STABLY THICK SUBCATEGORIES OF MODULES
OVER HOPF ALGEBRAS
MARK HOVEY AND JOHN H. PALMIERI
Abstract.We discuss a general method for classifying certain subcategori*
*es
of the category of finite-dimensional modules over a finite-dimensional *
*cocom-
mutative Hopf algebra B. Our method is based on that of Benson-Carlson-
Rickard [BCR96], who classify such subcategories when B = kG, the group
ring of a finite group G over an algebraically closed field k. We get a *
*similar
classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod*
* alge-
bra, with scalars extended to the algebraic closure of F2. Along the way*
*, we
prove a Quillen stratification theorem for cohomological varieties of mo*
*dules
over any B, in terms of quasi-elementary sub-Hopf algebras of B.
Introduction
The ultimate goal of modular representation theory is to classify all finite-
dimensional modules over the group ring kG of a finite group, up to isomorphism.
To begin such a project, one needs an invariant of such a module. The most pow-
erful such invariant, introduced by Quillen almost thirty years ago [Qui71], is*
* the
cohomological variety of a module. Since then, cohomological varieties have been
used extensively to study restricted Lie algebras as well as finite groups. Rec*
*ent
work of Benson-Carlson-Rickard [BCR96 , BCR97 ] has determined precisely how
faithful an invariant the cohomological variety is: when G is a finite group an*
*d k is
an algebraically closed field, two finite-dimensional kG-modules have the same *
*co-
homological variety if and only if they belong to the same stably thick subcate*
*gory
of kG-modules. Here a full subcategory is stably thick if it is closed under re*
*tracts
and tensoring with any simple module, and also if two out of three modules in a
short exact sequence are in the subcategory, so is the third. As a consequence *
*of
this, Benson-Carlson-Rickard are able to classify all stably thick subcategorie*
*s of
finite-dimensional modules.
The object of this paper is to extend the theory of cohomological varieties t*
*o gen-
eral finite-dimensional cocommutative Hopf algebras. Naturally the theory becom*
*es
somewhat more complicated. The usual induction method from elementary abelian
subgroups is replaced by induction from quasi-elementary sub-Hopf algebras; the
precise statement of this induction is called the Quillen stratification theore*
*m. Un-
fortunately, very little is known about quasi-elementary Hopf algebras, so we c*
*an
go no further in general. However, in situations where the quasi-elementary sub-
Hopf algebras are well understood, it should be possible to continue and classi*
*fy
all stably thick subcategories of finite-dimensional modules. We carry out this*
* plan
for finite sub-Hopf algebras of the mod 2 Steenrod algebra, with scalars extend*
*ed
to the algebraic closure of F2.
____________
Date: January 4, 2000.
1991 Mathematics Subject Classification. 20C05, 20J05, 18E30, 18G35, 55P42.
1
2 MARK HOVEY AND JOHN H. PALMIERI
In more detail, suppose that B is a finite-dimensional cocommutative Hopf al-
gebra over a field k of characteristic p > 0. Given a finitely generated B-modu*
*le
M, define the ideal I = I(M) in Ext*B(k; k) to be the kernel of Ext*B(k; k) !
Ext*B(M; M). This ideal determines a subvariety of SpecExt *B(k; k), which we
write VB (M) and call the cohomological variety of M. As mentioned above, in
the cases when B is a group algebra or restricted enveloping algebra, these va-
rieties have been studied for some time, and they are known to have a number
of nice properties in these cases; for example, there is the "tensor product th*
*e-
orem", VB (M N) = VB (M) \ VB (N). Recently, in the case B = kG where
k is an algebraically closed field and G is a finite group, Benson, Carlson, and
Rickard [BCR96 ] have extended the definition of cohomological variety to infin*
*itely
generated modules, and they have proved that these new varieties satisfy many of
the same properties. The tensor product theorem for infinitely generated modules
is one of the most important, because it leads to structural information about *
*the
category of kG-modules, such as a classification of the stably thick subcategor*
*ies
of finitely generated kG-modules [BCR97 ].
Benson, Carlson, and Rickard prove the tensor product theorem in several step*
*s.
First, they show that the variety for any kG-module M can be recovered from the
varieties over kE of M restricted to E, for all elementary abelian subgroups E *
*of
G; they do this with analogues for these new varieties of Quillen and Avrunin-S*
*cott
stratification. In particular, they show that if one knows the tensor product p*
*rop-
erty in the elementary abelian case, then one can conclude it in general. Secon*
*d, in
the elementary abelian case, they develop the theory of "rank varieties" for in*
*fin-
itely generated modules; they prove the tensor product theorem for rank varieti*
*es,
and they show that rank varieties agree with cohomological varieties.
In our more general setting, we define varieties as they do; see Section 2. We
are able to prove that we always have an analogue of Quillen stratification, fo*
*r any
finite-dimensional cocommutative Hopf algebra B. The role of elementary abelian
groups is played by "quasi-elementary" Hopf algebras. When B is graded and
connected, we reduce the question of whether B satisfies a sort of Avrunin-Scott
stratification to whether the quasi-elementary sub-Hopf algebras of B do. Still*
* in
the graded connected case, we are able to reduce the question of whether the te*
*nsor
product theorem holds to the quasi-elementary case: we show that varieties over*
* B
satisfy the desired tensor product formula as long as the varieties over the qu*
*asi-
elementary sub-Hopf algebras of B satisfy both Avrunin-Scott stratification and
the tensor product formula.
Let A be the mod 2 Steenrod algebra. In the setting of sub-Hopf algebras of
A, quasi-elementary Hopf algebras are accessible: a sub-Hopf algebra Q of A is
quasi-elementary if and only if it is isomorphic, as an ungraded algebra, to the
mod 2 group algebra of an elementary abelian 2-group. Because of this, we can
apply the theory of rank varieties. There are two main distinctions between a q*
*uasi-
elementary sub-Hopf algebra Q of A and the group algebra of an elementary abeli*
*an
2-group: first, Q is graded, so one needs to understand the effect of that on v*
*arieties.
Second, the coproducts are not the same. This is important because the coproduct
is used to define the module structure on a tensor product, so knowing that rank
varieties for elementary abelian 2-groups satisfy the tensor product formula is*
* not
apparently relevant to these sub-Hopf algebras of A. It turns out that one can *
*get
around both of these problems; hence one can prove the tensor product theorem
for graded Q-modules over an algebraically closed field of characteristic 2. O*
*ne
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 3
can also verify Avrunin-Scott stratification for these Hopf algebras Q, hence e*
*very
finite-dimensional sub-Hopf algebra of A satisfies the tensor product formula. *
*As a
result, if B is a finite-dimensional sub-Hopf algebra of A, we have a classific*
*ation of
stably thick subcategories of finitely generated B-modules, a complete descript*
*ion of
the Bousfield lattice in the category of all B-modules, and a proof of the tele*
*scope
conjecture in the category of B-modules. Note that the proof relies on the case
for elementary abelian 2-groups, and in particular requires us to work over an
algebraically closed field, rather than the field F2.
For example, we have the following result. If B is a finite-dimensional sub-H*
*opf
algebra of_the mod 2 Steenrod algebra, then after extending scalars by the alge*
*braic
closure F2 of F2, the stably thick subcategories of finitely generated B-module*
*s are
in one-to-one_correspondence_with sets T of bihomogeneous prime ideals of the r*
*ing
Ext**B(F 2; F2) which are closed under specialization: if p 2 T and p q, then
q 2 T . See Corollaries 3.7 and 8.6 for more details. The restriction that the *
*field
be algebraically closed has recently been removed by the authors [HP99 ], both *
*in
this case and in the group algebra case.
We point out that, when dealing with stably thick subcategories, it is most
convenient to work in an appropriate triangulated category of B-modules. Two
natural choices are the stable module category StMod (B) and the category S(B)
of unbounded chain complexes of projective B-modules. See Section 1 for descrip-
tions of these categories, as well as the relationship between thick subcategor*
*ies of
StMod (B) and stably thick subcategories of the abelian category of B-modules.
Many of the results of this paper are related to results in [NP98 ]. In that *
*paper,
though, the focus was on finitely generated modules, and in this paper, we need
to study varieties for arbitrary modules. Also, the results in [NP98 ] were giv*
*en in
terms of homogeneous prime ideals of Ext**B(k; k). For our purposes, we need to*
* use
bihomogeneous prime ideals, and this turns out to be a not insignificant differ*
*ence.
