SYMMETRIC RING SPECTRA AND TOPOLOGICAL
HOCHSCHILD HOMOLOGY
BROOKE SHIPLEY
1.Introduction
The category of symmetric spectra introduced by Jeff Smith is a closed sym-
metric monoidal category whose associated homotopy category is equivalent to the
traditional stable homotopy category, see [HSS]. In this paper, we study symmet*
*ric
ring spectra, i.e., the monoids in the category of symmetric spectra. The categ*
*ory
of symmetric ring spectra is closely related to the category of "functors with *
*smash
product defined on spheres" defined for instance in [HM, 2.7]. Actually, the ca*
*t-
egory of symmetric ring spectra is equivalent to the category of FSPs defined on
spheres if the usual connectivity and convergence conditions on FSPs are remove*
*d.
The choices of equivalences must also be changed when considering symmetric
ring spectra instead of FSPs on spheres. A map of FSPs on spheres is a weak
equivalence when the map is a ss*-isomorphism, i.e., when it induces an isomorp*
*hism
in the associated stable homotopy groups. As with symmetric spectra, one must
consider a broader class of equivalences called stable equivalences when working
with symmetric ring spectra, see 2.1.9. In section 2.2, the model category stru*
*cture
on symmetric ring spectra is defined with these stable equivalences. In [MMSS],
we show that the associated homotopy category is equivalent to the traditional
category of A1 -ring spectra.
Because there are more stable equivalences than ss*-isomorphisms, the classic*
*ally
defined stable homotopy groups are not invariants of the homotopy types of sym-
metric spectra or symmetric ring spectra. Hence stable equivalences can be hard
to identify. To remedy this we consider a detection functor, D, which turns sta*
*ble
equivalences into ss*-isomorphisms. Theorem 3.1.2, shows that X -! Y is a stable
equivalence if and only if DX -! DY is a ss*-isomorphism. Thus, the classical s*
*ta-
ble homotopy groups of DX are invariants of the stable homotopy type of X. There
is also a spectral sequence for calculating the classical stable homotopy group*
*s of
DX, see Proposition 2.3.4.
The category of FSPs was defined in [B] in order to define the topological
Hochschild homology for an associative ring spectrum R. In section 4, three dif*
*fer-
ent definitions of topological Hochschild homology for a symmetric ring spectrum
are considered; B"okstedt's original definition restated for symmetric ring spe*
*ctra,
4.2.6, a derived smash product definition, 4.1.1, and a definition which mimics*
* the
standard Hochschild complex from algebra, 4.1.2. Theorems 4.1.10 and 4.2.8 show
that under certain cofibrancy conditions these definitions all agree.
Perhaps the most surprising of these results is the agreement of B"okstedt's *
*def-
inition with the others without any connectivity or convergence conditions. Some
____________
Date: January 16, 1998.
Research partially supported by an NSF Postdoctoral Fellowship.
1
2 BROOKE SHIPLEY
conditions are indeed necessary to apply B"okstedt's approximation theorem, [B,
1.6], though the usual connectivity and convergence conditions can be weakened,
see Corollary 3.1.7. For spectra which do not satisfy these hypotheses the model
category structure on symmetric ring spectra is used instead to prove compari-
son results such as Theorem 3.1.2. Also, without any extra conditions, B"oksted*
*t's
original definition of THH takes stable equivalences of symmetric ring spectra *
*to
ss*-isomorphisms, see Corollary 4.2.10. See also Remark 2.2.2.
Outline. In the first section we recall various definitions from [HSS], defi*
*ne
symmetric ring spectra, and discuss homotopy colimits. Section 2.2 uses the res*
*ults
of [SS] and [HSS] to establish model category structures for symmetric ring spe*
*ctra,
for R-modules over any symmetric ring spectrum R, and for R-algebras over any
commutative symmetric ring spectrum R. In section 2.3 we define the homotopy
colimit of diagrams of symmetric spectra and state several comparison results f*
*or
homotopy colimits which are used in sections 3 and 4. A functor, D, which detec*
*ts
stable equivalences is defined in section 3. In section 4 three different defin*
*itions of
topological Hochschild homology are defined and compared. The detection functor
from section 3 is used in one of the comparisons in section 4.
Acknowledgments. I would like to thank Bill Dwyer and Jeff Smith for many
helpful conversations throughout this work. I also benefited from discussions w*
*ith
Lars Hesselholt, Mark Hovey, Mike Mandell, Haynes Miller, Charles Rezk, and
Stefan Schwede.
2.Basic definitions
In this section we state the basic definitions which are needed in sections 3*
* and 4.
Most of the definitions in the first subsection come from [HSS]. The second sub*
*sec-
tion, 2.2, considers symmetric ring spectra and model categories for R-modules *
*and
R-algebras. In the last subsection, 2.3, we consider the properties of the homo*
*topy
colimit needed for sections 3 and 4.
2.1. Symmetric spectra. We first define the symmetric monoidal category of
symmetric spectra. Next we define certain model category structures on symmetric
spectra. Then we consider a subcategory of symmetric spectra, the semistable sp*
*ec-
tra, between which stable equivalences are exactly the ss*-isomorphisms. Throug*
*h-
out this paper "space" means simplicial set, except in Remark 3.1.4.
Definition 2.1.1.Let be the skeleton of the category of finite sets and bijec-
tions with objects n = {1; . .;.n}. The category of symmetric sequences S* is t*
*he
category of functors from to S*, the category of pointed simplicial sets. Thus,
a symmetric sequence is a sequence Xn of spaces, where Xn is equipped with a
basepoint preserving action of the symmetric group n.
This category of symmetric sequences is a symmetric monoidal category with
the following definition of the tensor product of two symmetric sequences.
Definition 2.1.2 (HSS).Given two symmetric sequences X and Y , we define their
tensor product, X Y ,
_
(X Y )n = +n^pxq (Xp ^ Yq):
p+q=n
Note here, as elsewhere in the paper, we sometimes denote an extra basepoint as
X+ to make the notation more readable.
RING SPECTRA AND THH 3
Let S1 = [1]= _[1] and Sn = (S1)^n for n > 1. Then S = (S0; S1; . .;.Sn; . .).
is a symmetric sequence. In fact, S is a commutative monoid.
Definition 2.1.3.The category of symmetric spectra, Sp , is the category of left
S-modules in the category of symmetric sequences.
A symmetric spectrum is then a sequence of pointed spaces with a left, pointed
p action on Xp and associative, unital, pxq-equivariant maps Sp^Xq -!Xp+q.
The category of symmetric spectra is a symmetric monoidal category with the
following definition of the smash product of two symmetric spectra, see [HSS].
Definition 2.1.4.Given two symmetric spectra X and Y we define their smash
product X ^S Y as the coequalizer of the two maps
X S Y -!-!X Y:
We now describe certain symmetric spectra which play an important role in the
model category structures and in the later sections of this paper. Let I be the
skeleton of the category of finite sets and injections with objects n. Note th*
*at
homI(n; m) ~=m =m-n as m sets.
Definition 2.1.5.Define Fn :S* ! Sp by FnK = S GnK where GnK is the
symmetric sequence with homI(n; n)+ ^K in degree n and the basepoint elsewhere.
So (FnK)m = +m^m-n Sm-n ^ K ' homI(n; m)+ ^ Sm-n ^ K where Sn = * for
n < 0.
Fn is left adjoint to the nth evaluation functor Evn : Sp -! S* where Evn(X)*
* =
Xn. There is a natural isomorphism FnK ^S Fm L -!Fn+m (K ^ L).
Model category structures. There are two model category structures on symmet-
ric spectra which we consider; the injective model category and the stable model
category. The injective model category is a stepping stone for defining the sta*
*ble
model category. The homotopy category associated to the stable model category is
equivalent to the stable homotopy category of spectra, see [HSS]. Hence, the st*
*able
model category is the model category which we refer to most often. See [Q] or [*
*DS]
for the basic definitions for model categories.
Definition 2.1.6.Let f : X -! Y be a map in Sp . The map f is a level equiva-
lence if each fn : Xn -! Yn is a weak equivalence of spaces, ignoring the n act*
*ion.
It is a level cofibration if each fn is a cofibration of spaces.
With level equivalences, level cofibrations, and fibrations the maps with the*
* right
lifting property with respect to all maps which are trivial cofibrations, Sp f*
*orms a
simplicial model category referred to as the injective model category. A cofibr*
*ation
here is just a monomorphism. A fibrant object here is called an injective spect*
*rum.
An injective spectrum is a spectrum with the extension property with respect to
every monomorphism that is a level equivalence.
To define the equivalences for the stable model category we first need the fo*
*llow-
ing definition.
Definition 2.1.7.A spectrum X in Sp is an -spectrum if X is fibrant on each
level and the adjoint to the structure map S1 ^ Xn -! Xn+1 is a weak equivalence
of spaces for each n.
Define shifting down functors, shn : Sp -! Sp , by shn(X)k = Xn+k. Then
the adjoint of the structure maps give a map iX :X ! sh1X.
4 BROOKE SHIPLEY
Lemma 2.1.8. X is an -spectrum if and only if X is level fibrant and X !
sh1X is a level equivalence.
Definition 2.1.9.Let f : X -! Y be a map in Sp . The map f : X -! Y is a
stable equivalence if
ss0map (Y; Z) -!ss0map (X; Z)
is an isomorphism for all injective -spectra Z. The map f is a stable cofibrati*
*on if
it has the left lifting property with respect to each level trivial fibration, *
*i.e., a map
that is a trivial fibration on each level. The map f is a stable fibration if i*
*t has the
right lifting property with respect to each map which is both a stable cofibrat*
*ion
and a stable equivalence.
Theorem 2.1.10 (HSS). With these definitions of stable equivalences, stable cof*
*i-
brations, and stable fibrations, Sp forms a model category referred to as the *
*stable
model category. A map is a stable trivial fibration if and only if it is a leve*
*l triv-
ial fibration. Moreover, the fibrant objects are the -spectra and a map between
-spectra is a stable equivalence if and only if it is a level equivalence.
As shown in [HSS], the stable model category is in fact a cofibrantly generat*
*ed
model category. In particular, this means that a transfinite version of Quille*
*n's
small object argument, [Q, II 3.4], exists. This argument is central to the pro*
*ofs in
sections 3 and 4.
Proposition 2.1.11 (HSS).There is a set of maps J in Sp such that any stable
trivial cofibration is a retract of a directed colimit of pushouts of maps in J*
*. Sim-
ilarly, there is a set of maps I which generate the stable cofibrations. These *
*maps
are called the generating stable (trivial) cofibrations.
Let I = {Fn( _[k]+ ) -! Fn([k]+ )} and J0 = {Fn(l[k]+ ) -! Fn([k]+ )}. I is
the set mentioned in the proposition which generates the stable cofibrations. T*
*he
set J in the proposition is the union of J0 with a set K which we now describe.
