
We now give some examples of Adams Hopf algebroids. Most of the ones we
are interested in come from algebraic topology. Recall the notion of minimal we*
*ak
colimit from [HPS97 , Section 2.2].
Definition 1.4.5. A ring spectrum R is called topologically flat if R is the mi*
*nimal
weak colimit of a filtered diagram of finite spectra Xi such that R*Xi is a fin*
*itely
generated projective R*-module.
This definition is based on [Ada74 , Condition III.13.3].
Lemma 1.4.6. Suppose R is a ring spectrum that is topologically flat and such
that R*R is commutative. Then (R*, R*R) is an Adams Hopf algebroid.
Proof.Write R as the minimal weak colimit of the Xi. Then R*R = colimR*Xi.
In particular, R*R is flat over R* (and this is the reason for the term öt polo*
*gically
flat") and satisfies the Adams condition.
The reason for the hypothesis that R*R be commutative is that there could well
be non-commutative ring spectra R, such as Morava K-theory K(n) at the prime
2, where R*R is nevertheless commutative.
Adams gave several examples of topologically flat ring spectra in [Ada74 , Pr*
*opo-
sition III.13.4], which we restate here.
Theorem 1.4.7. The ring spectra MU, MSp, K, KO, HFp, and K(n) are topo-
logically flat.
Adams did not of course consider K(n), since it had not been discovered yet,
but his proof for HFp works for any field spectrum.
We can add another case to this list as well. Recall that BP is the Brown-
Peterson spectrum, and BP* ~=Z(p)[v1, v2, . .].. Given an invariant regular seq*
*uence
J = (pi0, vi11, . .,.vik-1k-1) in BP*, there is a spectrum BP J with BP J* ~=BP*
**=J
studied in [JY80 ].
Proposition 1.4.8. Let J = (pi0, vi11, . .,.vik-1k-1) be an invariant regular s*
*equence
of length k in BP*. Then BP J is topologically flat.
Proof.Write BP as a minimal weak colimit of spectra Xff, where Xffis a finite
spectrum with cells in only even degrees. Then BP J ^ BP is the minimal weak
colimit of the BP J ^Xff. On the other hand, we claim that BP J ^BP J is a wedge
of 2k copies of BP J ^BP . Indeed, BP J*BP J is a free module over BP J*BP , and
so we choose generators for the free module and use them to construct the desir*
*ed
splitting.
Hence BP J ^BP J is the minimal weak colimit of BP J ^Yff, where Yffis a fini*
*te
wedge of copies of Xff. Since each BP J*(Yff) is a free module, this completes *
*the
proof.
14 MARK HOVEY
The following theorem, generalizing Proposition 2.12 of [HS99b ], gives us ma*
*ny
other examples of topologically flat ring spectra. Recall that, if R is a ring *
*spectrum
and E is an R-module spectrum, then E is said to be Landweber exact over R if
the natural map
E* R* R*X -!R*X
is an isomorphism for all spectra X.
Theorem 1.4.9. Suppose R is a topologically flat ring spectrum, and E is a
Landweber exact R-module spectrum. Then E is topologically flat.
As mentioned in [HS99b ], a version of this theorem was certainly known to
Hopkins. Rezk also proves a version of this theorem as Proposition 15.3 of [Rez*
*98].
Proof.The proof is much like that of Proposition 2.12 of [HS99b ]. Write R as t*
*he
minimal weak colimit of finite spectra Xi such that R*Xi is finitely generated *
*and
projective over R*. Then E*Xi is finitely generated and projective over E*. We
show that E is the filtered colimit of a diagram of finite wedges of suspension*
*s of
Xi. To do so, consider the category F=E of all maps of finite spectra to E, and
consider the full subcategory F0=E of such maps whose domain is a (variable) fi*
*nite
wedge of suspensions of the Xj. We claim that this is cofinal in F=E. Since E is
the minimal weak colimit of the obvious functor from F=E to spectra, it will fo*
*llow
that E is the minimal weak colimit of the restriction of this functor to F0=E.
To show that F0=E is cofinal in F=E, it suffices to show that any map f from a
finite spectrum Z to E factors through a finite wedge of suspensions of the Xi.*
* By
Spanier-Whitehead duality
E*(Z) ~=E* R* R*Z.
P m
We can thus write f = i=1bi ci. Because R is the minimal weak colimit of the
Xj, each map ci has a factorization
ci= (Z gi-! -|ci|Xi-ei! -|ci|R).
W m
Let Y = i=1 -|ci|Xi, let g :Z -! Y be the map with components gi, and let
h: Y -! E be the map with components bi ei 2 E* R* R*Xi ~=E*(Xi). This
gives the desired factorization.
Of course, there are algebraic examples of Adams Hopf algebroids as well.
Proposition 1.4.10. Suppose is a Hopf algebra over a field k. Then (k, ) is
an Adams Hopf algebroid.
Proof.By Lemma 9.5.3 of [HPS97 ], every -comodule is the filtered colimit of i*
*ts
finite-dimensional sub-comodules. In particular, this is true for itself.
Proposition 1.4.11. Suppose (A, ) is an Adams Hopf algebroid, I is an invariant
ideal in A, and v is a primitive element in A. Then (A=I, =I) and (v-1A, v-1 )
are Adams Hopf algebroids.
Proof.Suppose that ~=colim j, where each j is finitely generated and project*
*ive
over A. Then =I = colim j=I and j=I is finitely generated and projective over
A=I. Similarly, v-1 = colimv-1 j.
Despite all these examples of Adams Hopf algebroids, there is a theoretical d*
*if-
ficuly with the notion.
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 15
Question 1.4.12. Suppose : (A, ) -! (B, ) is a weak equivalence of Hopf
algebroids. Is it true that (A, ) is Adams if and only if (B, ) is Adams?
Note that if is a weak equivalence, then the dualizable -comodules generate
if and only if the dualizable -comodules generate.
2. The projective model structure
In this section, we establish a preliminary model structure on Ch( ), the cat*
*egory
of unbounded chain complexes of -comodules.
2.1. Construction and basic properties. We recall the results of [CH02 ]. Be-
ginning with a set of objects S in a cocomplete abelian category A, there is a
projective class (P, E), where E consists of all maps f such that A(P, f) is on*
*to
for all P in S, and P consists of all retracts of direct sums of elements of S.*
* The
elements of P are called relative projectives, and the maps of E are called rel*
*ative
epimorphisms. This is [CH02 , Lemma 1.5], but it is also easy to see.
The main result of [CH02 ] associates a model structure on Ch(A), the category
of unbounded chain complexes in A, to a projective class (P, E), given some hy-
potheses. We recall that a chain map OE is a fibration in this model structure *
*when
A(P, OE) is a degreewise surjection for all P 2 P, and a weak equivalence when
A(P, OE) is a homology isomorphism for all P 2 P.
The hypothesis needed is that functorial cofibrant replacements exist. This *
*is
automatic, by [CH02 , Proposition 4.2], when A is complete and cocomplete, there
are enough ~-small P-projectives for some cardinal ~, and functorial P-resoluti*
*ons
exist. When P is generated by a set S as above, then functorial P-resolutions
obviously exist, since there is a functorial P-epic
M M
P -! M
P2Sf2A(P,M)
for any M 2 A. When each object of S is ~-small for some ~, then there are enou*
*gh
~-small P-projectives (take ~ to be the supremum of the ~'s).
Now, if A happens to be a Grothendieck abelian category, then it is automatic*
*ally
complete and cocomplete, and every object in A is ~-presented, and so a fortiori
~-small, for some ~. This latter statement is an immediate corollary of the fac*
*t that
Grothendieck abelian categories are locally presentable [Bek00, Proposition 3.1*
*0],
but a direct proof can be found in the Appendix to [Hov01 ].
We thus have the following result, which was inexplicably not stated in [CH02*
* ].
Theorem 2.1.1. Suppose A is a Grothendieck abelian category, and S is a set of
objects in A. Then there is a model structure on Ch(A) in which the fibrations *
*are
the maps OE such that A(P, OE) is a surjection for all P 2 S and the weak equiv*
*alences
are the maps OE such that A(P, OE) is a homology isomorphism for all P 2 S. More
generally, this model structure exists when A is complete and cocomplete, but n*
*ot
necessarily Grothendieck, as long as every object of S is ~-small for some ~.
Now we return to the case at hand, when A is the category of -comodules and
(A, ) is a Hopf algebroid. In the light of the results of Section 1.4, we shou*
*ld take
S to be the set of dualizable -comodules.
Definition 2.1.2. Suppose (A, ) is a flat Hopf algebroid. Let S be a set conta*
*ining
one comodule from each isomorphism class of dualizable -comodules. We refer to
16 MARK HOVEY
the retracts of direct sums of elements of S as relatively projective comodules*
*, and
to the maps f of comodules such that -comod(P, f) is surjective for all P 2 S
as relative epimorphisms. The resulting model structure on Ch( ) obtained from
Theorem 2.1.1 is called the projective model structure. Thus, a map OE is a pro*
*jective
fibration if OE is a degreewise relative epimorphism, and OE is a projective eq*
*uivalence
if -comod(P, OE) is a homology isomorphism for all P 2 S. The map OE is a proj*
*ective
cofibration, or simply a cofibration, if OE has the left lifting property with *
*respect
to all projective trivial fibrations. We refer to an chain complex F as project*
*ively
trivial if 0 -!F is a projective equivalence.
Goerss and Hopkins [GH00 ] put a model structure on the category of nonnega-
tively graded chain complexes over an Adams Hopf algebroid. Their model structu*
*re
gave us the idea for the projective model structure, but it is not the same, as*
* they
took S to be the set of all the D i(under the assumption that = colim i). We *
*do
not know whether the projective model structure (when restricted to nonnegative*
*ly
graded complexes) is Quillen equivalent to the Goerss-Hopkins model structure.
The answer would seem to depend on questions about the structure of dualizable
comodules that we are unable to answer. One obvious advantage of our definition*
* is
that the dualizable comodules are canonically attached to the symmetric monoidal
category of -comodules, while the D i are not.
We point out that we do not need (A, ) to be an Adams Hopf algebroid, or
even for the dualizable comodules to generate, for the projective model structu*
*re
to exist. Also note that every relatively projective comodule is projective as*
* an
A-module, but we do not know if the converse holds.
Note also that because the elements of S are finitely presented in the catego*
*ry
-comod (see Proposition 1.3.3), filtered colimits of projective equivalences (*
*resp.
projective fibrations) are again projective equivalences (resp. projective fibr*
*ations).
We then have the following theorem describing some of the properties of the
projective model structure.
Theorem 2.1.3. Suppose (A, ) is a flat Hopf algebroid. Then the projective mod*
*el
structure on Ch( ) is proper, finitely generated, stable, and symmetric monoida*
*l.
A map OE is a cofibration if and only if it is a degreewise split monomorphism *
*whose
cokernel is cofibrant. A chain complex X is cofibrant if and only if it is a re*
*tract of
a colimit of complexes
X0 -!X1 -!. .X.ff. . .
where each Xff-! Xff+1is a degreewise split monomorphism whose cokernel is a
complex of relative projectives with no differential. The homotopy relation bet*
*ween
cofibrant objects is the usual chain homotopy relation.
Proof.This all follows from the results of [CH02 ]. The characterization of cof*
*ibrant
objects follows from Corollary 4.4 of [CH02 ], and the characterization of cofi*
*brations
follows from Proposition 2.5 of [CH02 ]. The fact that the model structure is p*
*roper
is [CH02 , Proposition 2.18], and stability is [CH02 , Corollary 2.17] and obvi*
*ous.
The fact that homotopy is the usual chain homotopy is [CH02 , Lemma 2.13].
The generating cofibrations for the projective model structure are Sn-1P -!
DnP for P 2 S, and the generating trivial cofibrations are 0 -! DnP for P 2 S.