The paper is organized as follows. We start in Section 1 by recalling some ba*
*sic
facts and constructions regarding Hopf algebras and their modules; we also defi*
*ne
the module categories with which we work for the remainder of the paper. We
want to use many tools from stable homotopy theory here, so in the second secti*
*on,
we recall enough axiomatic stable homotopy theory for our needs. We also define
cohomological varieties; they are defined in terms of certain "idempotent" modu*
*les.
In the third section, we describe the tensor product property, and show that if
it holds, it leads to various important structural information about the module
categories under consideration, such as classifications of thick subcategories *
*and
Bousfield classes; Corollary 3.7 is one of the main results. In Section 4, we s*
*how
how restriction to sub-Hopf algebras behaves on the idempotent modules used to
define varieties. In Section 5, we prove the Quillen stratification theorem and*
* discuss
Avrunin-Scott stratification; we also relate these to the tensor product proper*
*ty_
see Corollary 5.10, for example. In the sixth section, we discuss rank varietie*
*s and
use them to prove the tensor product property in some special cases. One can vi*
*ew
rank varieties as ad hoc tools for computing cohomological varieties; in Sectio*
*n 7 we
present an outline of a general proof of the tensor product theorem, in an atte*
*mpt
to explain why rank varieties are easier to use in this setting than cohomologi*
*cal
varieties. In the last section, we apply the theory to graded connected Hopf al*
*gebras,
and in particular sub-Hopf algebras of the mod 2 Steenrod algebra.
The authors would like to thank Gilles Gnacadja for helpful discussions about
group cohomology and Chris Bendel for helpful discussions about rank varieties *
*and
4 MARK HOVEY AND JOHN H. PALMIERI
the tensor product theorem. Also, this paper arose from trying to understand the
work of Benson, Carlson, and Rickard on varieties for infinitely generated modu*
*les,
so we owe them our thanks both for their papers and for many discussions on the
subject.
1. Recollections on Hopf algebras, modules, and categories
Convention 1.1. Unless otherwise indicated, all Hopf algebras in this paper are
assumed to be graded, cocommutative, and defined and finite-dimensional over a
field k; the characteristic of k will be nonzero throughout. "Module" means "le*
*ft
module" throughout. All modules are graded modules, except in Section 6. The
spectrum SpecR of a bigraded ring will always denote the space of bihomogeneous
prime ideals in the Zariski topology, again except in Section 6, when we will c*
*onsider
primes which are only homogeneous with respect to one grading.
In particular, all Hopf algebras under consideration are self-injective_see [*
*LS69,
p. 85]. Hence projective and injective graded B-modules coincide. (Note also th*
*at
we view a group algebra as being graded trivially: kG is concentrated in degree
zero.)
In addition, because our Hopf algebras are cocommutative, the tensor product
over k makes the category of B-modules into a symmetric monoidal category, with
unit k.
We work with two categories in this paper. First, the stable module category
StMod (B) is defined as follows. Its objects are B-modules, and the morphisms
from M to N are written Hom__B(M; N) and are defined to be
Hom__B(M; N) = Hom B(M; N)= ';
where maps f and g are stably equivalent, written f ' g, if f - g factors throu*
*gh a
projective module. Because projectives and injectives are the same, this catego*
*ry
is well-behaved; for example, the stable module category is a triangulated cate-
gory [Mar83 , Chapter 14]. This means, in particular, that there is a suspensi*
*on
functor : StMod (B) -! StMod(B), where M is the cokernel of an embedding
of M into an injective module. We refer to this as "external" suspension. The
functor is an equivalence of categories. We can also suspend "internally", by
changing the grading of M. We then have functors i;j:StMod (B) -!StMod (B),
where the first grading is the external suspension and the second grading is the
internal suspension. Furthermore, Hom__B(M; i;jN) ~=Exti;jB(M; N) when i > 0.
The tensor product descends to make StMod (B) a symmetric monoidal category
with unit k.
The second category is written S(B), and it was defined in [HPS97 , Section 9*
*.5].
The objects of this category are unbounded chain complexes of projective B-
modules, and the morphisms of S(B) are chain homotopy classes of chain maps.
S(B) is also triangulated: the functor i;jshifts the chain complex degree up by
i and the internal degree up by j, with the usual sign conventions. Writing [X;*
* Y ]
for chain homotopy classes from X to Y , if X and Y are projective resolutions
of modules M and N, respectively, then we have [X; i;jY ] ~=Exti;jB(M; N). The
usual tensor product of chain complexes makes S(B) into a symmetric monoidal
category; the unit S is any projective resolution of k.
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 5
Whichever category we are working in, we tend to abuse notation and write "="
for isomorphism in that category; for instance, when working in StMod (B), P = 0
for any projective module P .
We are concerned with certain sorts of subcategories of these. To define one *
*of
these, we need to recallLthat an object`F of a triangulated category C is calle*
*d small
if the natural map iC(F; Xi) -!C(F; iXi) is an isomorphism for all sets {Xi}
with a copoduct in C.
Definition 1.2.A full subcategory of a (bigraded) triangulated category is call*
*ed
thick if it is (bigraded and) triangulated and closed under retracts. A thick s*
*ub-
category of a symmetric monoidal triangulated category is called tensor-closed *
*if
it is closed under the symmetric monoidal product with any small object. A thick
subcategory is called localizing if it is closed under coproducts as well.
In particular, from the stable homotopy theory perspective, it is natural to *
*ask
for a classification of the (tensor-closed) thick subcategories of small object*
*s in ei-
ther of the above categories: thick subcategories of finitely generated modules*
* in
StMod (B), or thick subcategories of bounded below chain complexes whose ho-
mology is finite-dimensional in S(B). We point out that this is equivalent to a
classification question in the abelian category of B-modules.
Definition 1.3.A full subcategory C of B-Mod is called stably thick if
(a) C is closed under summands.
(b) If 0 ! M1 ! M2 ! M3 ! 0 is a short exact sequence and two of M1, M2,
and M3 are in C, then so is the third.
(c) If M is in C, then S M is in C for every simple module S.
The empty category and the category containing only the object 0 are always sta*
*bly
thick, and we refer to them as trivial stably thick subcategories.
By induction on composition factors, if M is in C, then so is M N for any
finite-dimensional module N. In particular, M B is in C. But M B is a free
module; hence every non-trivial stably thick subcategory of B-Mod contains all
projective modules.
Note also that for many of the examples of this paper, for example when B
is graded connected, the trivial module k is the only simple module, so the thi*
*rd
condition is automatically satisfied.
It may be of interest to consider subcategories that only satisfy the first t*
*wo
conditions in Definition 1.3, but we know nothing about these in general.
There is a functor j :B-Mod ! StMod(B) which is the identity on objects, and
takes each map f to its stable equivalence class. It provides a link between t*
*he
stably thick subcategories of B-Mod and the thick subcategories of StMod (B).
Proposition 1.4.The functor j :B-Mod ! StMod (B) induces a bijection be-
tween nontrivial stably thick subcategories of finitely generated modules in B-*
*Mod
and nonempty tensor-closed thick subcategories of finitely generated modules in
StMod (B).
2. Noetherian stable homotopy theory
In this section, we recall some axiomatic stable homotopy theory from [HPS97 ]
and apply it to our situation.
6 MARK HOVEY AND JOHN H. PALMIERI
The categories C we consider are all bigraded in the following sense:0there0a*
*re
functors i;j:C -! C for pairs of integers0(i;0j), such that i;jO i ;jis natural*
*ly,
and coherently, isomorphic to i+i ;j+j, and 0;0is naturally isomorphic to the
identity. We denote C(X; i;jY ) by [X; Y ]i;j, and the symbol [X; Y ]** will st*
*and
for the entire bigraded set of morphisms.
Usually in stable homotopy theory, one writes [X; Y ]ifor C(iX; Y ) rather th*
*an
C(X; iY ) = C(-iX; Y ). We have it graded "backwards" because our homotopy
groups are essentially Ext, in which the boundary homomorphism raises degrees.
Because of this difference between Ext and ordinary homotopy groups, some part
of the grading has to look non-standard to topologists, and we have chosen to m*
*ake
it this part.
The following definition is based on [HPS97 , Definition 6.0.1], but we give *
*a more
general definition.
Definition 2.1.A bigraded Noetherian stable homotopy category is a bigraded cat-
egory C which has all products and coproducts, together with a triangulation wi*
*th
1;0as the suspension functor, and a closed symmetric monoidal structure com-
patible with the triangulation. In keeping with usual stable homotopy theoretic
notation, we write X _ Y for the coproduct of X and Y and call it the "wedge"
of X and Y ; we write X ^ Y for the symmetric monoidal structure, and call it t*
*he
"smash product". We denote the unit of the symmetric monoidal structure by S.