First, consider the map oe : F1(S1) -! F0(S0) which is adjoint to the identity
map S1 -! S1 = Ev1(F0S0). There is a factorization of oe as a level cofibration
c : F1(S1) -! C followed by a level trivial fibration r : C -! F0(S0). C is def*
*ined
as the pushout in the following square.
F1S1 ^S F0([0]+ )--oe--!F0S0
? ?
1^Si0?y i?y
F1S1 ^S F0([1]+ )--b--! C
Define c : F1S1 -! C as the composite b O (1 ^S i1). In [HSS], c is shown to be*
* a
stable trivial cofibration. The left inverse, r0, to i0 induces a map oe O (1 ^*
*S r0) :
F1S1^S F0([1]+ ) -!F0S0. Use this map and the identity map on F0S0 to induce
r :C ! F0S0 using the property of the pushout.
The generating stable trivial cofibrations are built from the map c: F1S1 ! C
as follows. Let Pm;r be the pushout of the following square.
F1S1 ^S Fm ( _[r]+-)c^S1---!C ^S Fm ( _[r]+ )
? ?
1^SFm gr?y ?y
F1S1 ^S Fm ([r]+ )----! Pm;r
RING SPECTRA AND THH 5
The map P (c; Fm gr) : Pm;r -!C ^S Fm ([r]+ ) is induced by the property of the
pushout by the maps c ^S 1Fm ([r]+)and 1C ^S Fm gr. Let J = J0 [ K where
K = {P (c; Fm gr); m; r 0} and J0 is the set of generating level trivial cofib*
*rations
defined above.
Quillen's small object argument [Q, p. II 3.4] has an analogue which allows o*
*ne
to functorially factor maps whenever the model category is cofibrantly generate*
*d,
see [HSS].
Definition 2.1.12.Let L be the functorial stable fibrant replacement functor de-
fined by functorially factoring the map X -! * into a stable trivial cofibratio*
*n,
X -! LX and a stable fibration LX -! *. This is the factorization one defines
using the small object argument applied to the set of maps J. Using J0 instead,
one defines L0 as the functorial level fibrant replacement with X ! L0X a level
trivial cofibration and L0X ! * a level fibration.
This analogue of the small object argument provides a general method for prov*
*ing
that the class of cofibrations or trivial cofibrations has some property. One *
*only
needs to show that the generating maps have some property and that the property
is preserved under pushouts, colimits, and retracts. This general method is use*
*d in
sections 3 and 4.
We also need to know that the symmetric monoidal structure and the model
category structure fit together to give a monoidal model category structure, see
[HSS].
Proposition 2.1.13 (HSS).The stable model category is a monoidal model cat-
egory. That is, the symmetric monoidal structure satisfies the following pusho*
*ut
product axiom. Let f : A -!B and g : C -! D be two stable cofibrations. Then
Q(f; g) : (A ^S D) [A^SC (B ^S C) -!B ^S D
is a stable cofibration. If one of f or g is also a stable equivalence, then s*
*o is
Q(f; g).
This structure can also be restricted to a simplicial structure via the funct*
*or
F0 : S* -!Sp .
Semistable objects and ss*-isomorphisms. It is often useful to compare symmet*
*ric
spectra to the model category of spectra, SpN , defined in [BF]. There is a for*
*getful
functor U : Sp -! SpN which forgets the action of the symmetric groups and uses
the structure maps S1 ^ Xn -! X1+n.
Definition 2.1.14.Let ssk(X) = ssk(UX) = colimissk+iXi: A map f of symmetric
spectra is a ss*-isomorphism if it induces an isomorphism on these classical st*
*able
homotopy groups.
As seen in [HSS] these classical stable homotopy groups are NOT the maps in
the homotopy category of symmetric spectra of the sphere into X. For example
oe :F1S1 -!F0S0 is a stable equivalence but it is not a ss*-isomorphism. As sho*
*wn
in [HSS], though, a ss*-isomorphism is a particular example of a stable equival*
*ence.
Hence, to avoid confusion, we use the term ss*-isomorphism instead of stable ho-
motopy isomorphism and call these the classical stable homotopy groups instead
of just stable homotopy groups. In section 3 we construct a functor, D, which
converts any stable equivalence into a ss*-isomorphism between semistable spect*
*ra,
see Definition 2.1.16 below.
As in [BF], we define a functor Q for symmetric spectra.
6 BROOKE SHIPLEY
Definition 2.1.15.Define QX = colimnnL0shnX:
This functor does not have the same properties as in [BF]. For instance QX is
not always an -spectrum and X -! QX is not always a ss*-isomorphism. One
property that does continue to hold, however, is that a map f is a ss*-isomorph*
*ism
if and only if Qf is a level equivalence. Also, QX is always level fibrant.
Definition 2.1.16.A semistable symmetric spectrum is one for which the stable
fibrant replacement map, X -! LX, is a ss*-isomorphism.
Of course X -! LX is always a stable equivalence, but not all spectra are
semistable. For instance F1S1 is not semistable. Any stably fibrant spectrum,
i.e., an -spectrum, is semistable though. The following proposition shows that *
*on
semistable spectra Q has the same properties as in [BF] on SpN .
Proposition 2.1.17.The following are equivalent.
1. The symmetric spectrum X is semistable.
2. The map X -! L0sh1X is a ss*-isomorphism.
3. X -! QX is a ss*-isomorphism.
4. QX is an -spectrum.
Before proving this proposition we need the following lemma.
Lemma 2.1.18. Let X 2 Sp . Then ssk(QX)n and ssk+1(QX)n+1 are isomorphic
groups, and iQX : (QX)n ! (Q sh1X)n+1 induces a monomorphism ssk(QX)n -!
ssk+1(QX)n+1.
Proof.We can assume X is level fibrant. Both ssk(QX)n and ssk+1(QX)n+1 are
isomorphic to the (k - n)th classical stable homotopy group of X. However, the
map sskiQX need not be an isomorphism. Indeed, sskiQX is the map induced on the
colimit by the vertical maps in the diagram
sskXn ----! ssk+1Xn+1 ----! ssk+2Xn+2 ----! . . .
?? ? ?
y ?y ?y
ssk+1Xn+1----! ssk+2Xn+2 ----! ssk+3Xn+3 ----! . . .
where the vertical maps are not the same as the horizontal maps, but differ from
them by isomorphisms. The induced map on the colimit is injective in such a __
situation, though not necessarily surjective. |_*
*_|
Proof of Proposition 2.1.17.First we show that (1) implies (2) by using the fol*
*low-
ing diagram.
X ----! L0sh1X
?? ?
y ?y
LX ----! L0sh1LX
Since L0sh preserves ss*-isomorphisms both vertical arrows are ss*-isomorphisms.
The bottom map is a level equivalence since LX is an -spectrum. Hence the top
map is also a ss*-isomorphism.
Also, (2) easily implies (3). Since and sh commute and both preserve ss*-
isomorphisms, X ! QX is a colimit of ss*-isomorphisms provided X ! L0sh1X
is a ss*-isomorphism.
RING SPECTRA AND THH 7
Next we show that (3) and (4) are equivalent. The map ss*X ! ss*QX factors
as ss*X ! ss*(QX)0 ! ss*QX where the first map here is an isomorphism by
definition. Then by Lemma 2.1.18 we see that ss*(QX)0 ! ss*QX is an isomorphism
if and only if ss*(QX)n ! ss*+1(QX)n+1 is an isomorphism for each n.
To see that (3) implies (1), consider the following diagram.
X ----! LX
?? ?
y ?y
QX ----! QLX
By (3) and (4) the left arrow is a ss*-isomorphism to an -spectrum, QX. Since
LX is an -spectrum the right arrow is a level equivalence. Since X ! LX is a
stable equivalence, the bottom map must also be a stable equivalence. But a sta*
*ble
equivalence between -spectra is a level equivalence, so the bottom map is a lev*
*el_
equivalence. Hence the top map is a ss*-isomorphism. |__|
Two classes of semistable spectra are described in the following proposition.*
* The
second class includes the connective and convergent spectra.
Proposition 2.1.19. 1. If the classical stable homotopy groups of X are all
finite then X is semistable.
2. Suppose that X is a level fibrant symmetric spectrum and there exists some
ff > 1 such that Xn -! Xn+1 induces an isomorphism sskXn -! ssk+1Xn+1
for all k ffn for sufficiently large n. Then X is semistable.
Proof.By Lemma 2.1.18, ssk(QX)n ! ssk+1(QX)n+1 is a monomorphism between
two groups which are isomorphic. In the first case these groups are finite, so *
*this
map must be an isomorphism. Hence QX is an -spectrum, so X is semistable.
For the second part we also show that QX is an -spectrum. Since for fixed
k the maps ssk+iXn+i ! ssk+1+iXn+1+i are isomorphisms for large i, ssk(QX)n_!_
ssk+1(QX)n+1 is an isomorphism for each k and n. |__|
The next proposition shows that stable equivalences between semistable spectra
are particularly easy to understand.
Proposition 2.1.20.Let f : X -! Y be a map between two semistable symmetric
spectra. Then f is a stable equivalence if and only if it is a ss*-isomorphism.
Proof.Since X ! LX and Y ! LY are ss*-isomorphisms, Lf is a ss*-isomorphism
if an only if f is a ss*-isomorphism. But Lf is a map between -spectra so it is*
* a
ss*-isomorphism if an only if it is a level equivalence. But, in general, f is *
*a_stable
equivalence if and only if Lf is a level equivalence. |*
*__|
Finally, we show that any spectrum ss*-isomorphic to a semistable spectrum is
itself semistable.
Proposition 2.1.21.If f :X ! Y is a ss*-isomorphism and Y is semistable then
X is semistable.
Proof.Since Lf and Y ! LY are ss*-isomorphisms, X ! LX is also a ss*-_
isomorphism. |__|
8 BROOKE SHIPLEY
2.2. Symmetric ring spectra. In this section, rings, modules, and algebras are
defined for symmetric spectra. We also discuss the model category structures on
these categories.
Definition 2.2.1.A symmetric ring spectrum is a monoid in the category of sym-
metric spectra. In other words, a symmetric ring spectrum is a symmetric spectr*
*um,
R, with maps : R ^S R -! R and j : S -! R such that they are associative and
unital, i.e., O ( ^S id) = O (id^S ) and O (j ^S id) ~=id ~= O (id^S j).
R is called commutative if O tw = where tw: R ^S R ! R ^S R is the twist
isomorphism.
Since symmetric ring spectra are the only type of ring spectra in this paper *
*we
also refer to them as simply ring spectra. Using formal properties of symmetric
monoidal categories, one can show that a monoid in the category of S-modules is
the same as a monoid in the category of symmetric sequences with a monoid map
j : S -! R which is central in the sense that O (id^S j) O tw= O (j ^S id).