Here Sn-1P denotes the complex which is P in degree n - 1 and zero elsewhere,
and DnP denotes the complex which is P in degrees n and n - 1 and 0 elsewhere.
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 17
This is proved in Section 5 of [CH02 ]. Each of Sn-1P , 0, and DnP are finitely
presented, so the projective model structure is finitely generated.
Finally, the fact that the projective model structure is symmetric monoidal f*
*ol-
lows from Corollary 2.21 of [CH02 ], the fact that A, the unit of ^, is a relat*
*ive
projective, and the fact that relative projectives are closed under ^.
Note that, when (A, ) is discrete, a -comodule is the same thing as an A-
module. In this case, the relative projectives are just the projective A-modul*
*es,
and we see that the projective model structure agrees with the usual projective
model structure on Ch(A), in which the fibrations are the surjections and the w*
*eak
equivalences are the homology isomorphisms.
We point out that there is another model structure on Ch( ) given by [CH02 ,
Example 3.4] called the absolute model structure. In this model structure, the
weak equivalences are the chain homotopy equivalences, the cofibrations are the
degreewise split monomorphisms, and the fibrations are the degreewise split epi-
morphisms. Since the generating cofibrations of the projective model structure
are degreewise split monomorphisms, and the generating trivial cofibrations are
chain homotopy equivalences, we conclude that the identity functor is a left Qu*
*illen
functor from the projective model structure to the absolute model structure. In*
* par-
ticular, a trivial cofibration in the projective model structure is a chain hom*
*otopy
equivalence, and all chain homotopy equivalences are projective equivalences.
The symmetric monoidal product behaves particularly well with respect to the
projective model structure.
Proposition 2.1.4. Suppose (A, ) is a flat Hopf algebroid. Then the projective
model structure satisfies the monoid axiom. Furthermore, if X is cofibrant and f
is a projective equivalence, then X ^ f is a projective equivalence.
This proposition is important because of the work of Schwede and Shipley [SS0*
*0],
who introduced the monoid axiom. As a consequence of their work and Proposi-
tion 2.1.4, given a monoid R in Ch( ), which is just a differential graded como*
*dule
algebra, there is a model structure on (differential graded) R-modules in which
the fibrations are underlying projective fibrations and the weak equivalences a*
*re
underlying projective equivalences. There is also a similar model structure on
differential graded comodule algebras, and a projective equivalence R -!R0of di*
*f-
ferential graded comodule algebras induces a Quillen equivalence from R-modules
to R0-modules.
Proof.The monoid axiom, introduced by Schwede and Shipley in [SS00], asserts
that, if K is the class of maps {j ^ X} where j is a generating trivial cofibra*
*tion
and X is arbitrary, then all transfinite compositions of pushouts of maps of K *
*are
projective equivalences. In the case at hand, j is one of the maps 0 -!DnP , wh*
*ere
P 2 S. It follows easily that j ^ X is a dimensionwise split monomorphism and a
chain homotopy equivalence, so a trivial cofibration in the absolute model stru*
*c-
ture. Thus, all transfinite compositions of pushouts of maps of K are also triv*
*ial
cofibrations in the absolute model structure, and so in particular chain homoto*
*py
equivalences. Hence they are also projective equivalences.
Now suppose f :Y -! Z is a projective equivalence and X is cofibrant. We want
to show that X ^ f is a projective equivalence. Since X is cofibrant, X is a re*
*tract
of a colimit of a sequence of complexes {Xi}i<~, where Xi-! Xi+1is a degreewise
split monomorphism whose cokernel Ci is a complex of relative projectives with
18 MARK HOVEY
zero differential. Since projective equivalences are closed under filtered col*
*imits,
it suffices to show that Xi^ f is a projective equivalence for all i ~, where
X~ = colimXi. We prove this by transfinite induction on i. We assume the base
case i = 0 for the moment. The limit ordinal case follows from the fact the fil*
*tered
colimits of projective equivalences are projective equivalences. For the suces*
*sor
ordinal case, we have the commutative diagram below.
0 ----! Xi^ Y ----! Xi+1^ Y ----! Ci^ Y ----! 0
? ? ?
Xi^f?y Xi+1^f?y ?yCi^f
0 ----! Xi^ Z ----! Xi+1^ Z ----! Ci^ Z ----! 0
The rows of this diagram are degreewise split and short exact, since Xi-! Xi+1is
degreewise split. Now apply the functor -comod(P, -) to this diagram for a fix*
*ed
P 2 S. The rows of the resulting diagram will still be short exact. The inducti*
*on
hypothesis tells us that -comod(P, Xi^f) is a homology isomorphism, and the ba*
*se
case of the induction (which we have postponed) tells us that -comod(P, Ci^ f)*
* is
a homology isomorphism. The long exact sequence in homology then implies that
-comod(P, Xi+1^ f) is a homology isomorphism.
We are left with showing that C ^ f is a projective equivalence,Lwhere C is a
complex of relative projectives with zero differential. Then C ~= n SnCn. Again
using the fact that the objects in S are finitely presented, we find that it su*
*ffices to
show that SnCn ^ f is a projective equivalence. But Cn is a retract of a direct*
* sum
of elements of S. Another use of the fact that objects in S are finitely presen*
*ted
reduces us to showing that SnQ ^ f is a projective equivalence, for Q 2 S. Assu*
*me
P 2 S. Then, since Q is strongly dualizable in -comod by Proposition 1.3.4, we
have
-comod(P, SnQ ^ f) ~= -n -comod(P ^ DQ, f)
which is a homology isomorphism since P ^ DQ is also (isomorphic to something)
in S.
An obvious drawback with the projective model structure is that is difficult *
*to
tell what the weak equivalences look like. We do have the following proposition.
Proposition 2.1.5. Suppose (A, ) is a flat Hopf algebroid for which the dualiz*
*able
-comodules generate the category of -comodules. Then every projective fibrati*
*on
is surjective, and every projective equivalence is a homology isomorphism.
Proof.Suppose p: X -! Y is a projective fibration, and y 2 Yn. By Proposi-
tion 1.4.1, there is a comodule P in S and a map f :P -! Yn whose image contains
y. Suppose f(t) = y. Since p is a projective fibration, there is a map g :P -! *
*Xn
such that pg = f. In particular, pg(t) = y, so p is surjective.
Now suppose p is a projective equivalence. We wish to show that p is a homolo*
*gy
isomorphism. Every projective trivial cofibration is a chain homotopy equivalen*
*ce,
so a homology isomorphism. We can thus assume that p is a projective trivial
fibration. In particular, p is surjective. Thus it suffices to show that kerp i*
*s exact.
We know that kerp -! 0 is a projective trivial fibration, so -comod(P, kerp) is
exact for all P 2 S. Suppose that x is a cycle in kerpn. Let Zn denote the
comodule of cycles in kerpn. Then there is a P 2 S, a t 2 P , and a comodule
map f :P -! Zn such that f(t) = x, by Proposition 1.4.1. The map f is a cycle
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 19
in -comod(P, kerp), so there is a map g :P -! kerpn+1 such that dg = f. In
particular, dg(t) = x, so x is a boundary.
2.2. Naturality. We now show that the projective model structure is natural in
(A, ).
Proposition 2.2.1. Suppose : (A, ) -!(B, ) is a map of flat Hopf algebroids.
Then induces a left Quillen functor *: Ch( ) -!Ch ( ) of the projective model
structures.
Proof.We have seen in Proposition 1.2.3 that induces an adjunction
( *, *): -comod -! -comod.
This prolongs to an adjunction ( *, *): Ch ( ) -! Ch ( ) by defining * and
* degreewise. Since * is symmetric monoidal, it preserves dualizable comodule*
*s.
This is easy to see directly in this case, since if M is finitely generated and*
* projective
over A, then B M is finitely generated and projective over B. It follows easi*
*ly
that * takes the generating (trivial) cofibrations of the projective model str*
*ucture
on Ch( ) to (trivial) cofibrations in the projective model structure on Ch( ).
Note that there is a map of Hopf algebroids : (A, ) -! (A, A) which is the
identity on A and ffl on . The functor * is just the forgetful functor from *
* -
comodules to A-modules, and the right adjoint * is the extended comodule funct*
*or.
Hence, if f is a homology isomorphism of complexes of A-modules, then f is a
projective equivalence.
Theorem 2.2.2. Suppose : (A, ) -!(B, ) is a weak equivalence of flat Hopf al-
gebroids. Then *: Ch( ) -!Ch ( ) is a Quillen equivalence of the projective mo*
*del
structures. In fact, both * and * preserve and reflect projective equivalence*
*s.
Proof.Since * is a right Quillen functor, it preserves weak equivalences (betw*
*een
fibrant objects, but everything is fibrant). Since * is an equivalence, the u*
*nit
X -! * *X is an isomorphism, so * reflects projective equivalences. On the
other hand, * is a symmetric monoidal left adjoint since * is an equivalence *
*of
categories. In particular, * preserves dualizable comodules. Thus * is a left*
* (and
right) Quillen functor of the projective model structures. Hence * is also a l*
*eft
and right Quillen functor, so * preserves projective equivalences. Thus * ref*
*lects
projective equivalences.
To show that * is a Quillen equivalence, we need to show that, for X cofibra*
*nt
and Y fibrant, a map f : *X -! Y is a projective equivalence if and only if its
adjoint g :X -! *Y is a projective equivalence. Recall that g is obtained from f
as the composite
*f
X -! * *X --! *Y.
The first map in this composite is an isomorphism. Thus g is a projective equiv*
*a-
lence if and only if *f is so. But * preserves and reflects projective equiva*
*lences,
so *f is a projective equivalence if and only if f is so.
2.3. The cobar resolution. The projective model structure is clearly not the
model structure we want, because
hoCh( )(S0A, S0A)* ~=A
20 MARK HOVEY
concentrated in degree 0, because A is both cofibrant and fibrant in the projec*
*tive
model structure. Recall that we want
Stable( )(S0A, S0A)* ~=Ext*(A, A).
Therefore, we have to get an injective, or at least relatively injective, resol*
*ution of
A involved. See Section 3.1 for a description of relatively injective comodules.
The resolution we choose is the (reduced) cobar resolution [Rav86 , A1.2.11],
though we offer a simpler construction of it. Suppose M is a -comodule. Then _
is a natural comodule embedding M -! M of M into an extended comodule,
which is split over A by ffl 1. We can iterate this to construct a resolution*
* of M
by extended A-comodules. The most important case is when M = A. We begin
with the A-split short exact sequence of comodules
__
0 -!A jL--! -! -!0.
__
Here is of course the cokernel of jL, but it is easily seen to be isomorphic *
*to kerffl.
When we think of it as kerffl, the coaction_is defined by _(x) = (x) - x 1. *
*We
can then tensor this sequence with ^s to get the A-split short exact sequence *
*of
comodules __ __ __
0 -! ^s -! ^ ^s-! ^(s+1)-!0.
We splice these_short exact sequences together to obtain a complex LA, where
(LA)-n = ^ ^n for n 0 and (LA)-n = 0 for n < 0, and the differential is the
composite
__^n __^(n+1) __^(n+1)
^ -! -! ^ .
In particular, there is a homology isomorphism S0A -!LA induced by jL, so that_
LA is a resolution of A, and the cycle comodule Z-n (LA) is isomorphic to ^n f*
*or
n > 0 (and A for n = 0). Furthermore, the A-splittings patch together to show
that S0A -!LA is a chain homotopy equivalence of complexes of A-modules.
The complex LA will be very important in the rest of this paper, but LA is not
cofibrant in the projective model structure, since (LA)0 = is not even projec*
*tive
over A in general. The following proposition is then crucial for us.
Proposition 2.3.1. Suppose that (A, ) is a flat Hopf algebroid,_that the duali*
*zable
-comodules generate the category of -comodules, and that ^ X is projectively
trivial when X is so. Let LA denote the cobar resolution of A.