We also require that there be a finite set G of objects of C such that S and G *
*satisfy
the following properties
(a) S and every element of G are small. That is, the natural map
M _
[A; Xi] -![A; Xi]
i i
is an isomorphism for all A 2 G, or for A = S, and all sets {Xi} of object*
*s of
C.
(b) G is a set of bigraded weak generators. That is, X is isomorphic to the 0
object of C if and only if [A; X]**= 0 for all A 2 G.
(c) The ring [S; S]** is commutative and Noetherian as a bigraded ring. For any
objects Y and Z of C, [Y; Z]**is a module over [S; S]**via the smash produ*
*ct;
when Y and Z are small, we require that [Y; Z]** be a finitely generated
module over [S; S]**.
When we can take G = {S}, we say that C is monogenic. Monogenic Noetherian
stable homotopy categories are studied in [HPS97 , Section 6]. Straightforward *
*ana-
logues of the results proved there hold in any Noetherian stable homotopy categ*
*ory,
with only trivial modifications to the statements and proofs. In general, the w*
*ord
"thick" in [HPS97 , Section 6] needs to be replaced by "tensor-closedLthick" wh*
*erever
it appears, and the expression "ss*X" needs to be replaced by " A2G [A; X]*.
If C is a bigraded Noetherian stable homotopy category, we usually denote
[S; X]** by ss**X, in analogy with stable homotopy theory. A small object of
C, as in part (a) of Definition 2.1, is also called finite.
Example 2.2. Let B be a finite-dimensional graded cocommutative Hopf algebra
over a field k. We followPthe usual convention of denoting the conjugation by O*
* and
the diagonal by (b) = b0b00. Then the stable module category StMod(B) is al-
most a bigraded Noetherian stable homotopy category. Indeed, as mentioned above,
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 7
StMod (B) is triangulated. The monoidal structure takes (M; N) to M kN, given
the diagonal B-action. This is symmetric monoidal since B is cocommutative.PThe
closed structure takes (M; N) to Hom k(M; N), where (bf)(m) = b0f(O(b00)m).
The generating set G is the set of simple modules (or, more accurately, a set c*
*on-
taining one element of each isomorphism class of simple modules). Every simple
module is finite-dimensional, so is small. There are only finitely many of them*
*, since
every simple module occurs as a composition factor in B. The finite objects are
the modules which are stably equivalent to finitely generated modules. However,
[k; k]**is isomorphic to the Tate cohomology of B, and this may not be Noetheri*
*an.
For this and other reasons, we prefer to work with the bigraded Noetherian
stable homotopy category S(B), described in Section 1. The triangulated structu*
*re
comes from short exact sequences of chain complexes, and the symmetric monoidal
structure is the usual tensor product (over k) of chain complexes. The unit S i*
*s a
projective resolution of k. A generating set G can be obtained by taking projec*
*tive
resolutions of the simple modules. In S(B) we have [S; S]** ~=Ext**B(k; k), whi*
*ch
is a bigraded commutative Noetherian ring (see [FS97 ] and [Wil81]). One can al*
*so
find in [FS97 ] a proof that Ext**B(M; N) is finitely generated over [S; S]** f*
*or any
finite-dimensional modules M and N, so in particular for M; N 2 G. It follows by
thick subcategory arguments that [X; Y ]** is finitely generated over [S; S]** *
*for all
finite X and Y .
One good way of studying stable homotopy categories is via "Bousfield localiz*
*a-
tion"; this is also a good way of producing new categories from old ones. For a
general discussion of localization, see [HPS97 , Chapter 3]. In this paper, we *
*will
use one kind of Bousfield localization, called "finite localization".
Theorem 2.3 (Theorem 3.3.3 in [HPS97 ]).Let C be a bigraded Noetherian stable
homotopy category, and let A be a tensor-closed thick subcategory of finite obj*
*ects
of C. There is a functor LA :C ! C with the following properties:
(a) LA is exact.
(b) There is a natural transformation 1 -!LA .
(c) LA is idempotent_for any X, the map LA X -! LA LA X induced by the
natural transformation in (b) is an equivalence.
(d) For any X, LA X = X ^ LA S. Hence by idempotence, LA S ^ LA S = LA S.
(e) For any finite X, LA X = 0 if and only if X 2 A. For any X, LA X = 0 if
and only if X is in the localizing subcategory generated by A.
Definition 2.4.In the situation of the theorem, LA is called finite localization
away from A, and A is the category of finite acyclics for LA . For each X, defi*
*ne
CA X by the exact triangle
CA X -! X -! LA X:
Then CA is a functor; it is called the acyclization functor associated to A and*
* LA .
If LA is a finite localization functor on a category C, one calls the full su*
*bcategory
consisting of objects {LA X | X 2 C} the "finite localization of C away from A".
The inclusion of that subcategory into C is right adjoint to LA .
Example 2.5. We will normally work with S(B) rather than StMod (B), as it has
better formal properties. In addition, StMod (B) is equivalent to the full subc*
*ate-
gory of S(B) consisting of chain complexes with no homology [HPS97 , Section 9.*
*6].
This subcategory is the finite localization of S(B) away from the tensor-closed*
* thick
8 MARK HOVEY AND JOHN H. PALMIERI
subcategory generated by the chain complex B concentrated in degree 0. We can
thus recover results about StMod (B) from results about S(B), via finite locali*
*za-
tion.
Now suppose that C is a bigraded Noetherian stable homotopy category, and
suppose that p is a bihomogeneous prime ideal in ss**S. Then there is a finite *
*local-
ization functor Lp on C, which is called "p-localization" and is described in [*
*HPS97 ,
Proposition 6.0.7]. We often denote LpX = Xp. For our purposes here, its key
properties are the following:
o ss**(Xp) = (ss**X)p.
o Xp = X ^ Sp, and Sp ^ Sp = Sp (since Lp is a finite localization functor).
o Xp = 0 if and only if Xq = 0 for all q p.
There is another finite localization functor L of an object X in a stable homotopy
category is the collection of all Y such that X ^ Y = 0. We order Bousfield cla*
*sses
by reverse inclusion, so that <0> is the least Bousfield class and the unit ~~*
* of the
smash product is the greatest.
The partially ordered set of Bousfield classes carries a lot of information a*
*bout
structure of the category C, so one would like to understand as much as possible
about it.
In any bigraded Noetherian stable homotopy category C, we have the following
equality of Bousfield classes, from Theorem 6.1.9 of [HPS97 ]:
_
~~~~ = < Mp>;
p
as p runs through the bihomogeneous prime ideals of ss**S. In other words, an
object X is isomorphic to zero if and only if Mp ^ X = 0 for all primes p. We a*
*lso
note that if p 6= q, then Mp ^ Mq = 0, by [HPS97 , Proposition 6.1.7].
Hence if each Mp were a skew field object (a ring object such that every modu*
*le
is free), and G = {S}, then the tensor product property would follow_see [HPS97*
* ,
Section 3.7]. The Mp themselves are not even ring objects, however. They do
have the same Bousfield class (see below) as a ring object, as explained in [HP*
*S97 ,
Section 6], but this ring is almost never a skew field.
Corollary 3.3.Suppose C is a bigraded Noetherian stable homotopy category with
the tensor product property. Then the Bousfield class is minimal in C.
Proof.Suppose < . Then there is an object Y such that E ^ Y = 0 but
Mp^ Y is nonzero. Since Mp^ E ^ Y = 0, the tensor product property implies that
Mp ^ E = 0. On the other hand, if q 6= p, then Mp ^ Mq = 0; hence E ^ Mq = 0 *
* __
for all q 2 Specss**S. But then Theorem 6.1.9 of [HPS97 ] implies that E = 0. *
* |__|
Corollary 3.4.Suppose C is a bigraded Noetherian stable homotopy category with
the tensor product property. Then the Bousfield lattice of C is isomorphic to *
*the
complete Boolean algebra on the atoms Mp.
This is in contrast with the Bousfield lattice in ordinary stable homotopy th*
*eory,
which is much more complicated. See [HP98 ] for more information.