Remark 2.2.2. This description of a symmetric ring spectrum agrees with the
definition of a functor with smash product defined on spheres as in [HM, 2.7]. *
*The
centrality condition mentioned above, and in [HM, 2.7.ii], is necessary but was*
* not
included in some earlier definitions of FSPs defined on spheres. Note, however,*
* that
there are no connectivity (e.g. F (Sn+1) is n-connected) or convergence conditi*
*ons
(e.g. the limit is attained at a finite stage in the colimit defining ssn for e*
*ach n)
placed on symmetric ring spectra. These conditions are usually assumed although
not always explicitly stated when using FSPs. In particular, these conditions a*
*re
necessary for applying B"okstedt's approximation theorem, [B, 1.6]. Corollary 3*
*.1.7
shows that a special case of this approximation theorem holds for any semistable
spectrum. To consider non-convergent spectra we use Theorem 3.1.2 in place of
the approximation theorem. This theorem does not require any connectivity or
convergence conditions.
Proposition 2.1.19 shows that the connectivity and convergence conditions on
an FSP ensure that the associated underlying symmetric spectrum is semistable.
Proposition 2.1.20 shows that stable equivalences between such FSPs are exactly
the ss*-isomorphisms. As with the category of symmetric spectra, inverting the
ss*-isomorphisms is not enough to ensure that the homotopy category of symmetric
ring spectra is equivalent to the homotopy category of A1 -ring spectra. So once
the connectivity and convergence conditions are removed one must consider stable
equivalences instead of just ss*-isomorphisms.
We also need the following definitions of R-modules and R-algebras in later
sections.
Definition 2.2.3.Let R be a symmetric ring spectrum. A (left) R-module is a
symmetric spectrum M with a map ff : R^S M -! M that is associative and unital.
Definition 2.2.4.Let R be a commutative ring spectrum. An R-algebra is a
monoid in the category of R-modules. That is, an R-algebra is a symmetric spec-
trum A with R-module maps : A ^R A -! A and R -! A that satisfy the usual
associativity and unity diagrams.
Note that symmetric ring spectra are exactly the S-algebras.
Model category structures for rings and modules We developed techniques in
[SS] to form model category structures for algebras and modules over a cofibran*
*tly
RING SPECTRA AND THH 9
generated, monoidal model category. To apply the results from [SS] the smash
product is required to satisfy the monoid axiom, which is verified in [HSS]. He*
*nce,
the category of modules over a given symmetric ring spectrum R and the category
of R-algebras for any given commutative ring spectrum R are model categories,
see Theorems 2.2.5 and 2.2.7. Each of these model category structures uses the
underlying stable equivalences and underlying stable fibrations as the new weak
equivalences and fibrations. A map is then a cofibration if it has the left li*
*fting
property with respect to the underlying stable trivial fibrations. These model *
*cat-
egory structures are used in section 4.
After establishing these model category structures we state certain comparison
theorems which show that a weak equivalence of ring spectra induces an equivale*
*nce
of the homotopy theories of the respective modules and algebras. These comparis*
*on
theorems show that the cofibrancy condition that appears in certain theorems in
section 4 is not too restrictive.
In [HSS] Sp is shown to be a cofibrantly generated, monoidal model category
which satisfies the monoid axiom. Hence Theorem 4.1 in [SS] applies to give the
following results in the category of symmetric spectra.
Theorem 2.2.5. Let R be a symmetric ring spectrum. Then the category of R-
modules is a cofibrantly generated model category with weak equivalences and fi*
*bra-
tions given by the underlying stable model category structure of Sp . The gener*
*ating
cofibrations and trivial cofibrations are given by applying R ^S - to the gener*
*ating
maps in Sp .
Proof.This follows from Theorem 4.1 (1) in [SS], once we note that the domains
of the generating cofibrations and trivial cofibrations in Sp are small with_r*
*espect
to the whole category, see [HSS]. |__|
The next lemma is used in proving Theorem 2.2.7. It shows, for any commutative
monoid R, that the category of R-modules has properties similar to the underlyi*
*ng
category of S-modules.
Lemma 2.2.6. Let R be a commutative ring in Sp . Then the model category
structure on R-modules given above is a monoidal model category which satisfies
the monoid axiom.
Proof.This follows from [SS, 4.1 (2)]. |___|
Theorem 2.2.7. Let R be a commutative monoid in Sp . Then the category of
R-algebras is a cofibrantly generated model category with weak equivalences and
fibrations given by the underlying stable model category structure of Sp . The
generating cofibrations and trivial cofibrations are given by applying the free*
* monoid
functor to the generating (trivial) cofibrations of R-modules. Moreover, if f :*
* A -!
B is a cofibration of R-algebras with A cofibrant as an R-module then f is also*
* a
cofibration in the underlying category of R-modules. In particular, this shows *
*that
any cofibrant R-algebra is also cofibrant as an R-module.
Proof.This follows from Theorem 4.1 (3) in [SS]. The facts about cofibrant obje*
*cts
and cofibrations of R-algebras follow from [SS, 4.1 (3)] because S is_cofibrant*
*_in
Sp . |__|
Since symmetric ring spectra are exactly the S-algebras, the following is a c*
*orol-
lary of Theorem 2.2.7.
10 BROOKE SHIPLEY
Corollary 2.2.8.The category of symmetric ring spectra, S-alg, is a cofibrantly
generated model category with weak equivalences and fibrations the underlying s*
*table
equivalences and stable fibrations of Sp .
The following lemma is needed to apply the comparison theorems of [SS, 4.3-4]
which show that a weak equivalence of symmetric ring spectra induces an equiva-
lence on the respective homotopy theories of modules and algebras. This lemma is
also needed for section 4.
Lemma 2.2.9 (HSS). Let R be a symmetric ring spectrum and M a cofibrant R-
module. Then M ^R - takes level equivalences of R-modules to level equivalences
in Sp and it takes stable equivalences of R-modules to stable equivalences in *
*Sp .
The following two comparison theorems follow from Lemma 2.2.9, [SS, 4.3-4],
and the fact that S is cofibrant in Sp .
Theorem 2.2.10. If A -~! B is a map of symmetric ring spectra which is an
underlying stable equivalence, then the total derived functors of restriction a*
*nd ex-
tension of scalars induce equivalences of homotopy theories
Ho (A-mod ) ~= Ho (B-mod ) :
Theorem 2.2.11. If A -~!B is a map of commutative ring spectra which is an
underlying stable equivalence, then the total derived functors of restriction a*
*nd ex-
tension of scalars induce equivalences of homotopy theories
Ho (A-alg) ~= Ho (B-alg) :
The following two lemmas from [HSS] are used to verify some of the properties*
* of
the smash product that we have mentioned here. They are also needed in section *
*4.
Lemma 2.2.12. Let X -! Y be a level equivalence. Then A ^S X -! A ^S Y is a
level equivalence for any cofibrant spectrum A.
Lemma 2.2.13. Let X -! Y be a stable equivalence. Then A ^S X -! A ^S Y is
a stable equivalence for any cofibrant spectrum A.
2.3. Homotopy colimits. In this section we list some of the properties of the h*
*o-
motopy colimit functor for symmetric spectra which are used in the latter parts*
* of
this paper. The most important property is that the homotopy colimit in symmet-
ric spectra can be defined by using the homotopy colimit of spaces at each leve*
*l, see
Definition 2.3.1. This is useful not only because the homotopy colimit of space*
*s is
well understood, but also to show that the homotopy colimit preserves level equ*
*iv-
alences of symmetric spectra, see Proposition 2.3.2. We use the basic construct*
*ion
of the homotopy colimit for spaces from [BK].
Definition 2.3.1.Let B be a small category and F : B -! Sp a diagram of
symmetric spectra. Let Fldenote the diagram of spaces at level l. Then
(hocolimBSpF )l= hocolimBS*Fl:
This definition makes sense because any stable cofibration is a level cofibra*
*tion
and colimits in Sp are created on each level. Also, we show that this homotopy
colimit has the usual properties of a homotopy colimit. Namely, a map between
diagrams which is objectwise a level equivalence, a ss*-isomorphism, or a stable
equivalence induces the same type of equivalence on the homotopy colimit. The
RING SPECTRA AND THH 11
next two propositions consider the first two cases. The case of stable cofibrat*
*ions
could be proved by generalizing [BK, XII 4.2] to arbitrary model categories. In*
*stead,
here we use the detection functor developed in section 3 to verify this propert*
*y in
Lemma 4.1.5.
Proposition 2.3.2.Let F; G : B -! Sp be two diagrams of symmetric spectra
with a natural transformation j : F -! G between them. If j(b) : F (b) -! G(b) *
*is
a level equivalence at each object b 2 B, then hocolimBF -! hocolimBG is a level
equivalence.
Proof.Using Proposition 2.3.1 this statement reduces to asking that the homotopy
colimit preserve objectwise weak equivalences of simplicial sets. This is the *
*dual
of [BK, XI 5.6]. Cofibrancy conditions are not required here since any space_(i*
*.e.,_
simplicial set) is cofibrant. |__|
We also need to know that the homotopy colimit of an objectwise ss*-isomorphi*
*sm
is a ss*-isomorphism. For this we form a spectral sequence for calculating the
classical stable homotopy groups of the homotopy colimit. Following [BK, XII 5]
one can form a spectral sequence for calculating any homology theory applied to
the homotopy colimit of spaces. The spectral sequence is associated to the filt*
*ration
of the homotopy colimit given by the length of the sequence of maps in B. So for
F :B ! S* this spectral sequence converges to h*hocolimB F and has E2-term
E2s;t= colimsB(htF ):
We use the following lemma to go from the homology theory sss*defined on spac*
*es
by sss*K =ss*F0K to one on Sp .
Lemma 2.3.3. For X a symmetric spectrum, ss*X = colimnsss*Xn
Proof.Consider the lattice of spaces i(jL0jXi) indexed over (i; j) 2 N x N
with maps for fixed j using the adjoint structure maps of jL0jX or for fixed i
using ij applied to the adjoint structure maps of L0F0Xi. Applying homotopy
and taking colimits in the two different directions finishes the proof. In one *
*direc-
tion, one gets colimjss*jL0jX, but each of these terms and hence the colimit is*
* __
isomorphic to ss*X. In the other direction, one has the colimisss*Xi. *
* |__|
So applying the homology theory ss*, the above spectral sequence calculates s*
*s*
of each level of the homotopy colimit. Since ss*X = colimnsss*Xn and a sequenti*
*al
colimit of spectral sequences is a spectral sequence, taking the colimit of the*
*se level
spectral sequences produces a spectral sequence.
Proposition 2.3.4.For F :B ! Sp , there is a spectral sequence converging to
ss*hocolimBSpF with E2-term
E2s;t= colimsB(sstF ):
Using this spectral sequence we can show that homotopy colimits preserve ob-
jectwise ss*-isomorphisms.
Proposition 2.3.5.Let F; G : B -! Sp be two diagrams of symmetric spectra
with a natural transformation j : F -! G between them. If j(b) : F (b) -! G(b)
induces a ss*-isomorphism at each object b 2 B then hocolimBF -! hocolimB G
induces a ss*-isomorphism.