(a) If p is a projective fibration, then LA ^ p is a projective fibration.
(b) If p is a projective equivalence, then LA ^ p is a projective equivalenc*
*e.
Note that, for part (a), it is sufficient to assume that dualizable comodules
generate.
In view of Proposition 2.3.1, we make the following definition.
Definition 2.3.2. A Hopf algebroid (A, ) is amenable when it_is_flat, the dual-
izable -comodules generate the category of -comodules, and ^ (-) preserves
projectively trivial complexes.
For Proposition 2.3.1 to be of use, we need to know that amenable Hopf alge-
broids do exist.
Proposition 2.3.3. Every Adams Hopf algebroid is amenable.
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 21
The rest of this section will be devoted to proving Propositions 2.3.1 and 2.*
*3.3.
Proposition 2.3.3 is an immediate consequence of the following lemma and Propo-
sition 1.4.4.
Lemma 2.3.4. Suppose (A, ) is a flat Hopf algebroid.
(a) If M is a filtered colimit of dualizable comodules, then M ^ (-) preserv*
*es
projectively trivial complexes._
(b) If (A, ) is Adams, then is a filtered colimit of dualizable comodules.
Proof.For part (a), recall that filtered colimits of projective equivalences ar*
*e projec-
tive equivalences. We can therefore assume that M itself is a dualizable comodu*
*le.
Suppose then that X is projectively trivial. We must show that -comod(P, M ^X)
has no homology for all dualizable comodules P . But
-comod(P, M ^ X) ~= -comod(P ^ DM, X)
since M is dualizable. Furthermore, P ^ DM is again dualizable, so since X is
projectively trivial, we are done.
For part (b), since (A, ) is an Adams Hopf algebroid, we have = colimi2I i
for a filtered small category I of arrows i: i-! such that iis dualizable. *
*Let J
denote the category of factorizations A -! i-i! of jL through an arrow of I. By
abuse of notation, we write the map A -! i as jL as well; note that this jL must
be a split monomorphism of A-modules, since the usual jL is so. We claim that J
is filtered and that the obvious functor from J to I is cofinal (see Definition*
* 2.3.8
of [HPS97 ] for a reminder of what this means). This is a straightforwad conseq*
*uence
of the fact that A is itself finitely presented_as_a -comodule._It follows_the*
*n that
colimj2J j ~= , and therefore that colim j~= , where j = cokerjL. Each j
is finitely generated and projective over A, and hence dualizable.
Part (a) of Proposition 2.3.1 is an immediate consequence of the following le*
*mma
and Proposition 2.1.5.
Lemma 2.3.5. Let (A, ) be a flat Hopf algebroid.
(a) If M is an extended comodule, then M ^ (-) takes surjections of comodules
to relative epimorphisms.
(b) If X is a complex of extended comodules, then X ^ (-) takes surjections *
*of
complexes to projective fibrations.
Proof.Suppose f is a surjection of complexes, and M ~= N is an extended
comodule. Let P be a dualizable comodule. Then, using Lemma 1.1.5, we have
-comod(P, M ^ f) ~= -comod(P, ( N) ^ f)
~= -comod(P, (N f)) ~=Hom A(P, N f).
Since f is surjective, so is N f. Since P is projective over A, Hom A(P, N *
*f) is
also surjective, so M ^ f is a relative epimorphism.
Now suppose X is a complex of extended comodules. Then, in degree n, we have
M
(X ^ f)n ~= Xm ^ fn-m .
m
Each map fn-m is surjective, so part (a) assures us that Xm ^ fn-m is a relat*
*ive
epimorphism. Since the dualizable complexes are finitely presented, direct sums*
* of
relative epimorphisms are again relative epimorphisms. Hence X ^ f is a project*
*ive
fibration.
22 MARK HOVEY
We are left with proving part (b) of Proposition 2.3.1. Our approach is simil*
*ar
to that of Lemma 2.3.5.
Lemma 2.3.6. Suppose (A, ) is a flat Hopf algebroid.
(a) Suppose N is a flat A-module. Then ( N) ^ (-) takes exact complexes
to projectively trivial complexes.
(b) Suppose X is a bounded below complex such that Xn ^ (-) preserves pro-
jectively trivial complexes for all n. Then X ^ (-) preserves projectiv*
*ely
trivial complexes.
(c) Suppose X is a complex of comodules such that Xn ^ (-) and ZnX ^ (-)
preserve projectively trivial complexes for all n. Then X ^ (-) preserv*
*es
projectively trivial complexes.
Here ZnX denotes the cycles in degree n, as usual.
Proof.For part (a), suppose Y is a projectively trivial complex and P is a dual*
*izable
comodule. Then, using Lemma 1.1.5, we have
-comod(P, ( N) ^ Y ) ~= -comod(P, (N Y )) ~=Hom A(P, N Y ).
Since Y is exact and N is flat, N Y is also exact. Since P is projective over*
* A,
Hom A(P, N Y ) is also exact, as required.
For part (b), let Y be a projectively trivial complex. Without loss of genera*
*lity,
we can assume that Xn = 0 for n < 0. Suppose P 2 S, and z :P -! (X ^ Y )n is a
cycle in -comod(P, X ^ Y ). Since P is finitely presented as a -comodule,
M1
-comod(P, (X ^ Y )n) ~= -comod(P, Xi^ Yn-i).
i=0
We can therefore write z = (z0, z1, . .,.zi, . .)., where zi: P -! Xi^Yn-i and *
*zi= 0
for large i. Define the degree of z to be the largest i such that ziis nonzero.*
* We will
show that every cycle z is homologous to a cycle of smaller degree; since there*
* are
no cycles of degree -1 this will complete the proof. Indeed, suppose z has degr*
*ee
k. Then zk has to be a cycle in the complex Xk ^ Y . By assumption, Xk ^ Y is
projectively trivial, so zk must be a boundary in this complex. This means that
there is a w :P -! Xk^ Yn-k+1 such that (1 ^ d)w = zk. But then z is homologous
to z0= z + (-1)k+1dw, and one can easily check that w has degree < k.
For part (c), again assume that Y is projectively trivial. Let Xi be the subc*
*om-
plex of X such that Xin= Xn for n > -i, Xin= 0 for n < -i, and Xi-i= Z-iX. By
part (b), each of the complexes Xi ^ Y is projectively trivial. But X = colimXi,
so X ^ Y = colimXi ^ Y . Since filtered colimits of projective equivalences a*
*re
projective equivalences, X ^ Y is therefore projectively trivial.
We can now prove Proposition 2.3.1(b).
Proof of Proposition 2.3.1(b).We need to show that LA ^ (-) preserves projec-
tive equivalences. Since the projective trivial cofibrations are in particular*
* chain
homotopy equivalences, LA ^ (-) certainly takes them to projective equivalences.
It therefore suffices to show that LA ^ (-) preserves projective trivial fibrat*
*ions.
By part (a), LA ^ (-) preserves projective fibrations, so it suffices to show t*
*hat
LA ^ (-) preserves projectively trivial complexes.
In view of Lemma 2.3.6, it suffices to show that (LA)n ^ (-)_and ZnLA ^ (-) *
*__
preserve projectively trivial complexes. Since (LA)-n = ^ ^n for n 0, and
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 23
is flat as an A-module since is so, Lemma 2.3.6(a) guarantees that (LA)n_^ (-)
preserves projectively trivial complexes. On the_other hand, Z-n (LA) ~= ^n for
n 0. The amenable assumption guarantees that ^ (-) preserves projectively
trivial complexes, and then iteration shows that Z-n (LA)^(-) does so as well.
3.Homotopy groups
When the dualizable -comodules generate the category of -comodules, we
know from Proposition 2.1.5 that projective equivalences are homology isomor-
phisms. But homology is not the most important functor of complexes of comod-
ules; homotopy is. In this section we define and study the homotopy groups of
a chain complex of comodules. We show that these homotopy groups are closely
related to Ext in the category of -comodules and have similar properties. When
(A, ) is amenable, every projective equivalence is a homotopy isomorphism and
every homotopy isomorphism is a homology isomorphism. The object of Section 5
will then be to construct a model structure on Ch( ) in which the weak equivale*
*nces
are the homotopy isomorphisms.
3.1. Relatively injective comodules. To explain homotopy groups, we need to
remind the reader of some of the basic results on relatively injective comodule*
*s.
Some of this can be found in [Rav86 , Appendix 1].
Definition 3.1.1. Suppose (A, ) is a flat Hopf algebroid. A comodule I is call*
*ed
relatively injective if -comod(-, I) takes A-split short exact sequences to sh*
*ort
exact sequences.
Lemma 3.1.2. Suppose (A, ) is a flat Hopf algebroid. The relatively injective
comodules are the retracts of extended comodules. In particular, there is a nat*
*ural
A-split embedding of any comodule into a relatively injective comodule.
Proof.We have -comod(-, N) ~=Hom A (-, N). Thus extended comodules,
and so also retracts of extended comodules, are relatively injective. Conversel*
*y, if I
is a relative injective, the map I _-! I must have a retraction, since it is*
* a map
of comodules that is split over A by ffl 1. Thus I is a retract of I. The*
* natural
A-split embedding of the statement of the lemma is just _ :M -! M.
Of course, there are (absolutely) injective comodules as well. A similar argu*
*ment
shows that the injective comodules are retracts of extended comodules I, wher*
*e I
is an injective A-module. But relatively injective comodules are much easier to*
* work
with than injective comodules, partly because injective A-modules are complicat*
*ed,
and partly because of the following lemma.
Lemma 3.1.3. Suppose (A, ) is a flat Hopf algebroid.
(a) Relatively injective comodules are closed under coproducts and products.
(b) If M is an arbitrary comodule and I is relatively injective, then I ^ M *
*and
F (M, I) are relatively injective.
Proof.For part (a), it suffices to show that extended comodules are closed under
coproducts and products. But we have
M M Y Y
( Mi) ~= ( Mi) and ( Mi) ~= ( Mi),
the latter by the construction of products in Proposition 1.2.2.
24 MARK HOVEY
For part (b), we first prove that F (M, I) is relatively injective. We must s*
*how
that -comod(-, F (M, I)) takes A-split short exact sequences to short exact se-
quences. But -comod(-, F (M, I)) is naturally isomorphic to -comod(- ^ M, I),
so this is clear.
To show that I^M is relatively injective, note that I^M is a retract of ( I)*
*^M,
which is isomorphic to ^ I ^ M by Lemma 1.1.5. On the other hand, another
use of Lemma 1.1.5 shows that ^ I ^ M is isomorphic to the extended comodule
(I M), completing the proof.
Relatively injective comodules can be used to compute Ext when the source is
projective over A.
Lemma 3.1.4. Suppose (A, ) is a flat Hopf algebroid, P is a -comodule that is
projective over A, and I is a relatively injective comodule. Then Extn(P, I) = *
*0 for
all n > 0. Hence, if I* is a resolution of M by relatively injective comodules,
Extn(P, M) ~=H-n ( -comod(P, I*)).
This lemma is proved as Lemma A1.2.8(b) of [Rav86 ].
3.2. Homotopy groups. Now recall that LA denotes a specific resolution of A by
relatively injective comodules, defined in Section 2.3, such that the map S0A -*
*!LA
is a chain homotopy equivalence over A. It follows that LA^M is a resolution of*
* M
for any comodule M. It is in fact a resolution by relative injectives by Lemma *
*3.1.3.
Thus we have
Extn(P, M) ~=H-n ( -comod(P, LA ^ M))
for any comodule P that is projective over A.
We now extend this definition, replacing M by a complex X.
Definition 3.2.1. Suppose (A, ) is a flat Hopf algebroid, X 2 Ch ( ), P 2 S,
and n 2 Z. Define the nth homotopy group of X with coefficients in P , ßPn(X), *
*by
ßPn(X) = H-n ( -comod(P, LA ^ X)).