Proof.Again using Theorem 6.1.9 of [HPS97 ], we find that
_
= :
p
By Corollary 3.3, is either <0> or . Therefore, every Bousfield cla*
*ss is
a wedge of various of the . Any two such wedges are distinct, since_Mp^Mq_=*
* 0
when p 6= q. |__|
Corollary 3.5.Suppose C is a bigraded Noetherian stable homotopy category with
the tensor product property. Then there is an isomorphism of partially ordered *
*sets
between nonempty tensor-closed thick subcategories of finite objects in C and s*
*ubsets
of Specss**SC that are closed under specialization.
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 11
This corollary follows immediately from [HPS97 , Theorem 6.2.3] and [HPS97 ,
Proposition 9.6.8]. TheSisomorphism in question takes a tensor-closed thick su*
*b-
category D to V(D) = X2D V(X); the inverse takes a subset V closed under spe-
cialization to the thick subcategory consisting of all finite X such that V(X) *
* V .
Corollary 3.6.Suppose C is a bigraded Noetherian stable homotopy category with
the tensor product property. Then the telescope conjecture holds for C. That *
*is,
every smashing localization is a finite localization.
This corollary follows immediately from [HPS97 , Theorem 6.3.7].
In general, one hopes that the Bousfield lattice is (anti-)isomorphic to the *
*lat-
tice of tensor-closed localizing subcategories (tensor-closed thick subcategori*
*es also
closed under coproducts); see [HPS97 , Corollary 6.3.4]. Every Bousfield class *
*is a
tensor-closed localizing subcategory, but it is not known that every localizing*
* sub-
category arises in this way. In order to prove this, it would suffice to show *
*that
the tensor-closed localizing subcategory generated by Mp is minimal. We do not
know whether this holds, even in the cases of S(kG) and StMod (kG) for G a fini*
*te
p-group.
For the reader's convenience, we restate these results in terms of Hopf algeb*
*ras.
Corollary 3.7.Suppose that B is a finite-dimensional graded cocommutative Hopf
algebra over a field k. Assume that B has the tensor product property.
(a) The Bousfield lattice of S(B) is isomorphic to the complete Boolean algebra
on the atoms Mp.
(b) The Bousfield lattice of StMod (B) is isomorphic to the complete Boolean a*
*l-
gebra on the atoms Mp for p not equal to the unique maximal ideal.
(c) There is a poset isomorphism between nonempty tensor-closed thick subcate-
gories of finite objects in S(B) and subsets of SpecExt**B(k; k) that are *
*closed
under specialization.
(d) There is a poset isomorphism between nonempty tensor-closed thick subcat-
egories of finitely generated modules in StMod (B) and nonempty subsets of
SpecExt**B(k; k) that are closed under specialization.
(e) The telescope conjecture holds for S(B) and StMod (B): every smashing lo-
calization is a finite localization.
Proof.Most of this is immediate from the Corollaries 3.3-3.6. Part (b) follows *
*from
the fact that B ^ Mq = 0 for all non-maximal q, for dimensional reasons. Part (*
*d)_
follows from part (c) and [HPS97 , Proposition 9.6.8]. |*
*__|
We can be a bit more precise about certain aspects of the bijections; for exa*
*mple,
in parts (c) and (d), we have the following: the empty subset of SpecExt**B(k; *
*k)
corresponds to the thick subcategory of S(B) generated by the zero chain comple*
*x;
the subset consisting only of the unique maximal ideal corresponds to the thick
subcategory (both of S(B) and StMod (B)) generated by the free module B; the
subset consisting of all of Spec corresponds to the thick subcategory of all fi*
*nite
objects.
We will combine this with Corollary 5.10 and Corollary 6.13 to examine some
specific examples in Section 8.
12 MARK HOVEY AND JOHN H. PALMIERI
4. Certain finite localizations
Our goal for the next few sections is to reduce the question of whether a Hopf
algebra B has the tensor product property to whether certain of its sub-Hopf al-
gebras have it. In order to understand this, we need to understand how varieties
over B behave under restriction to sub-Hopf algebras_this is the content of the
Quillen stratification theorem, Theorem 5.2 below. To understand the restrictio*
*ns
of varieties, we must understand the restrictions of the objects Mp. This secti*
*on
is devoted to this problem. The results are similar to those of [BCR96 , Sectio*
*n 8];
the main results are Theorem 4.11 and its corollary, Corollary 4.12.
We point out that it appears to be more difficult to understand how the Mp
behave under induction. This difficulty is related to the difficulty in proving*
* the
tensor product property in general, as we explain in Section 7.
Stated rather generally, our goal here is to understand what happens to finite
localizations, and hence also to the Mp, under well-behaved functors of Noether*
*ian
stable homotopy categories. So first we discuss what sorts of functors to consi*
*der.
Definition 4.1.Recall from [HPS97 , Definition 3.4.1] that a functor F :C -!
D between stable homotopy categories is called a geometric morphism if F is a
left adjoint and preserves exact triangles, the smash product, and the unit (up*
* to
coherent natural isomorphism). A geometric morphism F is finitary if its right
adjoint G is also left adjoint to F and sends finite objects of D to finite obj*
*ects of
C.
Example 4.2. For example, if Q is a sub-Hopf algebra of B, the restriction func-
tor res:S(B) -! S(Q), applied dimension-wise to a chain complex, is a finitary
geometric morphism. The right adjoint of restriction is coinduction; it takes a*
* chain
complex X of projective Q-modules to the chain complex Hom (B; X) of B-modules.
Since the Hopf algebras we are considering are finite-dimensional and self-inje*
*ctive,
hence self-dual, coinduction actually coincides with the induction functor ind,*
* which
takes X to B Q X. So induction is both the left and right adjoint of restrictio*
*n;
since Q has finite index in B, induction will preserve finite objects.
Note that a geometric morphism F induces a map of rings F :ss**SC -!ss**SD ,
and so also a map F *:Spec ss**SD -! Spec ss**SC. If C is Noetherian and F is
finitary, then
ss**SD ~=[GSD ; SC];
which is a finitely generated ss**SC-module by the definition of a Noetherian s*
*table
homotopy category, Definition 2.1. In particular, F *is a closed mapping in th*
*is
case, by the going-up theorem [AM69 , Theorem 5.11].
We now discuss how geometric morphisms interact with support varieties; our
immediate goal is Corollary 4.7. We use the alternative definition of the suppo*
*rt
for finite objects given in Lemma 2.7.
Lemma 4.3. Suppose F :C -! D is a geometric morphism of bigraded Noether-
ian stable homotopy categories, and X is a finite object of C. Then VD (F X)
(F *)-1VC(X).
Proof.Suppose a 2 I(X). Then the self-map a ^ 1 of X is nilpotent, so the self-
map F a ^ 1 = F (a ^ 1) of F X is nilpotent. Thus F (I(X)) I(F X). Now suppose
q 2 VD (F X). Then q I(F X) F (I(X)), by Lemma 2.7. Hence F *(q) I(X),__
so F *(q) 2 VC(X), by Lemma 2.7 again. |__|
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 13
We now determine precisely the support variety of GY when F is a finitary
geometric morphism with two-sided adjoint G. This is a special case of the foll*
*owing
more general lemma. To state the lemma, let us denote the support of a module
M over a ring A by VA (M).
Lemma 4.4. Suppose f :A -! B is an integral homomorphism of commutative
rings, and M is a finitely generated B-module. Then VA (M) = f*VB (M), where
f* :Spec B -! SpecA is the induced map.
Proof.For any finitely generated A-module N, we have VA (N) = V (annA N).
Hence
VA (M) = V (annA N) = V (f-1 annB M):
Similarly, f*VB (M) = f*V (annB M). By definition, any prime in f*V (annB M)
contains f-1 annB M, so is in V (f-1 annB M). On the otherphand,_if p is a
prime ideal containing f-1 annB M, then it contains f-1 annB M , so must con-
tain f-1 q for some prime ideal q containing annB M. Hence p is in the closure_*
*of_
f*V (annB M), but f* is a closed map since f is integral. |_*
*_|
Corollary 4.5.Suppose F :C -! D is a finitary geometric morphism of bigraded
Noetherian stable homotopy categories, with left adjoint G. Then for all finite
objects Y of D, we have VC(GY ) F *VD (Y ). Equality holds if C is monogenic.
Proof.We have
[ [
VC(GY ) = Vss**SC([A; GY ]**) = Vss**SC([F A; Y ]**);
A2GC A2GC
since G is also the right adjoint of F . The lemma implies that
Vss**SC([F A; Y ]**) = F *Vss**SD([F A; Y ]**):
Since VD (Y ) is the union of the Vss**SD([B; Y ]**) for all finite B, we find *
*that
VC(GY ) F *VD (Y ), as required. If C is monogenic, then we need only consider_
A = S, so equality holds. |__|
We can now prove the desired result on the behavior of finite localizations u*
*nder
certain geometric morphisms.