12 BROOKE SHIPLEY
Proof.Since j induces a ss*-isomorphism between the two diagrams in question, it
induces an E2-isomorphism. Thus it induces an isomorphism on the E1 -term and_
hence a ss*-isomorphism on the homotopy colimits. |__|
For section 3 we also need the following two cofinality results which are from
[BK, XI 9.2].
Given a functor between two small categories f : A -!B one has a natural map
hocolimAf*F -! hocolimB F . A functor f : A -! B is called terminal or right
cofinal, if for every object b 2 B the under category (b # f) is contractible s*
*ee [BK,
XI 9].
Proposition 2.3.6.Let f : A -! B be a functor which is terminal. Then for any
functor F : B -! Sp , hocolimAf*F -! hocolimBF is a level equivalence.
Proof.The dual of [BK, XI 9.2] states this property for objectwise cofibrant di*
*a-
grams of spaces. Applying this on each level and using the fact that spaces_are
cofibrant proves this statement. |__|
In section 3, we consider diagrams over the skeleton of the category of finit*
*e sets
and injections, I, with objects n. Let Im denote the full subcategory of I whose
objects are n where n is greater than or equal to m. The following lemma states
the cofinality information relating these categories.
Lemma 2.3.7. Let F :I ! Sp be a diagram of spectra. The inclusion um :
Im -! I is terminal, hence hocolimImu*mF ! hocolimIF is a level equivalence.
Proof.Consider the functor - + m : I -! Im which induces a functor on the under
categories. There is a natural transformation from the identity functor to both
um O (- + m) and (- + m) O um . Hence the under categories are each homotopy
equivalent to (i # I). But (i # I) is contractible because it has an initial o*
*bject_
1 : i -!i. The homotopy colimit statements follow from Proposition 2.3.6. *
*|__|
Using these cofinality results we can prove the following proposition.
Proposition 2.3.8.Let F; G: I -! S* be two diagrams of spaces with a natural
transformation j : F -! G between them. Assume that j(n): F (n) -! G(n) is
a (n) connected map, where (n) (n + 1) and limn(n) is infinite. Then
hocolimIF -! hocolimIG is a weak equivalence.
Proof.We show that the map is an N-equivalence for every N > 0: Choose an n
such that (n) > N. Then for every object m in I0nthe map j :F (m) ! G(n) is an
N-equivalence, and so we can conclude that j :hocolimInu*nF ! hocolimInu*nG __
is an N-equivalence. The proposition follows by Lemma 2.3.7. |__|
We also need the following proposition which shows that the homotopy colimit
of a diagram of level equivalences over I is level equivalent to its value on 0.
Proposition 2.3.9.Let F :I ! Sp be a diagram of spectra. Assume that for
each morphism f in I, F (f) is a level equivalence. Then the inclusion F (0) !
hocolimIF is a level equivalence.
Proof.Consider the constant functor C :I ! Sp with constant value F (0). Then
at each object the map C(n) = F (0) ! F (n) induced by the unique map 0 ! n is
a level equivalence. Hence, by Proposition 2.3.2, it induces a level equivalenc*
*e_on
the homotopy colimits, F (0) ! hocolimIF . |__|
RING SPECTRA AND THH 13
Finally, we need the following proposition due to Jeff Smith, [S]. Let T be t*
*he
category with objects n = {1; : :;:n} and morphisms the standard inclusions. Ho-
motopy colimits over T are weakly equivalent to telescopes. Let ! be the ordered
set of natural numbers and I! be the category whose objects are the finite sets*
* n
and the set ! and whose morphisms are inclusions. Let LhF :I! ! S* be the left
homotopy Kan extension of F :I ! S* along the inclusion of categories i: I ! I!.
Proposition 2.3.10.Let M be the monoid of injective maps i: ! ! ! under
composition. Given any functor F :I ! S*, then
1. hocolimIF is weakly equivalent to (LhF (!))hM where (-)hM is the homotopy
orbits with respect to the action of M, and
2. LhF (!) is weakly equivalent to hocolimTF .
Proof.For the convenience of the reader we sketch Smith's proof of this proposi*
*tion.
Since LhF is the homotopy Kan extension, hocolimIF ' hocolimI!LhF . Next,
consider the full subcategory, A of I! with just one object, !. Since the inclu*
*sion
of A in I! is terminal, hocolimI!LhF is weakly equivalent to hocolimALhF . Since
Hom A(!; !) = M, hocolimALhF is the homotopy orbit space (LhF (!))hM .
For the second statement, LhF (!) = hocolimi(n)!!2(i#!)F (n). Here i is the
inclusion I ! I!. The category T described above is equivalent to the category
(i O ff # !) for the inclusion ff: T ! I. This category (i O ff # !) is termin*
*al in
(i # !), because every under category has an initial object. So by Proposition *
*2.3.6,
LhF (!) is weakly equivalent to hocolimTF . |___|
3. Detecting stable equivalences
In this section we introduce a functor, D, which detects stable equivalences *
*in
the sense that a map X -! Y is a stable equivalence if and only if DX -! DY is a
ss*-isomorphism. Of course the stable fibrant replacement functor, L, also has *
*this
property. It even turns stable equivalences into level equivalences. The drawba*
*ck of
L is that its only description is via the small object argument. Hence it is di*
*fficult to
say much about L apart from its abstract properties. The advantage of the funct*
*or
D is that it has a more explicit definition. In particular, there is a spectral*
* sequence
for calculating the classical stable homotopy groups of DX, see Proposition 2.3*
*.4.
Moreover, these groups are invariants of the stable equivalence type of X becau*
*se
D takes stable equivalences to ss*-isomorphisms.
In Section 4 we see that D fits into a sequence of functors used to define TH*
*H in
[B]. We use the notation D instead of THH 0because D is defined on any symmetric
spectrum, not just on ring spectra.
3.1. Main statements and proofs. The detection functor D is defined as a
homotopy colimit over the diagram category of the skeleton of finite sets and i*
*n-
jections, I. Given a symmetric spectrum X, define a functor DX :I -! Sp whose
value on the object n is nL0F0Xn. Recall L0 is just a level fibrant replacement
functor. For a standard inclusion of a subset ff: n m the map DX (ff) is just
nL0 applied to the composition of maps F0Xn -! F0m-n Xm -! m-n F0Xm
induced by the structure maps of X. For an isomorphism, the action is given by
the conjugation action on the loop coordinates and on Xn. All morphisms in I are
compositions of isomorphisms and these standard inclusions.
14 BROOKE SHIPLEY
Definition 3.1.1.The detection functor D : Sp -! Sp is defined by
DX = hocolimISpDX :
As defined in definition 2.3.1, the homotopy colimit of symmetric spectra is *
*given
by a level homotopy colimit of spaces. Hence
(DX)n = hocolimk2IS*kL0nXk:
As mentioned above, the main reason for considering D is that it detects stab*
*le
equivalences. This is stated in the next theorem.
Theorem 3.1.2. The following are equivalent.
1. X -! Y is a stable equivalence.
2. DX -! DY induces a ss*-isomorphism.
3. D2X -! D2Y is a level equivalence.
4. QDX -! QDY is a level equivalence.
Remark 3.1.3. Notice that one can apply the forgetful functor U :Sp ! SpN
after applying D. In that case, this theorem says that although the usual for-
getful functor does not detect and preserve stable equivalences, the composition
of this detection functor with the forgetful functor does detect and preserve w*
*eak
equivalences. Note also that although the classical stable homotopy groups are *
*not
invariants of stable equivalence types in symmetric spectra this theorem shows *
*that
after applying D the classical stable homotopy groups are invariants.
Remark 3.1.4. We could also consider symmetric spectra over topological spaces
instead of simplicial sets here, see [HSS]. In that case, Theorem 3.1.2 and all*
* of the
statements leading up to it in this section and in section 2.3 which do not inv*
*olve
the functor Q hold when the objects involved are levelwise non-degenerately bas*
*ed
spaces. Hence, D also detects stable equivalences between symmetric spectra bas*
*ed
on topological spaces. More precisely, let c be a cofibrant replacement functor
of spaces applied levelwise, then X ! Y is a stable equivalence if and only if
DcX ! DcY is a ss*-isomorphism.
The only fact that is needed to modify all of these statements for topological
spaces is that homotopy colimits of non-degenerately based spaces are invariant
under weak homotopy equivalences. For the statements involving Q one needs
to consider stably cofibrant symmetric spectra because these statements require
that homotopy groups commute with directed colimits. But these statements are
separate from the statements involving D.
Theorem 3.1.2 considers the properties of D with respect to morphisms. The
following theorem considers the properties of D on objects.
Theorem 3.1.5. Let X be a symmetric spectrum.
1. DX is semistable.
2. If X is semistable, then the level fibrant replacement of DX, L0DX, is an
-spectrum.
Since stable equivalences between semistable spectra are ss*-isomorphisms and
between -spectra are level equivalences, Theorem 3.1.5 shows that the second and
third statements of Theorem 3.1.2 really just say that D and D2 preserve and de*
*tect
stable equivalences.
RING SPECTRA AND THH 15
Theorem 3.1.2 shows that the classical stable homotopy groups of DX are a
stable equivalence invariant. In the next theorem we show that they are in fact*
* the
derived classical stable homotopy groups, i.e., they are isomorphic to ss*LX.
Theorem 3.1.6. Let X be a symmetric spectrum.
1. There is a natural zig-zag of functors inducing ss*-isomorphisms between LX
and DX.
2. There are natural zig-zags of functors inducing level equivalences between*
* LX,
D2X, and QDX.
This theorem shows that the fibrant replacement functor is determined up to
ss*-isomorphism by D or up to level equivalence by D2 or QD. The spectral se-
quence for calculating the classical stable homotopy groups of DX, Proposition
2.3.4, calculates the derived stable homotopy groups ss*DX ~=ss*LX.
Corollary 3.1.7.For X any semistable spectrum, X and DX are ss*-isomorphic.
Remark 3.1.8. This corollary is a special case of [B, 1.6] where the convergence
and connectivity conditions are replaced by the semistable condition. By Propos*
*i-
tion 2.1.19 we recover a statement with convergence conditions but no connectiv*
*ity
conditions.
The proofs of Theorems 3.1.2 and 3.1.5 use the following properties of the fu*
*nctor
D.
Proposition 3.1.9.Let f : X -! Y be a map of symmetric spectra.
1. If f is a stable equivalence then Df is a ss*-isomorphism.
2. If f is a ss*-isomorphism then Df is a level equivalence.
3. For any semistable spectrum X, there is a natural zig-zag of functors indu*
*cing
level equivalences between LX and DX.
We assume Proposition 3.1.9 to prove Theorems 3.1.2, 3.1.5, and 3.1.6. The
proof of Proposition 3.1.9 is technical, so it is delayed until the next subsec*
*tion.
Proof of Theorem 3.1.6.By Proposition 3.1.9 (3) applied to LX there is a zig-
zag of level equivalences between LLX and DLX. By Proposition 3.1.9 (1) since
X ! LX is a stable equivalence DX ! DLX is a ss*-isomorphism. Hence, putting
these equivalences together with the fact that LLX is level equivalent to LX, we
get a zig-zag of ss*-isomorphisms between LX and DX.