These homotopy groups will be equal to the graded maps from P to X in the
stable homotopy category of -comodules that we are trying to construct. Since
that category will be closed symmetric monoidal, it will also turn out that ßPn*
*(X) ~=
Hn(RF (P, X)), where RF (-, -) is the right derived functor, using the homotopy
model structure we will develop in Section 5, of the chain complex Hom functor
F (-, -).
We need to say a few words about grading. We have essentially two choices; we
can grade homotopy as if it were the homotopy groups of a space, or we can grad*
*e it
as if it were the Extgroups of a comodule. Either way has problems; grading it *
*like
Ext means the exact sequences on homotopy go up instead of down in dimension,
but grading it like homotopy means the homotopy groups of A will be concentrated
in negative degrees. Following Palmieri's work on the Steenrod algebra [Pal01],*
* we
choose to grade it like Ext. This extends to bigrading as well; if (A, ) is a *
*graded
Hopf algebroid, as it always is in algebraic topology, we define
ßPs,t(X) = H-s,t( -comod(P, LA ^ X)).
The homotopy groups are of course functorial in X, and they satisfy the expec*
*ted
properties, correcting for the strange grading.
Lemma 3.2.2. Suppose (A, ) is a flat Hopf algebroid.
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 25
(a) A short exact sequence of complexes
0 -!X -!Y -! Z -!0
induces a natural long exact sequence
. .-.!ßPn(X) -!ßPn(Y ) -!ßPn(Z) -!ßPn+1(X) -!. . .
(b) If X is a filtered colimit of complexes Xi, then ßPn(X) ~=colimßPn(Xi).
Proof.For part (a), since LA is degreewise flat over A, the sequence
0 -!LA ^ X -!LA ^ Y -! LA ^ Z -!0
remains exact. By Lemma 3.1.3, LA ^ X is a complex of relative injectives. Ther*
*e-
fore, the sequence of complexes
0 -! -comod(P, LA ^ X) -! -comod(P, LA ^ Y ) -! -comod(P, LA ^ Z) -!0
remains exact, by Lemma 3.1.4. The long exact sequence in homology of this short
exact sequence finishes the proof of part (a).
For part (b), we simply note that LA ^ -, -comod(P, -), and homology all
commute with filtered colimits.
3.3. Homotopy isomorphisms. A chain map OE is called a homotopy isomorphism
if ßPn(OE) is an isomorphism for all n 2 Z and all P 2 S. Note that OE is a hom*
*otopy
isomorphism if and only if LA ^ OE is a projective equivalence. We claim that
homotopy isomorphisms are the natural notion of weak equivalence in Ch( ).
Proposition 3.3.1. Suppose (A, ) is an amenable Hopf algebroid. Then every
projective equivalence is a homotopy isomorphism, and every homotopy isomor-
phism is a homology isomorphism.
Proof.Suppose p is a projective equivalence. Then Proposition 2.3.1 tells us th*
*at
LA ^ p is also a projective equivalence, so p is a homotopy isomorphism. Now
suppose p: X -! Y is a homotopy isomorphism. Then LA ^ p is a projective
equivalence, and hence a homology isomorphism by Proposition 2.1.5. But A -!LA
is a chain homotopy equivalence over A, so X -!LA ^ X and Y -! LA ^ Y are also
chain homotopy equivalences over A, and in particular homology isomorphisms.
Hence p is a homology isomorphism.
Homotopy isomorphisms have the properties one would hope for in a collection
of weak equivalences.
Proposition 3.3.2. Suppose (A, ) is a flat Hopf algebroid.
(a) Homotopy isomorphisms are closed under retracts and have the two out of
three property.
(b) Homotopy isomorphisms are closed under filtered colimits.
(c) If f is an injective homotopy isomorphism, and g is a pushout of f, then*
* g
is an injective homotopy isomorphism. Dually, if f is surjective homotopy
isomorphism, and g is a pullback of f, then g is a surjective homotopy
isomorphism.
(d) If f is a homotopy isomorphism, then any pushout of f through an injecti*
*ve
map is again a homotopy isomorphism. Dually, any pullback of f through
a surjective map is again a homotopy isomorphism.
26 MARK HOVEY
(e) Suppose f :X -!Y is an injective homotopy isomorphism, and g :A -!B
is a cofibration. Then the pushout product
f g :(X ^ B) qX^A (Y ^ A) -!Y ^ B
is an injective homotopy isomorphism.
Proof.We leave part (a) to the reader. Part (b) is immediate from the fact that*
* ho-
motopy groups commute with filtered colimits. For part (c), suppose g is a push*
*out
of the injective homotopy isomorphism f. Then g is injective, with cokernel cok*
*erf.
Since f is a homotopy isomorphism, the long exact sequence of Lemma 3.2.2 shows
that cokerf has zero homotopy. Another use of that long exact sequence shows
that g is a homotopy isomorphism. The dual case is similar.
For part (d), suppose that g :B -! D is the pushout of the homotopy isomor-
phism f :A -!C through the injection i: A -!B. Then we have the map of short
exact sequences below.
0 ----! A --i--! B ----! X ----! 0
? ? fl
f?y g?y flfl
0 ----! C ----! D ----! X ----! 0
The long exact sequence in homotopy and the five lemma show that g is a homotopy
isomorphism. The dual case is similar.
For part (e), note that parts (b) and (c) imply that injective homotopy isomo*
*r-
phisms are closed under pushouts and filtered colimits, hence transfinite compo*
*si-
tions. Thus it suffices to prove part (e) when g is one of the generating cofib*
*rations
Sn-1P -! DnP of the projective model structure, by Lemma 4.2.4 of [Hov99 ]. We
leave to the reader the check that f g is injective in this case, and just pr*
*ove
it is a homotopy isomorphism. Since f is a homotopy isomorphism, LA ^ f is a
projective equivalence. Therefore, (LA ^ f) ^ Sn-1P is a projective equivalence
by Proposition 2.1.4, and so f ^ Sn-1P is a homotopy isomorphism. Similarly,
f ^ DnP is a homotopy isomorphism. Both such maps are also injective, since P
is flat over A. Part (c) implies that the pushout
X ^ DnP -! (X ^ DnP ) qX^Sn-1P (Y ^ Sn-1P )
is also an injective homotopy isomorphism. The two out of three property for
homotopy isomorphisms then implies that f g is a homotopy isomorphism.
Our next goal is to give some useful examples of homotopy isomorphisms that
are not projective equivalences. We begin with the following lemma.
Lemma 3.3.3. Let (A, ) be a flat Hopf algebroid. Suppose X 2 Ch( ) is bounded
above and contractible as a complex of A-modules, and Y 2 Ch( ) is a complex of
relatively injective comodules. Then every chain map f :X -!Y is chain homotopic
to 0.
Proof.We construct a chain homotopy Dn :Xn -!Yn+1 by downwards induction
on n. Getting started is easy, since Xn = 0 for large n. Suppose we have constr*
*ucted
Dn+1 and Dn+2 such that dDn+2 + Dn+1d = fn+2. We need to construct Dn such
that dDn+1 + Dnd = fn+1. One can readily verify that
(fn+1 - dDn+1)d = 0
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 27
and so fn+1 - dDn+1 defines a map g :Xn+1= imd = Xn+1= kerd -!Yn+1. On the
other hand, we are given that X is A-contractible, so there are A-module maps
sn :Xn -! Xn+1 such that ds + sd = 1. In particular, d: Xn+1= kerd -! Xn
is an A-split monomorphism. Since Yn+1 is relatively injective, there is a map
Dn :Xn -! Yn+1 such that Dnd = fn+1 - dDn+1. This completes the induction
step and the proof.
This gives the following proposition.
Proposition 3.3.4. Let (A, ) be a flat Hopf algebroid, and suppose f :X -!Y is*
* a
map of bounded above complexes in Ch( ) that is an A-split monomorphism in each
degree and a chain homotopy equivalence of complexes of A-modules. Then LA ^ f
is a chain homotopy equivalence. In particular, f is a homotopy isomorphism.
Proof.Let Z denote the cokernel of f. Then Z is bounded above and contractible *
*as
a complex of A-modules (one can check this directly, but it also follows becaus*
*e f is
a trivial cofibration in the absolute model structure on Ch(A) [CH02 , Example *
*3.4]).
Therefore LA^Z is a bounded above complex of relatively injective comodules that
is contractible over A. Lemma 3.3.3 implies that LA ^ Z is contractible. Since *
*LA
is degreewise flat over A, LA ^ Z is the cokernel of LA ^ f. Furthermore, LA ^ f
is a degreewise A-split monomorphism of relatively injective comodules, so it is
a degreewise split monomorphism. It follows that LA ^ f is a chain homotopy
equivalence.
Corollary 3.3.5. Suppose (A, ) is a flat Hopf algebroid. Then the map
jL ^ X :X -!LA ^ X
is a homotopy isomorphism for all complexes X. In fact, LA ^ jL ^ X is a chain
homotopy equivalence.
Proof.The map jL :A -! LA is a map of bounded above complexes that is a
degreewise A-split monomorphism and an A-chain homotopy equivalence. Propo-
sition 3.3.4 implies that LA ^ jL is a chain homotopy equivalence. One can easi*
*ly
check that this forces LA ^ jL ^ X to be a chain homotopy equivalence for any X,
and so jL ^ X is a homotopy isomorphism.
4.Localization
In the next section, we will localize the projective model structure to obtain
a model structure on Ch ( ) in which the weak equivalences are the homotopy
isomorphisms. To prove that this construction works, we need some general resul*
*ts
about Bousfield localization of model categories. The basic reference for Bousf*
*ield
localization is [Hir03]. The results we prove in this section all follow in rea*
*sonably
straightforward fashion from the techniques of [Hir03], but they seem not to ha*
*ve
been noticed before.
Suppose we have a model category M and a class of maps T . The Bousfield
localization LT M of M with respect to T is a new model structure on M, with the
same cofibrations as the given one, in which the maps of T are weak equivalence*
*s.
Futhermore, it is the initial such model category, in the sense that if F :M -!N
is a (left) Quillen functor that sends the maps of T to weak equivalences, then
F :LT M -!N is also a Quillen functor.
28 MARK HOVEY
The Bousfield localization is known to exist when T is a set, M is left prope*
*r,
and, in addition, M is either cellular [Hir03] or combinatorial (unpublished wo*
*rk of
Jeff Smith). The cellular condition is technical, but has the virtue of being w*
*ritten
down and of applying to topological spaces. The combinatorial condition is simp*
*ler;
it just means that M is cofibrantly generated and locally presentable as a cate*
*gory.
To describe the localized model structure, it is necessary to recall that any*
* model
category M possesses a unital action by the category SSet of simplicial sets. T*
*hat
is, there is a bifunctor M x SSet -! M that takes (X, K) to X K described
in [Hov99 , Chapter 5]. This is a unital action but is not associative; it indu*
*ces an
associative action of hoSSet on hoM. In fact, hoM is not only tensored over
hoSSet , but also cotensored and enriched over hoSSet [Hov99 , Chapter 5]. The
enrichment is denoted by map (X, Y ) 2 SSet.
Now, a fibrant object X in LT M, called a T -local fibrant object, is a fibra*
*nt object
X in M such that map (f, X) is a weak equivalence of simplicial sets for all f *
*2 T .
Adjointness gives an equivalent description, as follows. Given a map f :X -!Y in
M, let ef:eX-!Yedenote a cofibration that is a cofibrant approximation to f. Th*
*is
means that eXand eYare cofibrant, efis a cofibration, and we have the commutati*
*ve
diagram below
Xe --ef--!eY
?? ?
y ?y
X ----!fY
where the vertical arrows are weak equivalences. Then a horn on f is one of the
maps fe in for n 0, where in :@ [n] -! [n] is the standard inclusion of
simplicial sets, and ef i is the pushout product map
(Xe [n]) qXe @ [n](eY @ [n]) -!Y [n].