Theorem 4.6. Suppose F :C -! D is a finitary geometric morphism of bigrad-
ed Noetherian stable homotopy categories. Let T Specss**SC be closed under
specialization. Then F takes the exact triangle
CTS -! S -! LTS
to the exact triangle
C(F*)-1TS -! S -! L(F*)-1TS:
For any bigraded Noetherian stable homotopy category C and any subset T of
Specss**SC closed under specialization, let AT denote the thick subcategory of *
*all
finite X 2 C such that V(X) T , as in [HPS97 , Section 6.2].
Proof.Recall that CTS is in the localizing subcategory generated by AT. Lem-
ma 4.3 implies that F takes AT to A(F*)-1T. Therefore F (CTS) is in the localiz*
*ing
subcategory generated by A(F*)-1T. To complete the proof, it suffices to show
that F (LTS) is local with respect to L(F*)-1T. But if Y 2 A(F*)-1T, we have
14 MARK HOVEY AND JOHN H. PALMIERI
D(Y; F (LTS))* ~= C(GY; LTS)*, where G denotes the left adjoint of F . Corol-
lary 4.5 implies that VC(GY ) F *VD (Y ) T , and so C(GY; LTS)* = 0, as_
required. |__|
This theorem tells us something about what a geometric morphism does to each
Mp. We need some more terminology.
Given a bihomogeneous prime ideal p of ss**S, we have seen how to form Mp as
the fiber of a map of finite localizations. In fact, given an arbitrary bihomog*
*eneous
ideal a in ss**S, we can form Ma in a somewhat analogous fashion. Let pi, for
i = 1 to k, denote the minimal prime ideals associated to a, so that the V (pi)*
* are
the irreducible components of the closed subvariety V (a). Define T (a) to be *
*the
set of all prime ideals p not contained in any pi. Define T 0(a) to be the set *
*of all
prime ideals not properly contained in any of the pi, so that T 0(a) contains t*
*he pi
themselves. Both T (a) and T 0(a) are closed under specialization, so give ris*
*e to
thick subcategories AT(a)and AT0(a)of finite objects: AT(a)is the collection of
all finite X 2 C such that V(X) T (a), and similarly for T 0(a). There are then
associated finite localization functors LT(a)and LT0(a). We define Ma to be the
fiber of the map LT(a)S -! LT0(a)S. Equivalently, Ma = CT0(a)S ^ LT(a)S, where
CT0(a)is the corresponding acyclization functor.
Corollary 4.7.Suppose F :C -! D is a finitary geometric morphism of bigraded
Noetherian stable homotopy categories, and p is a prime ideal in ss**SC. Define*
* a
to be the intersection of all prime ideals qi of ss**SD such that F *(qi) = p. *
*Then
the V (qi) are the irreducible components of V (a), and F Mp = Ma.
Proof.In view of Theorem 4.6, it suffices to identify T (a) with (F *)-1T (p) a*
*nd
T 0(a) with (F *)-1T 0(p). It is easy to check that q 62 T (a) implies that q *
*62
(F *)-1T (p). Therefore (F *)-1T (p) T (a), and similarly (F *)-1T 0(p) T 0(a*
*).
The rest of the corollary follows from the going-up theorem. In particular, t*
*he
going-up theorem implies that any two ideals q1 and q2 of ss**SD with F *(q1) =
F *(q2) must be incomparable under the containment relation, so the V (qi) are *
*the
irreducible components of V (a).
Now suppose q 62 (F *)-1T (p), so that F *(q) p. Then the going-up theo-
rem guarantees that there is a prime ideal qi such that q qi and F *(qi) = p.
Hence q 62 T (a). Thus T (a) = (F *)-1T (p), and the same argument shows_T_0(a)*
* =
(F *)-1T 0(p). |__|
So, if F :C ! D is a finitary geometric morphism between bigraded Noetherian
stable homotopy categories, to understand F Mp, we want to understand the objec*
*ts
Ma in terms of the Mq, for prime ideals q in ss**SD . Our goal here is Corollar*
*y 4.10.
We start by computing the Bousfield class of a finite localization.
Lemma 4.8. Suppose C is a bigraded Noetherian stable homotopy category, and
T Specss**S is closed under specialization. Let LT denote the associated finite
localization functor, and CT the associated acyclization functor. Then
_ _
= ; = :
p62T p2T
Proof.Combine [HPS97 , Theorem 6.1.9] with [HPS97 , Lemma 6.3.1]. |___|
So, for example, if p 62 T , then Mp is LT-local. We now prove an arithmetic
square proposition for finite localizations.
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 15
Proposition 4.9.Suppose C is a bigraded Noetherian stable homotopy category,
and T; U Spec ss**S are closed under specialization. Then there is a cofiber
sequence
LT\U S -! LTS _ LU S -! LT[U S
where the second map is the difference of the two evident maps.
Proof.Let Y denote the cofiber of the map LT\U S -! LTS _ LU S. Then there is a
map Y -! LT[U S, induced by the difference of the two maps LTS -! LT[U S and
LU S -! LT[U S. We claim that this map is an isomorphism.
To prove this, we first show that Y is LT[U -local. Since LT[U is a finite, a*
*nd
hence smashing, localization, it suffices to show that CT[U S ^ Y = 0, where CT*
*[U
is the associated acyclization functor. By Lemma 4.8, it suffices to show that
Mp ^ Y = 0 for all p 2 T [ U. Now, if p 2 T \ U, then, again using Lemma 4.8,
LT\U Mp = LTMp = LU Mp = 0, and so Mp ^ Y = 0. On the other hand, if p is
in exactly one of T and U, say T without loss of generality, then LTMp = 0 but
LT\U Mp = LU Mp = Mp. Hence we still have Mp ^ Y = 0, as required. Therefore
Y is LT[U -local.
Now, apply LT[U to the cofiber sequence defining Y . This is a further local-
ization, so we find that Y is the cofiber of the diagonal map LT[U S -! LT[U S_*
*__
LT[U S. This cofiber is clearly LT[U S, as required. |_*
*_|
Corollary 4.10.Suppose C is a bigraded Noetherian stable homotopy category, and
a is a bihomogeneous ideal in ss**S~with associated minimal prime ideals p1; : *
*:;:pk.
Then there is an isomorphism Ma =-!Mp1_ . ._.Mpk.
Proof.Let b = p2 \ . .\.pk. By induction, we know that Mb = Mp2 _ . ._.
~=
Mpk; we want to show that Ma -! Mp1_ Mb. Since T (a) = T (p1) \ T (b), then
Proposition 4.9 gives a cofiber sequence
LT(a)S -! LT(p1)S _ LT(b)S -! LT(p1)[T(b)S:
Now we smash this cofiber sequence with CT0(a)S. Lemma 4.8 implies that
_
= :
p2T0(a)
But if p 2 T 0(a), there are two possibilities. Suppose first that p 6= p1. The*
*n p is
not contained in p1, and so p 2 T (p1). Thus Mp ^ LT(p1)[T(b)S = 0. On the other
hand, if p = p1, then p is not contained in b, and so the same argument shows t*
*hat
Mp ^ LT(p1)[T(b)S = 0.
Thus we have an isomorphism
Ma -!(CT0(a)S ^ LT(p1)S) _ (CT0(a)S ^ LT(b)S):
We claim that CT0(a)S ^ LT(p1)S = Mp1 and CT0(a)S ^ LT(b)S = Mb. To see this,
smash the cofiber sequence
CT0(a)S -! S -! LT0(a)S
with LT(p1)S. It suffices to show that LT0(a)S ^ LT(p1)S = LT0(p1)S. But another
Bousfield class computation shows that LT0(a)S ^ Mp1= 0. Thus
LT0(a)S ^ LT(p1)S = LT0(a)S ^ LT0(p1)S = LT0(p1)S;
as required. |___|
16 MARK HOVEY AND JOHN H. PALMIERI
Combining Corollary 4.7 and Corollary 4.10, we obtain the following theorem,
which is our main result for this section.
Theorem 4.11. Suppose that F :C -! D is a finitary geometric morphism of
bigraded Noetherian stable homotopy categories, and let p be a prime ideal in s*
*s**SC.