Applying D to the zig-zag of ss*-isomorphisms between LX and DX shows that
DLX and D2X are level equivalent by Proposition 3.1.9 (2). Combining this with
the zig-zag of level equivalences between LX and DLX produces the level equiva-_
lence of LX and D2X. The equivalences for QDX are similar. |__|
Proof of Theorem 3.1.5.By Theorem 3.1.6 DX is ss*-isomorphic to LX. LX is an
-spectrum, hence it is semistable. So by Proposition 2.1.21, DX is semistable.
For X semistable, Proposition 3.1.9 shows that DX is level equivalent_to LX,
an -spectrum. Hence L0DX is an -spectrum. |__|
Proof of Theorem 3.1.2.Proposition 3.1.9 shows that (1) implies (2) and (2) im-
plies (3). A map f is a ss*-isomorphism if and only if Qf is a level equivalen*
*ce.
Hence the second and fourth statements are also equivalent.
16 BROOKE SHIPLEY
By Theorem 3.1.6 part 2, LX and D2X are naturally level equivalent. Hence if
D2X ! D2Y is a level equivalence then so is LX ! LY . But this is equivalent_to
X ! Y being a stable equivalence. |__|
3.2. Proof of Proposition 3.1.9. As mentioned above the proof of Proposition
3.1.9 is more technical. In this subsection we first prove the second part of P*
*roposi-
tion 3.1.9. Using this we prove the third part. Then, for the first part of Pro*
*position
3.1.9 we state and prove several lemmas which together finish the proof. Throug*
*h-
out this section we use several of the properties of the homotopy colimit devel*
*oped
in section 2.3.
For the proof of the second part of Proposition 3.1.9 we use Proposition 2.3.*
*10,
due to Jeff Smith.
Proof of Proposition 3.1.9 PartW2.e apply Lemma 2.3.10 to each level of D. Con-
sider the 0th level first. If f is a ss*-isomorphism then hocolimTnL0fn is a we*
*ak
equivalence, since ss*X =ss*hocolimTnL0Xn. Since taking homotopy orbits pre-
serves weak equivalences this shows that the 0th level of DX ! DY is a weak
equivalence, i.e., hocolimInL0fn is a weak equivalence.
The kth level of DX is the 0th level of DkX. Since kf is a ss*-isomorphism *
* __
if f is, this shows that each level is a weak equivalence. *
* |__|
For the third part of Proposition 3.1.9 we need the following functor. Recall*
* that
(shnX)k = Xn+k.
Definition 3.2.1.Define MX = hocolimInL0shnX.
Proof of Proposition 3.1.9 PartF3.irst we develop the transformations which play
a part in the zig-zag mentioned in the proposition. The inclusion of the object*
* 0
in I induces a natural map X -! MX. There is also a natural transformation of
functors D -! M. The structure maps on X induce a natural map of symmetric
spectra F0Xn -! shnX. Applying nL0 this map induces a map of diagrams
over I, and hence a natural map of homotopy colimits. So there is a natural zig*
*-zag
X -! MX- DX. The zig-zag mentioned in the proposition is this zig-zag applied
to LX along with the natural map DX ! DLX.
For semistable X, the map X ! LX is a ss*-isomorphism. So DX ! DLX is a
level equivalence by Proposition 3.1.9 part 2. So we only need to show that if *
*X is
an -spectrum, then both of the maps X ! MX- DX are level equivalences.
First we show that the map X -! MX is a level equivalence for any -spectrum
X. By definition an -spectrum is a level fibrant spectrum such that X -! sh1X
is a level equivalence. Using this and the fact that both shift and preserve l*
*evel
equivalences (on level fibrant spectra), one can show that each of the maps in *
*the
diagram over I used to define MX is a level equivalence. By Proposition 2.3.9 t*
*his
implies that X -! MX is a level equivalence.
To show that DX -! MX is a level equivalence for any -spectrum X, we
need to consider connective covers. Given a level fibrant spectrum X define its
kth connective cover, CkX, as the homotopy fibre of the map from X to its kth
Postnikov stage PkX. The kth Postnikov functor is the localization functor given
by localizing with respect to the set of maps {Fn@[m + n + k + 2] ! Fn[m +
n + k + 2]: m; n 0}. At level n, this functor is weakly equivalent to the (n +*
* k)th
Postnikov functor on spaces which is given by localization with respect to the *
*set of
maps {@[m + n + k + 2] ! [m + n + k + 2]: m 0}: See also [F2]. Then (CkT )n
RING SPECTRA AND THH 17
is n + k connected and ssi(CkT )n ! ssiTn is an isomorphism for i > n + k. Note
that any level fibrant spectrum is level equivalent to the homotopy colimit over
its connective covers. As -k decreases, the homotopy type of each level of C-kX
eventually becomes constant. So hocolimkC-kX ! X is a level equivalence.
Because m , L0and F0 commute up to level equivalence with directed homotopy
colimits and homotopy colimits commute, hocolimnDC-n X is level equivalent to
DX. The shift functor also commutes with homotopy colimits so hocolimnMC-n X
is level equivalent to MX. So we first show that for each n, DCnX and MCnX
are level equivalent.
In the diagrams creating these homotopy colimits, consider level l at the obj*
*ect
m 2 I. The map in question is m L0(Sl^ CnXm ) -! m L0CnXm+l. In general
the map m L0(llL0Y ) -! m L0Y is 2N - l - m + 1 connected when Y is N
connected. Hence for Y = CnXm+l the map in question is 2n+m+l+1 connected.
Using Proposition 2.3.8, we see that this connectivity implies that (DCnX)k -!
(MCnX)k is a weak equivalence. Homotopy commutes with directed colimits, so
taking colimits over n on both sides we get a weak equivalence DXk -!MXk. So
DX -! MX is a level equivalence. This is what we needed to finish the third part
of Proposition 3.1.9. Note also that since MX is an -spectrum this shows that_
the level fibrant replacement of DX is also an -spectrum. |__|
The proof of Proposition 3.1.9 part 1 breaks up into several parts. For the
case of stable trivial cofibrations we split the problem into showing that D of*
* any
generating stable trivial cofibration is a ss*-isomorphism and that D behaves w*
*ell
with respect to push outs, i.e., that the following two lemmas hold.
Lemma 3.2.2. Let j : A -! B be a generating stable trivial cofibration. Then
Dj : DA -!DB is a ss*-isomorphism.
Lemma 3.2.3. If
A ----! X
?? ?
y ?y
B ----! Y
is a pushout square with A -!B a cofibration, then
DA ----! DX
?? ?
y ?y
DB ----! DY
is a homotopy pushout square. I.e, if P is the homotopy colimit of DB- DA -!
DX, then P -! DY is a stable equivalence. In fact, P -! DY is a ss*-isomorphism.
Combining this lemma with the next shows that if DA -!DB is a ss*-isomorphism
then DX -! DY is also a ss*-isomorphism.
Lemma 3.2.4. Let
A ----! X
?? ?
y ?y
B ----! Y
18 BROOKE SHIPLEY
be a square in Sp with Y ss*-isomorphic to the homotopy pushout. Assume A -!B
is a ss*-isomorphism. Then X -! Y is a ss*-isomorphism.
For a proper model category this is a standard fact, that the homotopy pushout
of a weak equivalence is a weak equivalence. But no model category on symmetric
spectra has been written down with weak equivalences the ss*-isomorphisms.
Proof of Proposition 3.1.9 PartA1.ssuming Lemmas 3.2.2, 3.2.3, and 3.2.4, we
can finish this proof. First note that since any stable equivalence can be fact*
*ored
as a stable trivial cofibration followed by a level trivial fibration, we only *
*need to
show that D takes both stable trivial cofibrations and level equivalences to ss*
**-
isomorphisms.
A level equivalence induces a level equivalence at each object in the diagram*
* for
defining D and homotopy colimits preserve level equivalences, by Proposition2.3*
*.2.
Hence, D of a level equivalence is a level equivalence, and thus a ss*-isomorph*
*ism.
Any stable trivial cofibration is a retract of a directed colimit of pushouts*
* of maps
in J. Since retracts and directed colimits preserve ss*-isomorphisms we only ne*
*ed to
consider pushouts of generating stable trivial cofibrations. By Lemma 3.2.2, D *
*of
a generating trivial cofibration is a ss*-isomorphism. Hence, by Lemmas 3.2.3 a*
*nd
3.2.4, D of any map formed by a pushout of a generating stable trivial cofibrat*
*ion_
is a ss*-isomorphism. |__|
Proof of Lemma 3.2.4.Factor the map A -!X as a stable cofibration followed by
a level trivial fibration A ! Z -! X. Then form the pushout square as follows,
A ----! Z
?? ?
y ?y
B ----! P:
Since the top map is a level cofibration, P is the homotopy pushout of this squ*
*are.
Since A -! B is a ss*-isomorphism, Z -! P is a ss*-isomorphism because ss* is a
homology theory. Since Z -! X is a level equivalence, to see that X -! Y is a
ss*-isomorphism it is enough to know that P -! Y is a ss*-isomorphism. But_this*
*_is
assumed as part of the hypotheses. |__|
Now we proceed with the proof of Lemma 3.2.3.
Proof of Lemma 3.2.3.To see that P -! DY is a ss*-isomorphism, we use the fact
that homotopy colimits commute. P is the homotopy colimit of DB- DA -!DX,
so P is level equivalent to the homotopy colimit over I of the homotopy pushout*
* of
the squares at each object in I. In other words, let P nbe the homotopy pushout
at the object n 2 I. Then P is level equivalent to hocolimIP n.
Proposition 2.3.5 shows that a map of diagrams which is a ss*-isomorphism at
each object induces a ss*-isomorphism on the homotopy colimits. Hence, it is en*
*ough
to show that P n-! nL0F0(Yn) is a ss*-isomorphism for each n.
Since cofibrations induce level cofibrations and F0 preserves cofibrations and
pushouts, F0 applied to each level of the pushout square in the lemma is a homo*
*topy
pushout square. Since X ! L0X is a level equivalence it preserves homotopy
pushout squares up to level equivalence. Since n only shifts ss* by n, it prese*
*rves
homotopy pushouts up to ss*-isomorphism. Hence P n ! nL0F0(Yn) is a ss*-_
isomorphism. |__|
RING SPECTRA AND THH 19
We are left with proving Lemma 3.2.2. First we prove the following lemma which
identifies the stable homotopy type of DFm (K).
Lemma 3.2.5. There is a 2l-m-1 connected map l: m L0(Sl^K) ! (DFm K)l.
These maps fit together to give a map of symmetric spectra :m L0F0K ! DFm K
which is a ss*-isomorphism.
To prove this lemma we define another functor on the category I.
Definition 3.2.6.Define Fm K :I ! S* by (Fm K)(n) = homI(m; n)+ ^ K.