These maps are all cofibrations [Hov99 , Proposition 5.4.1]. Then Proposition 4*
*.2.4
of [Hir03] says that a fibrant object X is a T -local fibrant object if and onl*
*y if
X -! * has the right lifting property with respect to the horns on all the maps*
* of
T .
Having obtained the fibrant objects, one defines a map f to be a T -local equ*
*iv-
alence if map (f, X) is a weak equivalence of simplicial sets for all T -local *
*fibrant
objects X. These are the weak equivalences in LT M. The fibrations are then the
maps that have the right lifting property with respect to all maps that are both
cofibrations and T -local equivalences.
If M is a simplicial model category, then one can understand the horns of f by
using the simplicial structure. But unbounded chain complexes_are not simplicia*
*l.
It is still easy to understand the horns,_however. Let [n] be the chain compl*
*ex
of abelian groups defined by letting [n]kbe the free abelian group on the n+1*
*k+1
k + 1-element subsets of {0, 1, . .,.n}, for k 0. If 1S denotes the generator
corresponding to the set S = {s0 < s1 < . .<.sk}, we define
Xk
d(1S) = (-1)i1S-{si}.
i=0
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 29
This is the_obvious_chain complex corresponding to the nondegenerate simplices *
*of
[n]. Then @ [n]k denotes the subcomplex containing all_the 1S except the one
in_degree_n_corresponding to S = {0, 1, . .,.n}. Let indenote the obvious inclu*
*sion
@ [n]-! [n].
The following lemma is a consequence of the naturality of the action of hoSSet
on hoM, and can be deduced from [Hov99 , Chapter 5].
Lemma 4.1. Suppose M is a Ch (Z)-model category, and f 2 M with a cofi-
brant approximation efthat is a cofibration. Then in the description of_Bousfie*
*ld
localization above, one can replace the horns on f with the maps ef in.
In general, it is difficult to understand the weak equivalences in LT M; cert*
*ainly
the maps of T become weak equivalences, but many other maps do as well. The
following theorem, which is obtained by pulling together different results in [*
*Hir03],
is of some help.
Theorem 4.2. Suppose M is a left proper model category that is either cellular
or combinatorial, and T is a set of maps in M. Let W be a class of maps in M
satisfying the two out of three property, containing the horns on the maps of T
and every weak equivalence in M, and such that maps that are both cofibrations
and in W are closed under transfinite compositions and pushouts. Then every weak
equivalence in LT M is in W.
Proof.Suppose f :X -!Y is a weak equivalence in the Bousfield localization. Let
L: M -! M denote the functor obtained by applying the small object argument
based on T [ J, where J is the set of generating trivial cofibrations of M, to *
*the
map X -!*. Then the maps X -!LX and Y -! LY are transfinite compositions of
pushouts of maps of T [J. This means that they are weak equivalences in LT M, by
Propositions 3.3.10 and 4.2.3 of [Hir03], and also in W by our hypotheses. Hence
Lf is a weak equivalence in LT M whose domain and codomain are fibrant (by
Proposition 4.2.4 of [Hir03]) in LT M. Thus, by Theorem 3.2.13 of [Hir03], Lf is
a weak equivalence in M itself, and hence is in W. The two out of three property
for W now guarantees that f is in W.
In general, Bousfield localization causes one to lose control of the set of g*
*ener-
ating trivial cofibrations. Even if M itself has a very nice set of generating *
*trivial
cofibrations, all the theory tells you is that LT M has some, possibly gigantic*
*, set
of generating trivial cofibrations. The following proposition is at least of so*
*me help
in dealing with this.
Proposition 4.3. Suppose M is a left proper, cellular or combinatorial model
category, and T is a set of maps in M. Assume that M has a set of generating
trivial cofibrations whose domains are cofibrant. Then LT M has a set of genera*
*ting
trivial cofibrations whose domains are cofibrant.
Proof.Let J be a set of generating trivial cofibrations of M whose domains (and
hence codomains) are cofibrant, and let J0 be some set of generating trivial co*
*fi-
brations of LT M. For each map j 2 J0, choose a cofibration ^_of cofibrant obje*
*cts
30 MARK HOVEY
that is a cofibrant approximation to j, so that we have a commutative square
dom ^_ ----! dom j
? ?
^j?y ?yj
codom ^_----! codom j
where the horizontal maps are weak equivalences in M. Let J^0denote the set
of those ^_, and let K = J [ ^J0. Then K is a set of trivial cofibrations in L*
*T M
with cofibrant domains. We claim that K is a generating set of trivial cofibrat*
*ions.
Indeed, suppose p has the right lifting property with respect to K. Then p has *
*the
right lifting property with respect to J, so p is a fibration in M. Since p als*
*o has
the right lifting property with respect to ^J0and M is left proper, Proposition*
* 13.2.1
of [Hir03] implies that p has the right lifting property with respect to J0, an*
*d hence
that p is a fibration in LT M.
5.The homotopy model structure
The object of this section is to construct a model structure on Ch( ), when (*
*A, )
is an amenable Hopf algebroid, in which the weak equivalences are the homotopy
isomorphisms. Proposition 3.3.1 tells us that we need to add more weak equiva-
lences to the projective model structure. We do this by using Bousfield localiz*
*ation,
described in the previous section.
5.1. Construction and basic properties.
Definition 5.1.1. Suppose (A, ) is an amenable Hopf algebroid. Let S denote a
set containing one element from each isomorphism class of dualizable comodules.
Define the homotopy model structure on Ch( ) to be the Bousfield localization of
the projective model structure with respect to the maps
jL ^ SnP :SnP -! LA ^ SnP
for P 2 S and n 2 Z.
We have already seen that -comod is a locally (finitely) presentable cate-
gory 1.4.1. It follows easily that Ch ( ) is also locally (finitely) presentab*
*le, so
that Ch( ) is a combinatorial model category. Thus the (unpublished) work of Je*
*ff
Smith guarantees that the homotopy model structure exists. In fact, Ch( ) is al*
*so
cellular, so one can use Hirschhorn's theory [Hir03] as well.
Note that the cofibrations do not change under Bousfield localization, though*
* the
fibrations and weak equivalences will change. This means that the trivial fibra*
*tions
also do not change under Bousfield localization, and therefore that a cofibrant
replacement functor in the projective model structure is also a cofibrant repla*
*cement
functor in the homotopy model structure. Since Bousfield localization preserves*
* left
properness [Hir03, Theorem 4.1.1], the homotopy model structure is left proper.
Our first goal is to prove that the weak equivalences in the homotopy model
structure are the homotopy isomorphisms, explaining the name.
Proposition 5.1.2. Let (A, ) be an amenable Hopf algebroid. Then every weak
equivalence in the homotopy model structure is a homotopy isomorphism.
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 31
Proof.Proposition 3.3.1 and Proposition 3.3.2 tell us that the class of homotopy
isomorphisms has all the properties necessary for Theorem 4.2 to apply. It rema*
*ins
to check that the horns on jL ^ SnP are homotopy isomorphisms. Now Ch( ) is a
Ch(Z) model category; in fact, there is a symmetric monoidal left Quillen funct*
*or
Ch(Z) -!Ch ( ), induced by the trivial comodule functor M 7! A ZM._Lemma_4.1
implies that the horns on f can be_taken to be the maps f (A Z in). One can
easily check that each map A Z inis a projective cofibration. The lemma then
follows from Proposition 3.3.2(e).
To prove the converse, we need the following proposition.
Proposition 5.1.3. Let (A, ) be a flat Hopf algebroid, and suppose C is cofibr*
*ant
in Ch( ). Then jL ^ C :C -!LA ^ C is a weak equivalence in the homotopy model
structure.
Proof.Factor jL :S0A -!LA into a cofibration i: S0A -!QLA followed by a triv-
ial fibration q. It suffices to show that i ^ C is a trivial cofibration in the*
* homotopy
model structure, because q ^C is a projective equivalence by Proposition 2.1.4.*
* For
dualizable P , jL ^ SnP is a weak equivalence in the homotopy model structure by
construction, so i ^ SnP is a trivial cofibration in the homotopy model structu*
*re.
Since C is cofibrant, 0 -!C is a retract of a transfinite composition
0 = C0 -!C1 -!. .C.i-!. . .
where each map Ci -!Ci+1 is a pushout of a map Sn-1P -! DnP , where P 2 S
and n 2 Z. It suffices to show that i ^ Ci is a trivial cofibration in the homo*
*topy
model structure for all i, which we do by transfinite induction. The base case *
*is
trivial, since C0 = 0. For the successor ordinal step, suppose i ^ Ci is a tri*
*vial
cofibration. We have pushout diagrams
Sn-1P - ---! DnP
?? ?
y ?y
Ci - ---! Ci+1
and
QLA ^ Sn-1P - ---! QLA ^ DnP
?? ?
y ?y
QLA ^ Ci - ---! QLA ^ Ci+1
and i induces a map from one of these to the next. This map is a weak equiva-
lence on the upper left corners as mentioned above, a chain homotopy equivalence
on the upper right corners because DnP is contractible, and a weak equivalence
on the lower left corners by the induction hypothesis. It follows from the cube
lemma [Hov99 , Lemma 5.2.6] that i ^ Ci+1 is a weak equivalence as well. Because
the projective structure is monoidal, i ^ Ci+1is also a cofibration.
We are left with the limit ordinal step of the induction. So suppose that i ^*
* Ci
is a trivial cofibration in the homotopy model structure for all i < ff for some
limit ordinal ff. Then Proposition 17.9.1 of [Hir03] implies that i ^ Cffis a w*
*eak
equivalence as well, and hence a trivial cofibration.
Theorem 5.1.4. Suppose (A, ) is an amenable Hopf algebroid. Then the weak
equivalences in the homotopy model structure are the homotopy isomorphisms.
32 MARK HOVEY
Proof.We have already seen that every weak equivalence in the homotopy model
structure is a homotopy isomorphism. Conversely, suppose f :X -! Y is a homo-
topy isomorphism. By using a cofibrant replacement functor Q in the projective
model structure, we can construct the commutative diagram below,
QX --Qf--!QY
? ?
qX?y ?yqY
X ----!f Y
where qX and qY are projective equivalences, and QX and QY are cofibrant. In
particular, since projective equivalences are homotopy isomorphisms by Proposi-
tion 3.3.1, Qf is a homotopy isomorphism. Since every projective equivalence is*
* a
weak equivalence in the homotopy model structure, it suffices to show that the *
*ho-
motopy isomorphism Qf is a weak equivalence. Consider the commutative square
below.
QX jL^QX-----!LA ^ QX
? ?
Qf?y ?yLA^Qf
QY -----!j LA ^ QY
L^QY
Both of the horizontal maps are weak equivalences in the homotopy model structu*
*re
by Proposition 5.1.3. Since Qf is a homotopy isomorphism, LA^Qf is a projective
equivalence, and therefore a weak equivalence in the homotopy model structure. *
*It
follows that Qf is a weak equivalence in the homotopy model structure as well.
Many properties of the homotopy model structure follow immediately from The-
orem 5.1.4.
Theorem 5.1.5. Suppose (A, ) is an amenable Hopf algebroid. Then the ho-
motopy model structure is proper, symmetric monoidal, and satisfies the monoid
axiom. Moreover, if C is cofibrant, then C ^ - preserves weak equivalences.
The monoid axiom and the last statement of this theorem are important because
they guarantee the existence of homotopy invariant model categories of modules *
*and
monoids, as explained following the statement of Proposition 2.1.4.
Proof.Bousfield localization preserves left properness, as has already been ob-
served. The fact that the homotopy model structure is right proper follows imme-
diately from Theorem 5.1.4 and Proposition 3.3.2. The homotopy model structure
is symmetric monoidal by Proposition 3.3.2(e).