Then
_
F Mp = Mq:
q | F*q=p
Corollary 4.12.In particular, suppose that B is a finite-dimensional graded co-
commutative Hopf algebra over a field k. If Q is a sub-Hopf algebra of B and
p 2 SpecExt**B(k; k), then
_
resMp ~= Mq:
q | res*(q)=p
We point out that when B and Q are connected, then the function res*is one-
to-one by Lemma 8.1, so resMp is either a single Mq or zero, depending on wheth*
*er
p is in the image of res*or not.
5. Quillen stratification
In this section, we apply Theorem 4.11 to compute support varieties in a Noe-
therian stable homotopy category C in terms of geometric morphisms out of C. The
result is Theorem 5.2, an analogue of the Quillen stratification theorem. Then *
*we
try to strengthen this result to get "Avrunin-Scott stratification". We show th*
*at
Avrunin-Scott stratification lets us reduce the question of whether C has the t*
*ensor
product property to whether other, perhaps simpler, stable homotopy categories
Cffhave it. In the Hopf algebra setting, we show that Avrunin-Scott stratificat*
*ion
for certain sub-Hopf algebras of B (the "quasi-elementary" ones) implies it for*
* B
itself, and we conclude that if the quasi-elementary sub-Hopf algebras of B have
the tensor product property, so does B.
This first theorem says that if we can detect whether objects are nonzero by
applying certain geometric morphisms, then we can describe the support for any
object in terms of those geometric morphisms. This is a generalization of [BCR9*
*6 ,
Theorem 10.6].
Definition 5.1.Let C be a bigraded Noetherian stable homotopy category. A
set {Fff:C ! Cff} of finitary geometric morphisms of bigraded Noetherian stable
homotopy categories is said to exhaust C if, for all objects X of C, X = 0 if a*
*nd
only if FffX = 0 for all ff.
Theorem 5.2 (Quillen stratification).Let C be a bigraded Noetherian stable ho-
motopy category, and let {Fff:C ! Cff} be a set of finitary geometric morphisms
of bigraded Noetherian stable homotopy categories that exhaust C. Then
[
VC(X) = Ff*f(VCff(FffX)):
ff
Proof.Suppose p 2 VC(X), so that Mp ^ X 6= 0. Then for some ff,
FffMp ^ FffX = Fff(Mp ^ X) 6= 0:
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 17
By Theorem 4.11,
_
Fff(Mp) = Mq:
q | F*ff(q)=p
Writing Sfffor the sphere object in Cff, there must be a q in Specss**Sffsuch t*
*hat
Mq ^ FffX 6= 0.
Conversely, if Mq^ FffX 6= 0 for some q ss**Sff, then Theorem 4.11 shows_that
Fff(Mp ^ X) 6= 0, and so Mp ^ X 6= 0. |__|
In order to apply this to Hopf algebras, we need to recall the analog of Chou*
*inard's
theorem from [Pal97]. The definition of a quasi-elementary Hopf algebra can be
found in [Pal97]; these Hopf algebras are the replacements for elementary abeli*
*an
subgroups of p-groups. The definition itself is quite complicated, so we do not
reproduce it here. We understand quasi-elementary Hopf algebras well in certain
special cases; for example, a group algebra kG over a field of characteristic p*
* is
quasi-elementary if and only if G is an elementary abelian p-group. We discuss
more examples in Section 8.
Theorem 5.3 (Chouinard's theorem).Let B be a finite-dimensional graded co-
commutative Hopf algebra over a field k. Fix an object X 2 S(B). Then X 6=
0 if and only if there is a quasi-elementary sub-Hopf algebra Q of B such that
resQX 6= 0.
Proof.This follows from the corresponding result for StMod (B), [Pal97, Theo-
rem 1.3]: suppose that resX = 0 for all quasi-elementary sub-Hopf algebras Q of
B. Then certainly the homology of X is 0, since (ordinary) homology does not
change under restriction. By [HPS97 , Lemma 9.6.6], this means that X can be
identified with an object of StMod (B). Restriction is compatible with this ide*
*nti-_
fication. Then [Pal97, Theorem 1.3] completes the proof. |__|
Corollary 5.4.Suppose B is a finite-dimensional graded cocommutative Hopf al-
gebra over a field k. Fix X 2 S(B). Then
[
VB (X) = res*Q(VQ (resQX))
Q
as Q runs over the quasi-elementary sub-Hopf algebras of B.
Note that the Quillen stratification theorem determines VC(X) in terms of the
VCff(FffX), but it does not determine VCff(FffX) in terms of VC(X). This is the
subject of Avrunin-Scott stratification (cf. [BCR96 , Theorem 10.7]).
Definition 5.5.Given a bigraded Noetherian stable homotopy category C and
a family {Fff:C ! Cff} of finitary geometric morphisms of bigraded Noetherian
stable homotopy categories that exhaust C, we say that C satisfies Avrunin-Scott
stratification if, for all X 2 C and all ff, we have
VCff(FffX) = (Ff*f)-1VC(X):
Note that Lemma 4.3 says that
VCff(FffX) (Ff*f)-1VC(X)
for any geometric morphism Fff:C ! Cff. We are not able to prove the other
containment in general, but in the case C = S(B) for certain graded connected B,
we can reduce it to the quasi-elementary case.
18 MARK HOVEY AND JOHN H. PALMIERI
Definition 5.6.A quasi-elementary Hopf algebra Q is hereditary if every sub-Hopf
algebra of Q is itself quasi-elementary.
Greg Henderson (private communication) has provided examples of quasi-ele-
mentary Hopf algebras which are not hereditary, unfortunately. The group algebra
of an elementary abelian p-group is hereditary, as are the quasi-elementary sub*
*-Hopf
algebras of the mod 2 Steenrod algebra. We note that the hereditary assumption
in the following may not be necessary, but it is certainly convenient.
Proposition 5.7.Suppose that B is a finite-dimensional graded connected cocom-
mutative Hopf algebra such that every quasi-elementary sub-Hopf algebra Q of B *
*is
hereditary and satisfies Avrunin-Scott stratification. Then B satisfies Avrunin*
*-Scott
stratification.
Before proving this proposition, we need to recall the full power of the Quil*
*len
stratification theorem for X = k from [NP98 , Corollary 2.6].
Theorem 5.8. Suppose that B is a finite-dimensional graded connected cocom-
mutative Hopf algebra. Then the space SpecExt**B(k; k) is a disjoint union of t*
*he
subsets
res*Spec+Ext**Q(k; k);
as Q runs through the quasi-elementary sub-Hopf algebras of B. Here
[
Spec+ Ext**Q(k; k) = SpecExt**Q(k; k) \ res*Q;Q0SpecExt**Q0(k; k)
as Q0runs through all proper sub-Hopf algebras of Q.
Note that the statement in [NP98 ] is about the maximal ideal spectrum, but
since it is really a statement about a certain map being an F-isomorphism of ri*
*ngs,
it will hold for the prime ideal spectrum as well.
Note as well that this is the first place in the paper where we need to place
serious restrictions on the Hopf algebra B. A version of Theorem 5.8 for group
algebras is one of the main results of [Qui71], and for general graded Hopf alg*
*ebras,
a version of Theorem 5.8 has been conjectured by the second author in [Pal97].
Proof of Proposition 5.7.Suppose that Mp ^ X 6= 0, Q is a quasi-elementary sub-
Hopf algebra of B, and q 2 SpecExt**Q(k; k) has res*(q) = p. There must be some
quasi-elementary sub-Hopf algebra Q0 of B and prime ideal q02 SpecExtQ0(k; k)
such that res*Q0(q0) = p and Mq0^resQ0X 6= 0, by the Quillen stratification the*
*orem.
By Theorem 5.8 applied to Q0, there is a unique (necessarily quasi-elementary)
sub-Hopf algebra Q00of Q0 and prime ideal q002 Spec+ Ext**Q00(k; k) such that
res*Q0;Q00(q00) = q0. Applying Theorem 5.8 to Q and B as well, we find that Q00
must be a sub-Hopf algebra of Q, and that res*Q;Q0(q00) = q. Furthermore, Avrun*
*in-
Scott stratification for Q0 implies that Mq002 VQ00(resQ00X). Hence the Quille*
*n__
stratification theorem for Q implies that Mq 2 VQ (resQX), as required. *
*|__|
Recall from Section 3 that we would like to know when a category C as in
Definition 5.5 has the tensor product property. We finish this section by showi*
*ng
how Avrunin-Scott stratification reduces this to knowing that the categories Cff
have the tensor product property. Corollary 5.10 below is the application of th*
*is to
the Hopf algebra situation.