Fm (-) is left adjoint to the functor from I-diagrams over S* to S* which eva*
*luates
the diagram at m 2 I. Hence a natural transformation from Fm K into any diagram
over I is determined by a map from K to the diagram evaluated at m.
Proof of Lemma 3.2.5.Let DlFm K:I ! S* be the functor given by the lth level
of the functor DFm K. Then there is a map OEl: Fm m L0(Sl^ K) ! DlFm Kdeter-
mined by the inclusion of the wedge summand corresponding to the identity map,
m L0(Sl^ K) ! m L0(Sl^ homI(m; m)+ ^ K): Because the homotopy colimit of
a free diagram is weakly equivalent to the colimit, see [F1], the homotopy coli*
*mit
of this map is the map l: m L0(Sl^ K) ! (DFm K)lmentioned in the lemma.
We show that the map of diagrams is 2l -m-1 connected at each spot. At each
n 2 I, OEl(n), factors into two maps as follows,
hom I(m; n)+ ^ m L0(Sl^ K) ! m L0(hom I(m; n)+ ^ Sl^ K)
! m n-m L0n-m (hom I(m; n)+ ^ Sl^ K):
The first map is 2l - m - 1 connected by the application of the Blakers-Massey
theorem which shows that a wedge of loop spaces, X _ Y , is equivalent in the
stable range to the loop of the wedge, (X _ Y ). The second map is 2l - m - 1
connected by the Freudenthal suspension theorem, which for simplicial sets conc*
*erns
the map X ! L0X. Hence the map at each spot in the diagram, OEl(n) and thus
the map of homotopy colimits, lis 2l - m - 1 connected.
To see that these levels fit together, note that we can prolong Fm to a funct*
*or
from symmetric spectra to I-diagrams of symmetric spectra. Then there is a map
OE: Fm (m L0F0K) ! DFm Kwhich on level l is given by the map, OEl, above. Hence,
taking homotopy colimits, this induces a map :m L0F0K ! DFm K which is_a
ss*-isomorphism. |__|
Proof of Lemma 3.2.2.We must show that D of a generating trivial cofibration is
a ss*-isomorphism. First recall that the generating trivial cofibrations are th*
*e maps
P (c; Fm gr) : Pm;r-! C ^S Fm ([r]+ ) where Pm;r is the pushout below.
F1S1 ^S Fm ( _[r]+-)---!F1S1 ^S Fm ([r]+ )
?? ?
y ?y
C ^S Fm ( _[r]+ )----! Pm;r
To show that D of P (c; Fm gr) is a ss*-isomorphism it is only necessary to show
that D of cK : F1S1 ^S Fm K -! C ^S Fm K is a ss*-isomorphism for K = _[r]+ or
[r]+ . This is enough because Lemma 3.2.4 shows that if D(F1S1^SFm ( _[r]+ )) -!
D(C^SFm ( _[r]+ )) is a ss*-isomorphism then the pushout D(F1S1^SFm ([r]+ )) -!
D(Pm;r) is also a ss*-isomorphism. If D(F1S1^SFm ([r]+ )) -!D(C^SFm ([r]+ ))
20 BROOKE SHIPLEY
is also a ss*-isomorphism, this implies that D(Pm;r) -! D(C ^S Fm ([r]+ )) is a
ss*-isomorphism.
Since Fm K is cofibrant and C -! F0S0 is a level equivalence, the map C ^S
Fm K -! F0S0^S Fm K is a level equivalence, by Lemma 2.2.12. As already noticed,
D takes level equivalences to ss*-isomorphisms so we can assume that C is repla*
*ced
by F0S0 in cK for both values of K.
Thus we only need to show that DcK : DFm+1 (S1 ^ K) ! DFm K is a ss*-
isomorphism. To see that this map induces an isomorphism on the stable homo-
topy groups, note that Fm+1 (S1 ^ K) ! Fm K is induced by hom I(m + 1; n) !
homI(m; n) which in turn is induced by the inclusion of m in m + 1. Now con-
sider homotopy applied to the map of diagrams, DcK . Using the ss*-isomorphisms
from Lemma 3.2.5 above, this map is a map of free diagrams, hom I(m + 1; -)
sss*+m+1S1 ^ K ! hom I(m; -) sss*+mK. This map induces an isomorphism on
the colimits and all of the higher colimivanish. Hence, using the spectral se-
quence for calculating the homotopy of homotopy colimits, see section 2.3, DcK *
*is
a ss*-isomorphism. One can also see this by considering the associated map_of f*
*ree
diagrams directly. |__|
4.Topological Hochschild Homology
Let k be a commutative symmetric ring spectrum. Let R be a k-algebra. Define
Re = R ^k Rop. Let M be a k-symmetric R-bimodule, i.e., an Re-module. With
this set up we have two different definitions of topological Hochschild homolog*
*y, one
using a derived tensor product definition, the other mimicking the usual Hochsc*
*hild
complex. In Theorem 4.1.10 we see that these definitions construct stably equiv*
*alent
k-modules. Of course, since the smash product is only stably invariant for cofi*
*brant
spectra, the case where R is a cofibrant k-module is the only one of interest.
The idea to define topological Hochschild homology by mimicking algebra in th*
*is
way is due to Goodwillie. But because a symmetric monoidal category of spectra
was not available until recently, one could not simply implement this idea. B"o*
*kstedt
was the first one to define topological Hochschild homology by modifying this i*
*dea
to work with certain rings up to homotopy. This original definition of topologi*
*cal
Hochschild homology concerns the case when k = S. We restate the definition
of the simplicial spectrum THH .(R) and its realization, THH (R), from [B] for a
symmetric ring spectrum. See Definition 4.2.6. In Theorem 4.2.8 we show that for
k = S our new definitions are stably equivalent to the original definition when*
* R
is a cofibrant symmetric ring spectrum. As a corollary to this comparison theor*
*em
we see that B"okstedt's definition of THH takes stable equivalences of S-algebr*
*as
to ss*-isomorphisms. Hence it always determines the right homotopy type, even on
non-connective and non-convergent ring spectra, whereas the other two definitio*
*ns
give the right homotopy type only on cofibrant symmetric ring spectra.
4.1. Two definitions of relative topological Hochschild homology. The first
definition corresponds to the derived tensor product notion of algebraic Hochsc*
*hild
homology. The second definition mimics the Hochschild complex from algebraic
Hochschild homology. As we see in Theorem 4.1.10, these notions are stably equi*
*v-
alent when M is a cofibrant Re-module.
Definition 4.1.1.Define thhk(R; M) by M ^Re R.
RING SPECTRA AND THH 21
Let : R ^k R -! R and j : k -! R be the multiplication and unit maps on
R. Let OEr : M ^k R -! M and OEl: R ^k M -! M be the right and left R-module
structure maps of R acting on M. Let Rs be the smash product over k of s copies
of R, i.e., R ^k . .^.kR. The following definition mimics the Hochschild complex
as in [CE].
Definition 4.1.2.tHHk.(R; M) is the simplicial k-module with s-simplices M ^k
Rs. The simplicial face and degeneracy maps are given by
8
>(idM ) ^ (idR )i-1^ ^ (idR )s-i-1if1 i < s
:(OEl^ (idR )s-1) O o ifi = s
and si= idM ^ (idR )i^ j ^ (idR )s-1.
Each level of this simplicial symmetric spectrum is a bisimplicial set. Since
the realization of bisimplicial sets is equivalent to taking the diagonal, we u*
*se the
diagonal to define the realization of this simplicial symmetric spectrum.
Definition 4.1.3.Define the k-module tHHk(R; M) as the diagonal of the bisim-
plicial set at each level of this simplicial k-module. For the special cases of*
* k = S
or M = R we delete them from the notation.
Since the homotopy colimit of a diagram of symmetric spectra is determined
by the homotopy colimit of each level, the fact that the homotopy colimit of a
bisimplicial set is weakly equivalent to the diagonal simplicial set, see [BK, *
*XII
4.3], proves the following proposition.
op
Proposition 4.1.4.The map hocolimSp tHHk.(R; M) -! tHHk(R; M) is a level
equivalence.
Using D and this Proposition we can show that the realization of a map which
is a stable equivalence at each simplicial level is a stable equivalence.
Lemma 4.1.5. Let F; G: B ! Sp be two diagrams of symmetric spectra with
a natural transformation j :F ! G between them. If j(b): F (b) ! G(b) is a
stable equivalence for each object b in B then hocolimBF ! hocolimBG is a stable
equivalence.
Proof.Consider Dj :DF ! DG. By Theorem 3.1.2 this is a ss*-isomorphism at
each object, so by Proposition 2.3.5 the homotopy colimits are ss*-isomorphic. *
*Since
L0, F0 and homotopy colimits commute with homotopy colimits and n commutes
with homotopy colimits up to ss*-isomorphism, hocolimBDF is ss*-isomorphic to
D hocolimBF . Hence, D hocolimBF ! D hocolimBG is a ss*-isomorphism. Thus, __
by Theorem 3.1.2, hocolimBF ! hocolimBG is a stable equivalence. |__|
Corollary 4.1.6.A map between simplicial symmetric spectra which is a stable
equivalence on each level induces a stable equivalence on the realizations.
Proof.This just combines Lemma 4.1.5 and [BK, XII, 4.3]. |___|
Proposition 4.1.7.Let R -! R0 be a stable equivalence between k-algebras which
are cofibrant as k-modules, M a Re-module, N a (R0)e-module, and M -! N a
stable equivalence of Re-modules. Then tHHk (R; M) -! tHHk (R0; N) is a stable
equivalence. In particular, tHHk (R) -! tHHk (R0), tHHk (R; M) ! tHHk (R; N),
and tHHk(R; N) ! tHHk(R0; N) are stable equivalences.
22 BROOKE SHIPLEY
First note that a cofibrant k-algebra is also cofibrant as a k-module by Theo*
*rem
2.2.7, so there are many examples of k-algebras which are cofibrant as k-module*
*s.
Proof.Lemma 2.2.9 applied to k shows that P ^k - preserves stable equivalences
of k-modules if P is a cofibrant k-module. Hence, Rs ! R0sis a stable equivalen*
*ce
between cofibrant k-modules. So both M ^kRs ! N ^kRs and N ^kRs ! N ^kR0s
are also stable equivalences. Thus each simplicial level is a stable equivalenc*
*e._Then
Corollary 4.1.6 shows that this map induces a stable equivalence on tHHk. |*
*__|
To compare these two definitions of topological Hochschild homology we first
define certain bar constructions. Let N be a left R-module, with OEN :R^kN -! N,
and M a right R-module, with OEM : M ^k R -!M. We define the topological bar
construction Bk.(M; R; N) by mimicking algebra.
Definition 4.1.8.The bar construction Bk.(M; R; N) is the simplicial k-module
with s-simplices M ^k Rs ^k N. The face and degeneracy maps are given by
8
>idM ^ (idR )i-1^ ^ (idR )s-i-1^ idNif1 i < s
:idM ^ (idR )s-1^ OEN ifi = s
Let Bk(M; R; N) be the realization of this simplicial k-module.