Now for the monoid axiom, which we recall states that any transfinite compo-
sition of pushouts of maps of the form f ^ X, where f is a trivial cofibration *
*and
X is arbitrary, is a weak equivalence. Let us suppose we know that f ^ X itself*
* is
a weak equivalence for all trivial cofibrations f and all X. Since cofibrations*
* are
degreewise -split monomorphisms, it follows that f ^ X is also injective. Sin*
*ce
injective homotopy isomorphisms are closed under pushouts and filtered colimits
by Propostion 3.3.2, the monoid axiom will follow.
We are left with showing that f ^ X is a homotopy isomorphism for all trivial
cofibrations f in the homotopy model structure and all X. It suffices to show t*
*his
for a set of generating trivial cofibrations f, and Proposition 4.3 allows us t*
*o assume
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 33
those generating trivial cofibrations f have cofibrant domains and codomains. L*
*et
q :QX -!X be a cofibrant replacement of X, so that q is a projective equvalence
and QX is cofibrant. We have the commutative square below.
dom f ^ QX dom-f^q----!domf ^ X
? ?
f^QX?y ?yf^X
codomf ^ QX -------!codomf^qcodomf ^ X
By Proposition 2.1.4, both the horizontal maps are projective equivalences, and
hence homotopy isomorphisms. Since the homotopy model structure is symmetric
monoidal, f ^ QX is a homotopy isomorphism. It follows that f ^ X is a homotopy
isomorphism as well, completing the proof of the monoid axiom.
Now suppose C is cofibrant, and f is a weak equivalence. Then LA ^ f is a
projective equivalence. By Proposition 2.1.4, it follows that C ^ LA ^ f is st*
*ill
a projective equivalence. Hence C ^ f is a homotopy isomorphism, so a weak
equivalence.
5.2. Fibrations. We now characterize the fibrations in the homotopy model struc-
ture.
Proposition 5.2.1. Suppose (A, ) is an amenable Hopf algebroid. Then a map
p is a fibration in the homotopy model structure if and only if p is a projecti*
*ve
fibration and kerp is fibrant in the homotopy model structure.
Proof.Certainly, if p is a homotopy fibration, then p must be a projective fibr*
*ation
and kerp must be homotopy fibrant. Conversely, suppose p: X -!Y is a projective
fibration and kerp is homotopy fibrant. Form the commutative square below by
using factorization,
X --iX--!X0
?? ??
(5.2.2) py yq
Y ----!i Y 0
Y
where iX and iY are trivial cofibrations in the homotopy model structure, X0 and
Y 0are homotopy fibrant, and q is a homotopy fibration. We claim that this squa*
*re
is a homotopy pullback square in the projective model structure. Proposition 3.*
*4.7
of [Hir03] then implies that p is a fibration in the homotopy model structure.
0
To see that the square 5.2.2 is a homotopy pullback square, let P -q!Y be the
pullback of q through iY . Then q0is a projective fibration, and there is an in*
*duced
factorization
X s-!P -t!X0
of iX . Since t is the pullback of the homotopy isomorphism iY through the surj*
*ec-
tion q, Proposition 3.3.2 implies that t is a homotopy isomorphism. Hence s is a
homotopy isomorphism as well. Consider the commutative diagram below
0- ---! kerp ----! X --p--!Y ----! 0
? ? fl
r?y s?y flfl
0
0- ---! kerq ----! P --q--!Y ----! 0
34 MARK HOVEY
whose rows are short exact (since projective fibrations are surjective). The l*
*ong
exact sequence in homotopy implies that r is a homotopy isomorphism. But kerp
is homotopy fibrant by assumption, and kerq is homotopy fibrant since q is a
homotopy fibration. Theorem 3.2.13 of [Hir03] implies that r is a projective eq*
*uiv-
alence. Applying -comod(Q, -) for Q 2 S and considering the long exact homol-
ogy sequence shows that s is a projective equivalence as well. This means that *
*the
square 5.2.2 is a homotopy pullback square, completing the proof.
The characterization of fibrations we have just given would be more helpful if
we knew what the fibrant objects in the homotopy model structure are.
Theorem 5.2.3. Suppose (A, ) is an amenable Hopf algebroid. Then the following
are equivalent.
(a) jL ^ X :X -!LA ^ X is a projective equivalence.
(b) X is projectively equivalent to some complex of relative injectives.
(c) X is fibrant in the homotopy model structure.
Proof.It is clear that (a) implies (b). To see that (b) implies (c), our first *
*goal is to
show that if X is projectively equivalent to some complex of relative injective*
*s, then
jL ^X is a projective equivalence (incidentally proving (b) implies (a)). It ob*
*viously
suffices to show this for actual complexes of relative injectives X. Any such c*
*omplex
can be written as the colimit of the bounded above complexes Xn, where Xni= 0
for i > n and Xni = Xi for i n. Since colimits of projective equivalences
are projective equivalences, we can assume that X is a bounded above complex of
relative injectives. In this case, we will show that jL ^X is in fact a chain h*
*omotopy
equivalence. Indeed, since jL is a degreewise A-split monomorphism, jL ^ X is
a degreewise A-split monomorphism between complexes of relative injectives. It
follows that jL ^X is a degreewise split monomorphism of relative injectives. L*
*et Y
denote the cokernel of jL ^ X. Then Y is also a bounded above complex of relati*
*ve
injectives, and Y is contractible as a complex of A-modules since jL is a chain
homotopy equivalence of complexes of A-modules. Lemma 3.3.3 then implies that
Y is contractible as a complex of comodules. Given this, an elementary argument
then shows that jL ^ X is a chain homotopy equivalence.
Now suppose that X is projectively equivalent to a complex of relative inject*
*ives,
C is cofibrant and ß*C = 0. We claim that every chain map f :C -! X is chain
homotopic to 0. Indeed, the composite
C f-!X jL^X----!LA ^ X
factors through LA ^ C, which is projectively trivial. Hence the map (jL ^ X) O*
* f
is 0 in the homotopy category of the projective model structure. Since jL ^ X
is a projective equivalence by the previous paragraph, we conclude that f is 0 *
*in
the homotopy category of the projective model structure. Since C is cofibrant a*
*nd
everything is fibrant in the projective model structure, it follows that f is c*
*hain
homotopic to 0.
We can now complete the proof that (b) implies (c) by showing that X -! 0
has the right lifting property with respect to every cofibration i: A -! B that*
* is
a homotopy isomorphism. Indeed, suppose f :A -! X and let C be the cokernel
of i. Since i is a cofibration, it is a split monomorphism in each degree, and*
* so
Bn ~=An Cn. The differential on B must then be given by d(a, c) = (da + øc, d*
*c),
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 35
where ød = -dø. Thus ø is a chain map from the desuspension -1C of C to A.
The composition
-1C ø-!A f-!X
must be chain homotopic to 0, since -1C is cofibrant and ß* -1C = 0. Hence
there are maps Dn :Cn -!An such that -Dn-1d + dDn = fø. Define
g(a, c) = fa + Dnc.
Then g :B -!C is a chain map extending f, so X -!0 has the right lifting proper*
*ty
with respect to i as required.
We now show that (c) implies (a). So suppose X is fibrant in the homotopy
model structure. The map jL ^ X :X -! LA ^ X is a homotopy isomorphism by
Corollary 3.3.5. Since X is fibrant, and LA ^ X is also fibrant since (b) impli*
*es (c),
it follows from Theorem 3.2.13 of [Hir03] that jL ^X is a projective equivalenc*
*e.
Corollary 5.2.4. Suppose (A, ) is an amenable Hopf algebroid, and give Ch( )
the homotopy model structure. For any X 2 Ch( ), the map
jL ^ X :X -!LA ^ X
is a natural weak equivalence whose target is fibrant.
This follows immediately from Corollary 3.3.5 and Theorem 5.2.3. Note that
jL ^ X is not normally a cofibration, however.
We also note the following corollary.
Corollary 5.2.5. Suppose (A, ) is an amenable Hopf algebroid, and given Ch( )
the homotopy model structure. Then weak equivalences and fibrations are closed
under filtered colimits.
This corollary is saying that the homotopy model structure behaves as if it w*
*ere
finitely generated. We do not know if it is in fact finitely generated for a ge*
*neral
amenable Hopf algebroid, though for an Adams Hopf algebroid it is.
Proof.We have seen in Proposition 3.3.2 that homotopy isomorphisms are closed
under filtered colimits. Since homotopy fibrations are just projective fibratio*
*ns with
homotopy fibrant kernel, and projective fibrations are closed under filtered co*
*limits,
it suffices to show that homotopy fibrant objects are closed under filtered col*
*imits.
This follows from the characterization of homotopy fibrant objects in part (a) *
*of
Theorem 5.2.3, since projective equivalences are closed under filtered colimits.
5.3. Naturality. Like the projective model structure, the homotopy model struc-
ture is natural.
Proposition 5.3.1. Suppose : (A, ) -!(B, ) is a map of amenable Hopf alge-
broids. Then induces a left Quillen functor *: Ch( ) -!Ch ( ) of the homotopy
model structures.
Proof.By Proposition 2.2.1, * is a right Quillen functor of the projective mod*
*el
structures. Thus * preserves projective fibrations and projective equivalences
(since everything is fibrant in the projective model structure), and so will al*
*so
preserves trivial fibrations in the homotopy model structure, since these coinc*
*ide
with projective trivial fibrations. Suppose p is a homotopy fibration. Then p*
* is
a projective fibration such that kerp is projectively equivalent to a complex of
relative injectives K, by Proposition 5.2.1 and Theorem 5.2.3. Hence *p is als*
*o a
36 MARK HOVEY
projective fibration, and ker *p is projectively equivalent to *K. We will show
that *K is a complex of relative injectives. Hence *p is a homotopy fibration
by Theorem 5.2.3 and Proposition 5.2.1, and so * is a right Quillen functor as
required.
We are now reduced to showing that * preserves relative injectives. Suppose
I is a relatively injective -comodule, and E is an A-split short exact sequenc*
*e of
-comodules. Then
-comod(E, *I) ~= -comod( *E, I).
Since *E = B A E, *E is a B-split short exact sequence, so -comod( *E, I)
is exact.
The homotopy model structure is also invariant under weak equivalences, but
this is considerably harder to prove. We begin with a definition.
Definition 5.3.2. Suppose (A, ) is an amenable Hopf algebroid. Define a -
comodule I to be pseudo-injective if Extn(P, I) = 0 for all dualizable comodule*
*s P
and all n > 0.
Every relative injective is pseudo-injective, by Lemma 3.1.4. The reason for *
*in-
troducing pseudo-injectives is the following lemma, which would be false for re*
*lative
injectives.
Lemma 5.3.3. Suppose : (A, ) -! (B, ) is a weak equivalence of flat Hopf
algebroids. If I is a pseudo-injective -comodule, then *I is a pseudo-inject*
*ive
-comodule.
Proof.Suppose P is a dualizable -comodule. Because * is an equivalence of
categories whose right adjoint is naturally isomorphic to *, we have
Extn(P, *I) ~=Extn( *P, I).
As explained in the proof of Theorem 2.2.2, *P is a dualizable -comodule. The
lemma follows.
Lemma 5.3.4. Suppose (A, ) is a flat Hopf algebroid. Any bounded above complex
of pseudo-injectives with no homology is projectively trivial.
Proof.Suppose X is a bounded above complex of pseudo-injectives with no homol-
ogy. We claim that the cycle comodule ZnX is pseudo-injective for all n. This is
obvious for large n, since X is bounded above. We have a short exact sequence
0 -!ZnX -!Xn -!Zn-1X -!0
since X has no homology. The long exact sequence in Ext*(P, -) shows that Zn-1X
is pseudo-injective. By induction, ZnX is pseudo-injective for all n. One can t*
*hen
easily check that -comod(P, X) is exact for all dualizable comodules P .