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 19
Proposition 5.9.Let C be a bigraded Noetherian stable homotopy category C and
let {Fff:C ! Cff} be a set of finitary geometric morphisms of bigraded Noetheri*
*an
stable homotopy categories that exhaust C and satisfy Avrunin-Scott stratificat*
*ion.
If each Cffhas the tensor product property, then so does C.
Proof.As before, we write Sfffor the sphere object in Cff.
Suppose X and Y are objects of C, and suppose Mp ^ X 6= 0 and Mp ^ Y 6= 0.
Applying Theorem 5.2 to X = S, we find that there is an ff and a prime ideal q *
*of
ss**Sffso that p = Ff*f(q). Since C satisfies Avrunin-Scott stratification, Mq^*
*FffX 6=
0 and Mq ^ FffY 6= 0. The tensor product property for Cffthen implies that
Mq ^ Fff(X ^ Y ) ~=Mq ^ FffX ^ FffY 6= 0:
The Quillen stratification theorem 5.2 then implies that Mp ^ X ^ Y 6=_0,_as
required. |__|
Corollary 5.10.Suppose B is a finite-dimensional graded connected cocommuta-
tive Hopf algebra such that all quasi-elementary sub-Hopf algebras of B are her*
*ed-
itary, have the tensor product property, and satisfy Avrunin-Scott stratificati*
*on.
Then B has the tensor product property.
6.Rank varieties and the tensor product property
To verify the tensor product property in specific cases, we use rank varietie*
*s. In
this section, we define rank varieties for certain sorts of Hopf algebras, we v*
*erify
the tensor product property for them, and we discuss the relationship between r*
*ank
varieties and support varieties. Rank varieties have been used to study cohomol*
*og-
ical varieties in two settings: restricted Lie algebras in the work of Friedlan*
*der and
Parshall, among others (see [FP86 ], for example), and elementary abelian p-gro*
*ups
in the work of Carlson and others (as in [BCR96 ]). For the most part we follow*
* the
group theory version, but both approaches are potentially useful.
In this section, we mainly work with ungraded modules over ungraded Hopf
algebras. More precisely, in this section, we mainly work in the following cate*
*gories:
let StMod0(B) be the stable category of ungraded B-modules, and let S0(B) be the
category with objects chain complexes of (ungraded) projective B-modules, and
morphisms chain homotopy classes of maps.
Now, the chain complex Mp described in Section 2 can be constructed in any
Noetherian stable homotopy category_it need not be bigraded (the singly graded
case is described in [HPS97 ]). To distinguish the ungraded case from the graded
case, we write M0pfor the object in S0(B). Since StMod0(B) is the finite locali*
*zation
of S0(B) (see Example 2.5), then one can localize M0pto get a B-module, also wr*
*itten
M0p, which is well-defined up to projective summands. Then we have support
varieties VS0(B)(-) and VStMod0(B)(-), defined in terms of the M0p. We will exp*
*lain
how to relate the varieties in the ungraded case to the varieties in the graded*
* case
in Corollary 6.13.
Now that we have defined support varieties in the ungraded case, we move on to
rank varieties. From the Lie algebra perspective [FP86 ], we are led to this de*
*finition.
(Similar definitions also appeared in [NP98 ].)
Definition 6.1.Let k be a field of characteristic p > 0, and let B be a finite-
dimensional cocommutative Hopf algebra over k.
20 MARK HOVEY AND JOHN H. PALMIERI
(a) A rank structure on B is a restricted Lie algebra g = gB and an algebra
isomorphism between B and the restricted enveloping algebra V (g) of g. (N*
*ote
that B and V (g) need not be isomorphic as Hopf algebras.)
(b) Assume that B has rank structure g. Given any x 2 g, let alg(x) denote the
subalgebra of V (g) generated by x. The weak rank variety of a B-module M
with respect to g is this subset of g:
VBr;g(M) = {x 2 g | x 6= 0; x[p]= 0; M is not projective overalg(x)} [ {0}:
This is called the "weak" rank variety because it may give incomplete informa*
*tion
about the module M, at least when M is not finitely generated; for example, Dad*
*e's
lemma 6.3 may fail_see the discussion before Lemma 5.1 in [BCR96 ]. When M is
finitely generated, though, the work of Friedlander and Parshall [FP86 ] applie*
*s.
From the group theory point of view [BCR96 ], we are led to this definition.
Definition 6.2.Let k be an algebraically closed field of characteristic p > 0, *
*and
let B be a finite-dimensional cocommutative Hopf algebra over k.
(a) [Wil81] B is elementary if B is isomorphic, as an algebra, to k[X1; : :;:X*
*n]=(Xpi)
for some n. In other words, B is isomorphic, as an algebra, to the k-group
algebra of an elementary abelian p-group.
(b) Suppose that B is elementary, isomorphic as an algebra to kE for some el-
ementary abelian p-group E. For any B-module M, define the rank variety
VrB(M) to be VrE(M), where VrE(M) is defined in [BCR96 , Definition 5.4].
(c) The vector space of indecomposables of B is Q(B) = IB=(IB)2, where IB is
the augmentation ideal of B.
An elementary Hopf algebra B ~=k[X1; : :;:Xn]=(Xpi) has a rank structure which
is given by setting g = Span(X1; : :;:Xn), with trivial bracket and restriction*
*. Be-
cause the restriction is trivial, then VBr;g(k) is the rank structure g. There *
*are several
choices for rank structure, but in this situation one can re-define it to be th*
*e inde-
composables Q(B)_viewed this way, one no longer has a well-defined embedding
of the rank structure into B, but one gains functoriality of rank varieties. (S*
*ee also
[Car83, Lemma 6.4] and [FSB97 , Lemma 6.4] for results related to independence *
*of
the choice of embedding.)
Continuing with the case when B is elementary, if the field k is algebraical-
ly closed, then VrB(M) can be described as follows: let K be a field extension
of k of transcendence degree at least n = dimk g. Then VrB(k) is the subset of
VKrkB;Kkg (K) = Kn consisting of the generic points for the closed homogeneous
irreducible subvarieties of VBr;g(k); this is essentially independent of the fi*
*eld exten-
sion K_see Proposition 5.3, and the comments preceding it, in [BCR96 ]. Given
VrB(k), then VrB(M) is described as follows:
VrB(M) = {x 2 VrB(k) | x 6= 0; K k M is not projective overalg(x)}:
(We have omitted the condition xp = 0 since that is automatic for elementary Ho*
*pf
algebras.) One could define VrB;g(M) this way for any Hopf algebra B with rank
structure g. We would guess that it is well-defined and well-behaved, but we ha*
*ve
not verified it. Bendel has been working on related issues [Ben ].
Remark. The above definitions of rank variety are incomplete in several ways.
(a) The weak rank variety is not obviously functorial (except when B is elemen-
tary, by the sort of trick described above).
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 21
(b) There is a gap between the definitions of weak rank variety and rank varie*
*ty;
namely, rank varieties are not defined for every Hopf algebra with a rank
structure. One would like to extend Benson, Carlson, and Rickard's definit*
*ion
to the general case, but this has not been done, to our knowledge. As it n*
*ow
stands, the notion of weak rank variety is, as the name implies, too weak *
*for
our purposes, in particular when dealing with infinitely generated modules.
(c) From a topologist's point of view, there is the flaw that in odd character*
*istics,
we have only dealt with evenly-graded Hopf algebras. This flaw can likely *
*be
fixed, at least in the connected and commutative case: graded connected
commutative cocommutative Hopf algebras at an odd prime split as a tensor
product of an evenly-graded Hopf algebra and an exterior algebra generated*
* by
odd-dimensional classes, by [MM65 , Proposition 7.21]. So one could defin*
*e the
rank variety to be the product of the rank varieties for two pieces, as in*
* [NP98 ,
Section 4]. Aside from some grading issues (as dealt with in Corollary 6.1*
*3),
this is probably the right definition. Something similar might also work *
*in
the non-connected case.
(d) From an algebraist's point of view, we do not have a definition of weak ra*
*nk
variety for every finite-dimensional cocommutative Hopf algebra. Perhaps
work of Friedlander and Suslin, as in the proof of [FS97 , Theorem 1.1] in
which they essentially write a finite group scheme as a semi-direct product
of a finite group and an infinitesimal group scheme, will lead to a general
definition of weak rank variety.
Note that both the weak rank variety VBr;g(M) and the rank variety VrB(M) are
independent of the coalgebra structure on the Hopf algebra B. As a consequence,
many of the properties proved about these varieties for Lie algebras and elemen*
*tary
abelian groups carry over immediately to our situation. For example, we have the
following results.