Let c.(X) be the constant simplicial object with X in each simplicial degree.
Using the identification M ^R R ~=M, the map j : k -! R induces a simplicial
k-module map Bk.(M; R; N) -!c.(M ^R N).
Lemma 4.1.9. For M a cofibrant R-module, the simplicial map of k-modules,
Bk.(M; R; N) -! c.(M ^R N), induces a stable equivalence of Bk(M; R; N) -!
M ^R N.
Proof.Note that Bk.(M; R; N) ~=c.M ^R Bk.(R; R; N). Since realization commutes
with smash products, Bk(M; R; N) ~=M ^R Bk(R; R; N). So using Lemma 2.2.9
it is enough to show that Bk(R; R; N) -! N is a stable equivalence. The map
N ~=k ^k N -! R ^k N provides a simplicial retraction for Bk.(R; R; N). Hence
the spectral sequence for computing the classical stable homotopy groups of the
homotopy colimit of this simplicial k-module collapses. So the map Bk.(R; R; N)*
*_-!
c.N induces a ss*-isomorphism on the realizations. |__|
Using the bar construction we now show that the two definitions of topological
Hochschild homology are stably equivalent when M is a cofibrant Re-module.
Theorem 4.1.10. There is a natural map of k-modules tHHk(R; M) -!thhk(R; M)
which is a stable equivalence for M a cofibrant Re-module.
Proof.We show that tHHk (R; M) is naturally isomorphic to M ^Re Bk(R; R; R)
below. Then the map tHHk (R; M) -! thhk(R; M) is given by M ^Re OE for OE :
Bk(R; R; R) -!R. R is always a cofibrant R-module, hence OE is a stable equival*
*ence
by Lemma 4.1.9. Then Proposition 2.2.9 shows that M ^ReOE is a stable equivalen*
*ce
since M is a cofibrant Re-module.
To see that tHHk(R; M) is naturally isomorphic to M ^Re Bk(R; R; R) we show
that tHHk .(R; M) is naturally isomorphic to c:(M) ^Re Bk.(R; R; R). On each
simplicial level there are natural isomorphisms
M ^k Rs ~=M ^Re (Re ^k Rs) ~=M ^Re (R ^k Rs ^k R) = M ^Re Bks(R; R; R):
RING SPECTRA AND THH 23
These isomorphisms commute with the simplicial structure. Hence the simplicial
k-modules are naturally isomorphic, so their realizations are also naturally_is*
*omor-
phic. |__|
4.2. B"okstedt's definition of topological Hochschild homology. We now de-
fine the simplicial spectrum THH .(R; M) and its realization THH (R; M) followi*
*ng
B"okstedt's original definitions. Each of the levels of the simplicial spectrum*
* THH .
can be defined for a general symmetric spectrum X. A ring structure is only nec*
*es-
sary for defining the simplicial structure. In fact, each of the levels of THH *
*.can be
thought of as a functor which gives the correct ss*-isomorphism type for the sm*
*ash
product of symmetric spectra. We start by considering each of these levels as a
functor of several variables.
Let X denote a sequence of j + 1 spectra, X0; : :;:Xj. Define a functor DjX
from Ij+1 to Sp which at n = (n0; . .;.nj) takes the value,
DjX(n) = nL0F0(X0n0^ : :^:Xjnj)
where L0 is a level fibrant replacement functor and n = ni, the sum of the ni.
Note that D0(X) is DX , the functor defined at the beginning of section 3. To s*
*ee
that DjX is defined over Ij+1 one uses maps similar to those described for DX .
Definition 4.2.1.Let X0; : :;:Xj be symmetric spectra. Define
j+1j
TjX = hocolimI D X:
We now define a natural transformation OEjX: T jX ! D(X0 ^S : :^:SXj).
Let : Ij+1 -! I be the functor induced by concatenation of all of the factors *
*in
Ij+1. Then there is a natural transformation from DjX to *D0(X0 ^S : :^:SXj).
This natural transformation is induced by the map from X0n0^ . .^.Xjnjto the nth
level of X0 ^S : :^:SXj: This map is n0x . .x.nj equivariant, which is exactly
what is necessaryjover+Ij+1.1Hence,jon+homotopy1colimits there is a natural map
hocolimI DjX -!hocolimI *D0(X0 ^S : :^:SXj).
Definition 4.2.2.There is a natural transformation OEjX: T jX ! D(X0^S: :^:S
Xj). It is given by the composition
j+1j Ij+1* 0 0 j
hocolimI D X -!hocolim D (X ^S : :^:SX )
-! hocolimID0(X0 ^S : :^:SXj):
Proposition 4.2.3.For any cofibrant symmetric spectra, X0; : :;:Xj, the map
OEjX is a ss*-isomorphism.
This proposition is proved in subsection 4.3. It is used in proving the compa*
*rison
theorem between B"okstedt's definition of THH and our previous definition of t*
*HH.
As a corollary of this proposition, Tj gives the correct ss*-isomorphism type f*
*or the
derived smash product of j + 1 symmetric spectra. Recall that the smash product
is only homotopy invariant on cofibrant spectra, so the derived smash product is
the smash product of the cofibrant replacements. In the stable model category of
symmetric spectra, consider a cofibrant replacement functor, C, analogous to the
fibrant replacement functor L.
Corollary 4.2.4.ss*T jX is isomorphic to ss*L(CX0 ^S : :^:SCXj), the derived
homotopy of the derived smash product of X0; : :;:Xj.
24 BROOKE SHIPLEY
Proof.Since C is a cofibrant replacement functor, CX ! X is a level equivalence.
Hence Tj(CX0; : :;:CXj) ! Tj(X0; : :;:Xj) is a level equivalence by Proposition
2.3.2 because the map is a level equivalence at each object in the diagram defi*
*ning
Tj. So this corollary follows from Proposition 4.2.3 since ss*D(CX0 ^S : :^:SCX*
*j)_
is isomorphic to ss*L(CX0 ^S : :^:SCXj) by Theorem 3.1.6. |__|
We now use this functor Tj to define THH following B"okstedt's definition in*
* [B].
Definition 4.2.5.Let R be a symmetric ring spectrum with M an Re-module.
Define THH j(R; M) = Tj(M; R; : :;:R).
The functors THH j(R; M) fit together to form a simplicial symmetric spectrum
THH .(R; M). Although the definition of THH j(R; M) does not use the ring struc-
ture of R or the module structure of M, the simplicial structure of THH .(R; M)
does use both the multiplication and unit maps. The ith face map uses the funct*
*or
ffii: Ij+1 -!Ij defined by concatenation of the sets in factors i and i + 1. Th*
*e last
face map uses the cyclic permutation of Ij+1 followed by concatenation of the f*
*irst
two factors. For ease of notation let Dj(R; M) = Dj(M; R; : :;:R). The multipli-
cation of R and M defines a natural transformation of functors from Dj(R; M) to
ffi*iDj-1(R; M). So di is the composition
j+1j Ij+1 * j-1 Ij j-1
di: hocolimI D (R; M) -!hocolim ffiiD (R; M) -!hocolim D (R; M):
The degeneracy maps are similar.
Definition 4.2.6.Define THH (R; M) as the diagonal of the bisimplicial set at
each level of the simplicial symmetric spectrum THH .(R; M).
One can check that each level in this spectrum agrees with the definition in *
*[B]
when M = R.
As in Proposition 4.1.4 we have the following equivalence.
op
Proposition 4.2.7.The map hocolimSp THH .(R; M) -! THH (R; M) is a level
equivalence.
The next theorem shows that the definition of topological Hochschild homology
which mimics the Hochschild complex is stably equivalent to the original defini*
*tion
of topological Hochschild homology.
Theorem 4.2.8. Let R be a cofibrant ring spectrum. Then there is a natural zig-
zag of stable equivalences between tHH (R; M) and THH (R; M).
Proof.The zig-zag of functors between tHH and THH is induced by a zig-zag of
maps between the simplicial complexes defining tHH and THH . First one applies
the zig-zag of functors 1 -! L -!ML- DL D--D to each simplicial level of the
Hochschild complex defining tHH . Here, L is the fibrant replacement functor, M,
D, and the natural transformations are defined in section 3, see 3.1.1, 3.2.1, *
*and the
proof of 3.1.9 part 3. Then there is a natural map OEj: THH j(R; M) -!D(M^SRj)
To see that the OEj maps commute with the simplicial maps, one needs to note th*
*at
the multiplication maps commute with the first map in the composite defining OE*
*j.
This follows since the map Rn ^ Rm ! Rn+m is the map on the appropriate wedge
summand of the map R R ! R which induces the map R ^S R ! R. The maps
involving M are similar. Putting these simplicial levels together one gets a zi*
*g-zag
of natural transformations from tHH .(-) to THH .(-). Note that by composing
maps this zig-zag is only of length 2.
RING SPECTRA AND THH 25
The zig-zag of functors between 1 and D was investigated in section 3. indu*
*ces
a stable equivalence on any object by definition of the fibrant replacement fun*
*ctor
L. So Corollary 4.1.6 applies to show that induces a stable equivalence on the
realization of these simplicial S-modules.
Since is always a stable equivalence, D is a ss*-isomorphism on each simpl*
*icial
level by Theorem 3.1.2. The two natural transformations of middle functors L -!
ML- DL induce level equivalences by Proposition 3.1.9. Propositions 2.3.2 and
2.3.5 apply to these natural transformations to show that they also induce level
equivalences and ss*-isomorphisms on the realizations.
So the only part of the zig-zag between tHH (R; M) and THH (R; M) that is left
is THH .(R; M) -! D(tHH .(R; M)). Let CM ! M be a cofibrant replacement of
M as an Re-module. Then by Proposition 4.1.7, tHH j(R; CM) ! tHH j(R; M) is
a stable equivalence. Similarly, THH j(R; CM) ! THH j(R; M) is a stable equiva-
lence since CM ! M is a level equivalence and hence induces a level equivalence
on the homotopy colimits used to define THH j. So it is enough to prove that
THH .(R; M) -! D(tHH .(R; M)) is a stable equivalence in the case when M is
cofibrant.
Since R is cofibrant as an S-algebra, it is also cofibrant as an S-module. Si*
*nce
M is cofibrant as an Re-module and Re is cofibrant, M is also cofibrant as an S-
module. Proposition 4.2.3 shows that if R and M are any cofibrant S-modules then
THH j(R; M) -! D(M ^S Rj) is a ss*-isomorphism. Then Proposition 2.3.5 shows
that this is enough to ensure that the map on the realizations is a ss*-isomorp*
*hism._
Hence, assuming Proposition 4.2.3, this finishes the proof of Theorem 4.2.8. *
* |__|
Corollary 4.2.9.The derived stable homotopy groups of tHH (R; M) are isomor-
phic to ss*THH (R; M).