Corollary 5.3.5. Suppose (A, ) is a flat Hopf algebroid, and f :X -! Y is a
homology isomorphism of complexes of bounded above pseudo-injectives. Then f is
a projective equivalence.
Proof.Let C denote the mapping cylinder of f, and let Z = C=X denote the map-
ping cone of f. Then f = pi, where i: X -!C is a degreewise split monomorphism
and p is a chain homotopy equivalence. It therefore suffices to show that the h*
*omol-
ogy isomorphism i is a projective equivalence. Since i is degreewise split, it *
*suffices
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 37
to show that Z is projectively trivial. But Zn = Yn Xn-1, so Z is a bounded
above complex of pseudo-injectives with no homology. Lemma 5.3.4 implies that Z
is projectively trivial.
Theorem 5.3.6. Suppose : (A, ) -! (B, ) is a weak equivalence of amenable
Hopf algebroids. Then *: Ch( ) -! Ch( ) is a Quillen equivalence of the ho-
motopy model structures. In fact, both * and * preserve and reflect homotopy
isomorphisms.
Proof.We have seen in Theorem 2.2.2 that * is a Quillen equivalence of the
projective model structures, and that * and * preserve and reflect projective
equivalences. We first show that * preserves homotopy isomorphisms, and hence
that * reflects homotopy isomorphisms. Indeed, it follows from Proposition 5.3*
*.1
that * preserves homotopy trivial cofibrations. Thus it suffices to show that *
* *p
is a homotopy isomorphism when p is a homotopy trivial fibration. But then p is*
* a
projective equivalence, so *p is also a projective equivalence.
It is more difficult to show that * reflects homotopy isomorphisms. To see t*
*his,
we first construct a factorization
B = *A -*jL--! *LA ff-!LB
of jL :B -!LB. We have
__^n __^n
( *LA)n ~= * ^ ( * ) and (LB)n = ^ .
There is a natural map of comodules *_ =_B_ -! that takes b x to
jL(b) 1(x). This map induces a map * -! , which in turn induces the desired
map ff: *LA -!LB of complexes.
Now *LA and LB are both complexes of pseudo-injectives, by Lemma 5.3.3.
The map *jL is a homology isomorphism, since *, like any equivalence of abeli*
*an
categories, is exact. The map jL :B -! LB is a homology isomorphism by con-
struction. Thus, ff: *LA -! LB is a homology isomorphism of bounded above
complexes of pseudo-injectives, and so a projective equivalence, by Corollary 5*
*.3.5.
We can now show that * reflects homotopy isomorphisms between cofibrant
objects. Indeed, suppose that f :X -! Y is a map of cofibrant objects such that
*f is a homotopy isomorphism. Then LB ^ *f is a projective equivalence. But
we have the commutative square below.
*LA ^ *X --*LA^-*f----! *LA ^ *Y
? ?
ff^ *X?y ?yff^ *Y
LB ^ *X ------!LB^LB ^ *Y
*f
Proposition 2.1.4 implies that the vertical maps in this square are projective *
*equiv-
alences. Hence *LA ^ *f ~= *(LA ^ f) is a projective equivalence. But *
reflects projective equivalences, so LA ^ f is a projective equivalence. Hence *
*f is a
homotopy isomorphism as required.
We now claim that * reflects all homotopy isomorphisms, from which it follows
easily that * preserves all homotopy isomorphisms. So suppose f :X -! Y is a
map such that *f is a homotopy isomorphism. We have the commutative square
below, in which the vertical maps are projective equivalences and QX and QY are
38 MARK HOVEY
cofibrant.
QX --Qf--!QY
? ?
qX?y ?yqY
X ----!f Y
Since * preserves projective equivalences, we conclude that *Qf is a homotopy
isomorphism. Since * reflects homotopy isomorphisms between cofibrant objects,
Qf is a homotopy isomorphism, and hence f is a homotopy isomorphism as re-
quired.
It now follows easily that * is a Quillen equivalence, as in the proof of Th*
*eo-
rem 2.2.2.
5.4. Comparison with injective model structure. When A = k is a field, there
is a model structure on Ch( ) in which hoCh ( )(A, A)* ~=Ext-*(A, A) developed
in [Hov99 , Section 2.5]. In this model structure, which we call the injective *
*model
structure, the cofibrations are just the monomorphisms, and the fibrations are *
*the
degreewise surjections with degreewise injective kernels. (Remember that relati*
*vely
injective and injective coincide in case A is a field). The weak equivalences a*
*re the
homotopy isomorphisms, where homotopy is defined as in Definition 3.2.1 but usi*
*ng
only simple comodules (that is, those comodules with no nontrivial subcomodule)
as the source. For years, the author searched for a generalization of this mod*
*el
structure to Hopf algebroids without success. The injective model structure is
NOT a special case of the homotopy model structure. Indeed, cofibrations in the
homotopy model structure are degreewise split over , whereas cofibrations in t*
*he
injective model structure are split over A, but not necessarily . However, we *
*have
the following theorem.
Theorem 5.4.1. Suppose is a Hopf algebra over a field k = A. Then the identity
functor defines a Quillen equivalence from the homotopy model structure to the
injective model structure.
Proof.We claim that the two model structures have the same weak equivalences.
If we can prove this, then the identity functor will be a left Quillen functor *
*from the
homotopy model structure to the injective model structure, since every projecti*
*ve
cofibration is a monomorphism. It must be a Quillen equivalence since the weak
equivalences are the same.
Note that the dualizable comodules coincide with the finite-dimensional comod-
ules. Every simple comodule is finite-dimensional by Lemma 9.5.5 of [HPS97 ]. T*
*hus
every homotopy isomorphism is a weak equivalence in the injective model structu*
*re.
To prove the converse, it suffices to prove that if -comod(P, LA ^ f) is a hom*
*ology
isomorphism for all simple comodules P , then it is a homology isomorphism for *
*all
finite-dimensional comodules P . We do this by induction on the dimension of P .
Since every one-dimensional comodule is simple, the base case is easy. Now supp*
*ose
we know -comod(P, LA ^ f) is an isomorphism for all P of dimension < n, and P
has dimension n. If P is simple, there is nothing to prove. If P is not simple,*
* there
is a short exact sequence of comodules
0 -!Q -!P -! P=Q -!0
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 39
with dimQ < n. This sequence is necessarily split over k, since k is a field. T*
*here-
fore, the sequence
0 -! -comod(P=Q, LA ^ X) -! -comod(P, LA ^ X) -! -comod(Q, LA ^ X) -!0
is still short exact, as is the corresponding sequence with Y replacing X. The *
*map
f induces a map between the corresponding long exact sequences in homology.
By the induction hypothesis, -comod(Q, LA ^ f) and -comod(P=Q, LA ^ f) are
isommorphism. The five lemma implies that -comod(P, LA^f) is an isomorphism
as well, completing the proof of the induction step.
6.The stable category
We define the homotopy category of the homotopy model structure on Ch( ) to
be the stable homotopy category of (A, ), and we denote it by Stable( ), follo*
*w-
ing Palmieri [Pal01] in the case of the Steenrod algebra. The category Stable( )
is what we should mean by the derived category D(A, ) of the Hopf algebroid
(A, ). This is consistent with the usual notation, since D(A, A) = D(A), the
usual unbounded derived category of A. However, we must remember that to form
the derived category, we invert the homotopy isomorphisms, not the homology
isomorphisms. It is just that in the case of a discrete Hopf algebroid (A, A), *
*the
homotopy isomorphisms coincide with the homology isomorphisms.
We conclude the paper by establishing some basic properties of Stable( ). We
show that it is a unital algebraic stable homotopy category [HPS97 ]. This means
that it shares most of the formal properties of the derived category of a commu*
*tative
ring, or the ordinary stable homotopy category, except that it has several gene*
*rators
rather than just one. In certain cases of interest in algebraic topology, such*
* as
= BP*BP , we show that Stable( ) is monogenic, so that (bigraded) suspensions
of BP* weakly generate the category. We also show that
Stable( )(S0M, SkN) ~=Extk(M, N)
for certain -comodules M and N.
We begin with the following lemma.
Lemma 6.1. Suppose (A, ) is an amenable Hopf algebroid, and P is a dualizable
-comodule. Then SnP is dualizable in the homotopy category of the projective
model structure on Ch( ) for all n.
Proof.Recall that the symmetric monoidal product in the homotopy category is
the derived smash product X ^L Y = QX ^ QY , where Q denotes a cofibrant
replacement functor. Similarly, the closed structure is RF (X, Y ) = F (QX, RY*
* ),
where R is a fibrant replacement functor. To show that X is dualizable, we must
show that the unit
S0A -!RF (X, X)
factors through the composition map
RF (X, S0A) ^L X -!RF (X, X).
In the projective model structure, everything is fibrant, so we may as well tak*
*e R
to be the identity functor. Furthermore, SnP is cofibrant, so we conclude that
RF (SnP, SnP ) ~=S0F (P, P ), RF (SnP, S0A) ~=S-n F (P, A),
40 MARK HOVEY
and
RF (SnP.S0A) ^L SnP ~=S0(F (P, A) ^ P ).
It is now clear that SnP is dualizable, since P is so.
Theorem 6.2. Suppose (A, ) is an amenable Hopf algebroid. Then the homotopy
category of the projective model structure and Stable( ) are unital algebraic s*
*table
homotopy categories. A set of small, dualizable, weak generators is given by th*
*e set
of all SnP for P a dualizable comodule and n 2 Z.
Proof.It is easy to check that the ordinary suspension, defined by ( X)n = Xn-1
with d X = -dX , is a Quillen equivalence of both the projective and homotopy
model structures. One can also check that it induces the model category theo-
retic suspension on the homotopy categories. This means that both the projective
and homotopy model structures are stable in the sense of [Hov99 , Section 7.1],
and therefore that the homotopy category of the projective model structure and
Stable( ) are triangulated.
Since the projective and homotopy model structures are symmetric monoidal,
their homotopy categories are also symmetric monoidal in a way that is compatib*
*le
with the triangulation (see Chapters 4 and 6 of [Hov99 ]). In fact, they satisf*
*y much
stronger compatibility relations than those demanded in [HPS97 ]; see [May01 ].
The projective model structure is finitely generated, so the results of Secti*
*ons 7.3
and 7.4 of [Hov99 ] guarantee that the cofibers of the generating cofibrations,*
* namely
the SnP , form a set of small weak generators for the homotopy category. The
homotopy model structure may not be finitely generated, but fibrations and weak
equivalences are closed under filtered colimits by Corollary 5.2.5, and this is*
* all that
is needed for the arguments of Section 7.4 of [Hov99 ] to apply. Thus the SnP a*
*lso
form a set of small weak generators for Stable( ). Lemma 6.1 guarantees that th*
*ey
are dualizable in the homotopy category of the projective model structure; since
the functor from this category to Stable( ) is symmetric monoidal, they are also
dualizable in Stable( ).
We now investigate when the homotopy category of the homotopy model struc-
ture is monogenic. We first recall a definition from abelian categories.
Definition 6.3. A thick subcategory of an abelian category C is a full subcateg*
*ory
T that is closed under retracts and has the two-out-of-three property. This mea*
*ns
that if
0 -!M0 -!M -!M00-!0
is a short exact sequence, and two out of M0, M, M00are in T , so is the third.*
* If C
is graded, we also insist that thick subcategories be closed under aribtrary sh*
*ifts.
The reader used to BP*BP -comodules will be familiar with the following defi-
nition.
Definition 6.4. Suppose (A, ) is an amenable Hopf algebroid, and M is a -
comodule. We say that M has a Landweber filtration if there is a finite filtrat*
*ion
0 = M0 M1 . . .Mt= M
of M by subcomodules such that each quotient Mj=Mj-1 ~=A=Ij for some ideal Ij
of A that is generated by an invariant finite regular sequence.
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 41
We recall that a sequence x1, . .,.xn is an invariant regular sequence if xi *
*is a
primitive nonzero divisor in A=(x1, . .,.xi-1) for all i.
In case (A, ) is graded, we allow the filtration quotients Mi=Mi-1to be isom*
*or-
phic to some shift of A=Ij rather than A=Ij itself.
From a structural point of view, whether or not M has a Landweber filtration *
*is
not important. What matters is whether M is in the thick subcategory generated
by A.
Lemma 6.5. Suppose (A, ) is an amenable Hopf algebroid, and M is a -comodule
with a Landweber filtration. Then M is in the thick subcategory generated by A.
Proof.Since thick subcategories are closed under extensions, it suffices to che*
*ck that
A=I is in the thick subcategory generated by A, where I is generated by a finite
invariant regular sequence x1, . .,.xn. This follows from the short exact seque*
*nces
of comodules
0 -!A=(x1, . .,.xi-1) xi-!A=(x1, . .,.xi-1) -!A=(x1, . .,.xi) -!0
and induction.
Theorem 6.6. Suppose (A, ) is an amenable Hopf algebroid, and every dualiz-
able comodule P is in the thick subcategory generated by A. Then Stable( ) is
monogenic, in the sense that {SnA} form a set of small weak generators.
In the graded case, we would instead get that {Sn,mA} would form a set of weak
generators.
Corollary 6.7. Let E be a ring spectrum that is Landweber exact over MU or BP J
for some finite invariant regular sequence J, and suppose that E*E is commutati*
*ve.
Then Stable(E*E) is a bigraded monogenic stable homotopy category.
Proof.It is shown in [HS02 ] that every finitely presented E*E-comodule is a re*
*tract
of a comodule with a Landweber filtration, and hence in the thick subcategory
generated by E*.
To prove Theorem 6.6, we first need a lemma.
Lemma 6.8. Suppose (A, ) is an amenable Hopf algebroid, and
0 -!M0 f-!M -!M00-!0
is a short exact sequence of comodules. Then
S0M0 -!S0M -!S0M00
is a cofiber sequence in Stable( ).
Proof.Factor S0f into a projective cofibration i: S0M0 -!X followed by a projec-
tive trivial fibration p. Then we have the commutative diagram below, whose rows
are exact.
0 ----! S0M0 - -i--!S0M ----! C ----! 0
flfl ? ?
fl p?y ?yq
0 ----! S0M0 - ---! S0M ----! S0M00 ----! 0
S0f
42 MARK HOVEY
The long exact sequence in homotopy (Lemma 3.2.2) and the five lemma imply that
q is a homotopy isomorphism. Therefore the bottom row is isomorphic in Stable( )
to the top row, which is a cofiber sequence.
Proof of Theorem 6.6.Suppose that ßA*(X) = 0. Let us denote maps in the homo-
topy category of the homotopy model structure by [Y, Z]*, so that [S0A, X]* = 0.
Let T denote the full subcategory of all comodules M such that [S0M, X]* = 0.
We claim that T is a thick subcategory, and therefore contains the dualizable c*
*o-
modules. Theorem 6.2 then completes the proof.
It is clear that T is closed under retracts. To show that T is thick, suppose*
* we
have a short exact sequence
0 -!M0 -!M -!M00-!0
such that two out of M0, M, M00are in T . Lemma 6.8 then implies that
S0M0 -!S0M -!S0M00
is a cofiber sequence in Stable( ). The long exact sequence obtained by applying
[, X]* then shows that the other one is also in T .
Finally, we study Stable( )(S0M, S0N) for comodules M and N.
Definition 6.9. A full subcategory of an abelian category C is called localizin*
*g if
it is a thick subcategory closed under coproducts.
Proposition 6.10. Let (A, ) be an amenable Hopf algebroid, and M and N be
-comodules. Then there is a natural map
Extk(M, N) ffMN---!Stable( )(S0M, SkN).
This map is an isomorphism if M is in the localizing subcategory generated by t*
*he
dualizable comodules.
Note that the Ext groups that appear in this proposition are Ext groups in the
category of -comodules, not relative Ext groups.
Proof.A class in Extk(M, N) is represented by an exact sequence of comodules
0 -!N = E0 f0-!E1 f1-!E2 f2-!. .f.k-1---!Ek fk-!Ek+1 = M -!0.
We can split this into the short exact sequences
0 -!kerfi-! Ei-! cokerfi-! 0.
Each such short exact sequence is gives rise to a cofiber sequence
S0(kerfi) -!S0Ei-! S0(cokerfi) -!S1(kerfi)
in Stable( ) by Lemma 6.8. By composing the maps S0(cokerfi) -! S1(kerfi),
we get a map S0M -! SkN in Stable( ). One can check that this respects the
equivalence relation that defines Extk(M, N), and is natural.
Note that this map is an isomorphism when M = P is dualizable, for then
Stable( )(S0P, SkN) ~=ßPk(S0N) ~=Extk(P, N)
by Lemma 3.1.4. Let T be the full subcategory consisting of all M such that
ffMN :Extk(M, N) -!Stable( )(S0M, SkN)
HOMOTOPY THEORY OF COMODULES OVER A HOPF ALGEBROID 43
is an isomorphism for all N and all k 0. We claim that T is a localizing subc*
*at-
egory. Indeed, it is clear that T is closed under retracts and coproducts. To c*
*heck
that T is thick, we note that a short exact sequence
0 -!M0 -!M -!M00-!0
induces a long exact sequence in Ext*(-, N). Because short exact sequences are
also cofiber sequences in Stable( ) by 6.8, we also get a long exact sequence in
Stable( )(-, S*N). There is a map between these two long exact sequences (one
must check that ffMN is compatible with the map Extk(M00, N) -!Extk+1(M0, N)
but the construction of ffMN makes this easy to check). The five lemma tells t*
*hat
that if two out of M0, M, M00are in T , so is the third.
References
[Ada74] J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures*
* in Math-
ematics, University of Chicago Press, Chicago, Ill.-London, 1974, x+373*
* pp.
[Bek00] Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Cambrid*
*ge Philos.
Soc. 129 (2000), no. 3, 447-475. MR 1 780 498
[Boa82] J. M. Boardman, The eightfold way to BP-operations or E*E and all that,*
* Current
trends in algebraic topology, Part 1 (London, Ont., 1981), CMS Conf. Pr*
*oc., vol. 2,
Amer. Math. Soc., Providence, R.I., 1982, pp. 187-226. MR 84e:55004
[Bor94] Francis Borceux, Handbook of categorical algebra. 1, Cambridge Universi*
*ty Press,
Cambridge, 1994, Basic category theory. MR 96g:18001a
[BW85] Michael Barr and Charles Wells, Toposes, triples and theories, Grundleh*
*ren der Math-
ematischen Wissenschaften [Fundamental Principles of Mathematical Scien*
*ces], vol.
278, Springer-Verlag, New York, 1985. MR 86f:18001
[CH02] J. Daniel Christensen and Mark Hovey, Quillen model structures for rela*
*tive homo-
logical algebra, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 2, *
*261-293. MR
1 912 401
[FC90] Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Spr*
*inger-Verlag,
Berlin, 1990, With an appendix by David Mumford. MR 92d:14036
[GH00] Paul G. Goerss and Michael J. Hopkins, Andr'e-Quillen (co)-homology for*
* simplicial
algebras over simplicial operads, Une d'egustation topologique [Topolog*
*ical morsels]:
homotopy theory in the Swiss Alps (Arolla, 1999), Amer. Math. Soc., Pro*
*vidence, RI,
2000, pp. 41-85. MR 2001m:18012
[Hir03] Philip S. Hirschhorn, Model categories and their localizations, Mathema*
*tical Surveys
and Monographs, vol. 99, American Mathematical Society, Providence, RI,*
* 2003. MR
1 944 041
[Hol01] Sharon Hollander, Homotopy theory for stacks, Ph.D. thesis, MIT, 2001.
[Hov99] Mark Hovey, Model categories, American Mathematical Society, Providence*
*, RI, 1999.
MR 99h:55031
[Hov01] _____, Model category structures on chain complexes of sheaves, Trans. *
*Amer. Math.
Soc. 353 (2001), no. 6, 2441-2457 (electronic). MR 1 814 077
[Hov02a]_____, Chromatic phenomena in the algebra of BP*BP -comodules, preprint*
*, 2002.
[Hov02b]_____, Morita theory for Hopf algebroids and presheaves of groupoids, A*
*mer. J.
Math. 124 (2002), 1289-1318.
[HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable *
*homotopy
theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114.
[HS99a] Mark Hovey and Hal Sadofsky, Invertible spectra in the E(n)-local stabl*
*e homotopy
category, J. London Math. Soc. (2) 60 (1999), no. 1, 284-302. MR 2000h:*
*55017
[HS99b] Mark Hovey and Neil P. Strickland, Morava K-theories and localisation, *
*Mem. Amer.
Math. Soc. 139 (1999), no. 666, viii+100.
[HS02] _____, Comodules and Landweber exact homology theories, preprint, 2002.
[JY80] David Copeland Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson*
* homol-
ogy and Hurewicz homomorphisms, Osaka J. Math. 17 (1980), no. 1, 117-13*
*6. MR
81b:55010
44 MARK HOVEY
[Lan76] P. S. Landweber, Homological properties of comodules over MU*MU and BP**
*BP,
Amer. J. Math. 98 (1976), 591-610.
[LMSM86]L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivar*
*iant stable
homotopy theory, Springer-Verlag, Berlin, 1986, With contributions by J*
*. E. McClure.
MR 88e:55002
[Mar83] H. R. Margolis, Spectra and the Steenrod algebra. Modules over the Stee*
*nrod alge-
bra and the stable homotopy category, North-Holland Mathematical Librar*
*y, vol. 29,
North-Holland Publishing Co., Amsterdam-New York, 1983.
[May01] J. P. May, The additivity of traces in triangulated categories, Adv. Ma*
*th. 163 (2001),
no. 1, 34-73. MR 2002k:18019
[MR77] H. R. Miller and D. C. Ravenel, Morava stabilizer algebras and the loca*
*lization of
Novikov's E2-term, Duke Math. J. 44 (1977), 433-447.
[Pal99] John H. Palmieri, Quillen stratification for the Steenrod algebra, Ann.*
* of Math. (2)
149 (1999), no. 2, 421-449. MR 1 689 334
[Pal01] John H. Palmieri, Stable homotopy over the Steenrod algebra, Mem. Amer.*
* Math. Soc.
151 (2001), no. 716, xiv+172.
[Pup79] Dieter Puppe, Duality in monoidal categories and applications to fixed-*
*point theory,
Game theory and related topics (Proc. Sem., Bonn and Hagen, 1978), Nort*
*h-Holland,
Amsterdam, 1979, pp. 173-185. MR 82m:55011
[Qui67] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, N*
*o. 43,
Springer-Verlag, Berlin, 1967. MR 36 #6480
[Rav86] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres,*
* Academic
Press, 1986.
[Rez98] Charles Rezk, Notes on the Hopkins-Miller theorem, Homotopy theory via *
*algebraic
geometry and group representations (Evanston, IL, 1997), Contemp. Math.*
*, vol. 220,
Amer. Math. Soc., Providence, RI, 1998, pp. 313-366. MR 2000i:55023
[SS00] Stefan Schwede and Brooke E. Shipley, Algebras and modules in monoidal *
*model cat-
egories, Proc. London Math. Soc. (3) 80 (2000), no. 2, 491-511. MR 1 73*
*4 325
[Ste75] B. Stenström, Rings of quotients, Die Grundlehren der mathematischen Wi*
*ssen-
schaften, vol. 217, Springer-Verlag, Berlin, 1975.
Department of Mathematics, Wesleyan University, Middletown, CT 06459
E-mail address: hovey@member.ams.org