Since tHH (R; M) and THH (R; M) are stably homotopic their derived stable
homotopy groups must be isomorphic. So this corollary says that the derived sta*
*ble
homotopy groups of tHH (R; M) are isomorphic to the classical stable homotopy
groups of THH (R; M).
Proof.The proof of Theorem 4.2.8 shows that the map from THH (R; M) to the real-
ization of D tHH.(R; M) is a ss*-isomorphism. But this realization is ss*-isomo*
*rphic
to D tHH(R; M) as shown in the proof of Proposition 4.1.5. So the derived sta-
ble homotopy groups of tHH (R; M), i.e., ss*D tHH(R; M), are isomorphic to the_
classical stable homotopy groups of THH (R; M). |__|
Using this comparison we can show that B"okstedt's original definition of THH
takes stable equivalences of ring spectra to ss*-isomorphisms. This is a stron*
*ger
result than for tHH because no cofibrancy condition is needed here and the map *
*is
a ss*-isomorphism, not just a stable equivalence.
Corollary 4.2.10.Let R -! R0 be a stable equivalence of ring spectra, M a Re-
module, N a (R0)e-module, and M ! N a stable equivalence of Re-modules. Then
THH (R; M) -!THH (R0; N) is a ss*-isomorphism.
Remark 4.2.11. This corollary could also be proved without using these com-
parison results. One can show that each THH j takes stable equivalences to ss*-
isomorphisms following arguments similar to those for THH 0 = D in section 3.
Then Proposition 2.3.5 shows that the realization, THH , also takes stable equi*
*va-
lences to ss*-isomorphisms.
26 BROOKE SHIPLEY
Proof.In the category of symmetric ring spectra, define a functorial cofibrant *
*re-
placement functor, C. Applying this functor we have the following square.
CR ----! CR0
?? ?
y ?y
R ----! R0
Each of the vertical maps is a level trivial fibration and hence a level equiva*
*lence.
The bottom map is a stable equivalence by assumption. Hence the top map is
also a stable equivalence. To show that THH applied to the bottom map is a ss*-
isomorphism we show that THH applied to the other three maps in this square are
ss*-isomorphisms.
We also need to consider cofibrant replacements of the modules in question. M
is a (CR)e-module and N is a (CR0)e-module. We replace them by modules which
are cofibrant as underlying S-modules. Since CR is a cofibrant S-algebra it is*
* a
cofibrant S-module. Thus (CR)e is also cofibrant as an S-module by Proposition
2.1.13. Hence the cofibrations in the category of (CR)e-modules are also underl*
*ying
cofibrations. So let CM ! M be the cofibrant replacement of M in the category
of (CR)e-modules. Similarly, R0 let CN ! N be the cofibrant replacement of N
as a (CR0)e-module. Then both CM and CN are cofibrant as S-modules. Also,
in the category of (CR)e-modules by the lifting property in the model category *
*of
(CR)e-modules we have a map CM ! CN because CN ! N is a level trivial
fibration. This map CM ! CN is a stable equivalence by the two out of three
property.
The level equivalences CR ! R and CM ! M induce a level equivalence on
each object of the diagram defining THH j. So by applying Proposition 2.3.2 and
Lemma 4.2.7 this shows that THH (CR; CM) ! THH (R; M) is a level equivalence.
Similarly THH (CR0; CN) ! THH (R0; N) is a level equivalence.
For the top map, first consider applying tHH . Proposition 4.1.7 implies that
tHH(CR; CM) -!tHH (CR0; CN) is a stable equivalence. Hence by Theorem 3.1.2,
D tHH(CR; CM) -! D tHH(CR0; CN) is a ss*-isomorphism. But in the proof of
Theorem 4.2.8 we showed that THH -!D tHH induces a ss*-isomorphism if the ring
and module are cofibrant as S-modules. So THH (CR; CM) -! THH (CR0; CN) is
a ss*-isomorphism. Stringing these equivalences together finishes the proof of *
*this
corollary. Because Proposition 4.2.3 applies to each level, we have actually sh*
*own_
that each THH j(R; M) -!THH j(R0; N) is also a ss*-isomorphism. |__|
4.3. Proof of Proposition 4.2.3. To prove Proposition 4.2.3 we follow an outline
similar to the proof that D takes stable trivial cofibrations to ss*-isomorphis*
*ms.
We show that OEj is a ss*-isomorphism when it is evaluated only on free symmetr*
*ic
spectra, i.e., some FnK. Then we prove an induction step lemma which deals with
pushouts over generating stable trivial cofibrations. Using these lemmas we show
that OEj is a ss*-isomorphism on any collection of cofibrant spectra.
Lemma 4.3.1. OEj(Fn0K0; : :;:FnjKj) is a ss*-isomorphism.
RING SPECTRA AND THH 27
Lemma 4.3.2. Let A ! B be a stable cofibration and X0; : :;:Xj be cofibrant
S-modules. Consider the following pushout square.
A ----! X
?? ?
y ?y
B ----! Y
Assume that THH j+1(X0; : :;:Z; : :;:Xj) ! D(X0 ^S : :^:SZ ^S : :^:SXj) is a
ss*-isomorphism for Z = A; B; or X where Z is inserted between the ith and i + *
*1st
spots. Then THH j+1(X0; : :;:Y; : :;:Xj) ! D(X0 ^S : :^:SY ^S : :^:SXj) is a
ss*-isomorphism.
Using these two lemmas we can now prove Proposition 4.2.3.
Proof of Proposition 4.2.3.We prove this by induction on i with the induction
assumption that OEj is a ss*-isomorphism when j - i variables are free spectra *
*and
the other variables are cofibrant. Lemma 4.3.1 verifies this for i = 0. For t*
*he
induction step, in one variable we build up a cofibrant spectrum from the initi*
*al
spectrum by retracts, colimits, and pushouts over generating cofibrations. Sin*
*ce
retracts of ss*-isomorphisms are ss*-isomorphisms and OEj of a retract is a ret*
*ract we
only need to consider colimits and pushouts.
Because F0, smash products, L0, n, and homotopy colimits commute with
filtered colimits, Tj of a colimit in one of the variables is a colimit. This i*
*s also true
of D. Since a filtered colimit of ss*-isomorphisms is a ss*-isomorphism this me*
*ans
OEj of a colimit in one variable is a ss*-isomorphism if it is a ss*-isomorphis*
*m at each
spot in the sequence. Hence we are only left with pushouts.
Since OEj is a level equivalence between trivial spectra if one of the variab*
*les is
the initial spectrum, *, one can proceed by induction to verify the pushout pro*
*p-
erty. By induction the two corners in the pushout corresponding to the generati*
*ng
cofibration are ss*-isomorphisms. This is because generating cofibrations are o*
*f the
form FnK ! FnL, so these two corners have one extra variable a free spectrum and
hence fall into the case covered by the previous induction step. The third corn*
*er
is assumed to be a ss*-isomorphism by induction, hence OEj is a ss*-isomorphism*
*_on
the pushout corner by Lemma 4.3.2. |__|
Proof of Lemma 4.3.1.To show that OEj(Fn0K0; : :;:FnjKj) is a ss*-isomorphism
we first establish the stable homotopy type of Tj(Fn0K0; : :;:FnjKj). There is a
functor F(n0;:::;nj)X :Ij+1 ! Sp defined by
F(n0;:::;nj)X(m0; : :m:j) = homIj+1((n0; : :;:nj); (m0; : :m:j))+ ^ X:
Then F(n0;:::;nj)(-) is left adjoint to the functor from I-diagrams over Sp to*
* Sp
which evaluates the diagram at (n0; : :;:nj) 2 Ij+1. There is a map of diagrams
F(n0;:::;nj)(nL0F0(K0^ : :^:Kj)) ! Dj(Fn0K0; : :;:FnjKj) where n = ni. Each
spot in this diagram is a ss*-isomorphism. This is similar to the proof of Lemma
3.2.5, on each level the map is an equivalence in the stable range by the Blake*
*rs-
Massey and the Freudenthal suspension theorems. Hence the map on homotopy
colimits is also a ss*-isomorphism, nL0F0(K0^: :^:Kj)) ! Tj(Fn0K0; : :;:FnjKj).
By Lemma 3.2.5, nL0F0(K0 ^ : :^:Kj)) ! D(Fn(K0 ^ : :^:Kj)) is also a
ss*-isomorphism. To see that OEj induces a ss*-isomorphism, note that on the fr*
*ee
j+1
diagrams there are similar maps hocolimI F(n0;:::;nj)(nL0F0(K0 ^ : :^:Kj)) !
28 BROOKE SHIPLEY
j+1 I
hocolimI *Fn(nL0F0(K0 ^ : :^:Kj)) ! hocolim Fn(nL0F0(K0 ^ : :^:Kj)) __
which induce level equivalences on the homotopy colimits. |__|
To prove Lemma 4.3.2 we first need to show that Tj of a homotopy pushout in
one variable is a homotopy pushout.
Lemma 4.3.3. Let X0; : :;:Xj be cofibrant spectra. If
A ----! X
?? ?
y ?y
B ----! Y
is a pushout square with A -!B a cofibration, then
Tj+1(X0; : :;:A; : :X:j)----!Tj+1(X0; : :;:X; : :X:j)
?? ?
y ?y
Tj+1(X0; : :;:B; : :X:j)----!Tj+1(X0; : :;:Y; : :X:j)
is a homotopy pushout square. I.e, if P is the homotopy pushout of the second
square then P -! T j+1(X0; : :;:Y; : :X:j) is a stable equivalence. In fact, P*
* -!
Tj+1(X0; : :;:Y; : :X:j) is a ss*-isomorphism.
Proof.This proof is similar to the proof of Lemma 3.2.3. As with Lemma 3.2.3, it
is enough to consider each object in Ij+1 since homotopy colimits commute.
The following square is a pushout square with the left map a cofibration since
each Xi is cofibrant.
X0n0^ : :^:Ani^ : :X:jnj----!X0n0^ : :^:Xni^ : :X:jnj
?? ?
y ?y
X0n0^ : :^:Bni^ : :X:jnj----! X0n0^ : :^:Yni^ : :X:jnj
The first step in constructing Tj is just applying F0 to this square. F0 pres*
*erves
cofibrations and pushouts, hence F0 applied to this square is a homotopy pushou*
*t.
L0preserves homotopy pushout squares up to level equivalence and ni preserves
homotopy pushout squares up to ss*-isomorphism. Hence the map from the homo-
topy pushout to the bottom right corner is a ss*-isomorphism. Since the homotop*
*y __
colimit of ss*-isomorphisms is a ss*-isomorphism, this finishes the proof. *
* |__|
Proof of Lemma 4.3.2.Both Tj and D take homotopy pushouts in one variable to
homotopy pushouts where the map from the pushout to the bottom right corner is
a ss*-isomorphism by Lemmas 4.3.3 and 3.2.3. Hence, this lemma follows from the_
fact that homotopy colimits preserve ss*-isomorphisms, Lemma 2.3.5. |_*
*_|
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Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
E-mail address: bshipley@math.uchicago.edu