Theorem 6.3 (Dade's lemma, Corollary 5.6 in [BCR96S]).uppose that k is an al-
gebraically closed field, and let B be a finite-dimensional elementary Hopf alg*
*ebra
over k. Then a B-module M is projective if and only if VrB(M) = ;.
According to [FP87 , Proposition 1.5] (together with Proposition 6.4 below), *
*we
have this weaker result for a Hopf algebra B with rank structure g, when working
over an algebraically closed field: a finitely generated B-module M is projecti*
*ve if
and only if VBr;g(M) = {0}. The obvious conjecture is that, once one has defined
VrB;g(M) appropriately, Theorem 6.3 will hold in general. Bendel has proved some
related results [Ben ].
The following was first proved for elementary abelian groups by Avrunin and
Scott, and is reproduced as [BCR96 , Proposition 6.1]; it was proved for restri*
*cted
Lie algebras by Jantzen [Jan86].
Proposition 6.4.Let k be an algebraically closed field. Let B be a finite-dimen*
*sional
cocommutative Hopf algebra over k with rank structure g. Then there is an F -
isomorphism of varieties
fi*: MaxSpec ExtB (k; k) ! VBr;g(k):
For restricted Lie algebras, this map is usually called . We denote it fi* fo*
*llowing
the group theory situation, in which VEr(k) is dual to H1(E; k), the cohomologi*
*cal
22 MARK HOVEY AND JOHN H. PALMIERI
variety VE (k) is dual to the image of the Bockstein fi in H2(E; k), and the ma*
*p is
the dual of fi, extended from Fp to k via the Frobenius. See [BCR96 , Section 6*
*].
Looking at the closed homogeneous irreducible subvarieties of VBr;g(k), i.e. *
*look-
ing at VrB(k), captures exactly the passage from MaxSpec to Spec, so we have
the following. Recall that VStMod0(B)(-) is the support variety in the category
StMod 0(B); in particular VStMod0(B)(k) is the set of all homogeneous prime ide*
*als
in Ext*B(k; k), except for the unique maximal ideal consisting of all positive *
*dimen-
sional elements.
Corollary 6.5.Let k be an algebraically closed field, and let B be an elementary
Hopf algebra over k. Then there is an F -isomorphism of varieties
fi*: VStMod0(B)(k) ! VrB(k):
The coproduct structure on B comes into play when dealing with rank varieties
of tensor products of B-modules. We claim that frequently, although the B-module
structure on M N depends on the coproduct on B, the (weak) rank variety for
M N is somewhat independent of the coproduct structure.
Theorem 6.6. Let B be a finite-dimensional Hopf algebra over a field k with ra*
*nk
structure g, and suppose that the coproduct : B ! B B induces the diagonal
embedding
VBr;g(k),! VBrB;gxg(k) = VBr;g(k) x VBr;g(k):
v7! (v; v)
Then for any B-modules M and N,
VBr;g(M N) = VBr;g(M) \ VBr;g(N):
For example, if B is the enveloping algebra V (g), then the coproduct is indu*
*ced
by the diagonal map g ,! g x g, and hence the hypotheses are satisfied. We show
in Lemma 6.8 below that the hypotheses also hold for any elementary Hopf algebra
(regardless of the coproduct).
We need a lemma to prove Theorem 6.6.
Lemma 6.7. Let B1 and B2 be finite-dimensional Hopf algebras with rank struc-
tures g1 and g2, respectively. If B1 ,! Q2 is an inclusion of Hopf algebras whi*
*ch
induces an embedding g1 ,! g2, and if M is a B2-module, then
VBr1;g1(M) = VBr1;g1(k) \ VBr2;g2(M):
Proof.See [BCR96 , Proposition 7.3] and [FP86 , Proposition 2.1]. |*
*___|
Proof of Theorem 6.6.We follow the proof of the tensor product theorem for rank
varieties for elementary abelian groups in [BCR96 , Section 7]. First of all, t*
*he weak
rank variety for M N as a B B-module is independent of the coproduct on B,
so we have the following, by [BCR96 , Theorem 7.2]:
VBrB;gxg(M N) = VBr;g(M) x VBr;g(N)
as subsets of VBrB;gxg(k) = VBr;g(k) x VBr;g(k). Next, applying Lemma 6.7 to the
inclusion : B ,! B B gives
VBr;g(M N) = V(rB);(g)(k) \ VBrB;gxg(M N)
= V(rB);(g)(k) \ (VBr;g(M) x VBr;g(N))
= VBr;g(M) \ VBr;g(N);
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 23
where the last equality follows because the embedding of weak rank varieties_is*
* the
diagonal map. |__|
As mentioned above, the hypotheses of Theorem 6.6 are satisfied when B is
elementary.
Lemma 6.8. Let B be a finite-dimensional elementary Hopf algebra over a field
k, with rank structure g. Writing for the map on weak rank varieties induced by
the coproduct on B, then for all v 2 VBr;g(k), we have (v) = v x v.
Proof.We view VQr(k) as the indecomposables Q(B) of B. For any v 2 IB, the
coproduct on v is of the form
X
(v) = v 1 + 1 v + vi wi;
i
where viand wiare in IB for each i. In particular, viwiis decomposable in BB,
so : B ! B B induces this map on indecomposables: (v) = v 1 + 1 v. In
other words, as a map Q(B) ! Q(B) x Q(B), we have
(v) = v x 0 + 0 x v = v x v;
as desired. |___|
When k is algebraically closed (so that VrB(M) is defined), then the discussi*
*on
after Definition 6.2 applies, giving this corollary.
Corollary 6.9.Let B be a finite-dimensional elementary Hopf algebra over an
algebraically closed field k. For any B-modules M and N, we have
VrB(M N) = VrB(M) \ VrB(N):
We conclude that, not only does the tensor product theorem hold for rank vari-
eties, but the entire theory of rank varieties is independent of the coproduct *
*on the
elementary Hopf algebra B. For example, we have the following.
Theorem 6.10 (Theorem 10.5 in [BCR96 ]).Let B be a finite-dimensional elemen-
tary Hopf algebra over an algebraically closed field k. For every B-module M, t*
*he F -
isomorphism fi* from Proposition 6.4 induces an F -isomorphism VStMod0(B)(M) !
VrB(M).
We have been working in the setting of ungraded modules; we need to convert
our results from ungraded to graded, and also from modules to chain complexes.
First, in the ungraded case, we have the following.
Corollary 6.11.Let B be an elementary Hopf algebra over an algebraically closed
field. Then the category StMod 0(B) has the tensor product property. That is, f*
*or
all B-modules M and N,
VStMod0(B)(M N) = VStMod0(B)(M) \ VStMod0(B)(N):
First, we convert from StMod 0(B) to S0(B). We recall from [HPS97 , Definition
6.0.8] the construction of an object K(p) which is Bousfield equivalent to M0p,*
* by
[HPS97 , Theorem 6.1.8]: given a prime ideal p Ext*B(k; k), choose generators
y1; : :;:yn for p. For each i, let S=yibe the cofiber in S0(B) of the map yi: S*
* ! S.
Then define S=p by
S=p = S=y1 ^ : :^:S=yn;
24 MARK HOVEY AND JOHN H. PALMIERI
and let K(p) = LpS=p. The objects S=p and K(p) depend on the choice of genera-
tors, but their Bousfield classes ~~~~ and = do not.
Corollary 6.12.Let B be an elementary Hopf algebra over an algebraically closed
field. Then the category S0(B) has the tensor product property.
Proof.We recall from [HPS97 , Theorem 9.6.4] the relationship between S0(B) and
StMod 0(B): StMod 0(B) may be obtained from S0(B) by finite localization LB away
from the thick subcategory generated by the module B, viewed as a chain complex
concentrated in degree zero. (See Theorem 2.3 and Definition 2.4 for information
about finite localization.)
To verify the tensor product property for S0(B), we need to verify that for a*
*ny
homogeneous prime ideal p in Ext*B(k; k) and chain complex X with X ^ M0p6= 0,
then = .
There are two cases: either p is the unique maximal ideal m consisting of all
elements in positive degree, or not. If p 6= m, then B ^ M0p= 0; indeed, one can
check that since B is concentrated in degree zero and p does not contain every
element of positive degree, then both LpB and L = ~~:
But since both M0pand M0p^ X are both LB -local, we see that
= ;
as desired.
If p = m, then we claim that = **; since B is a field object, if X ^ B*
* 6= 0,
then = ****. It remains to verify the equality = ****, or equivalen*
*tly,
= ****. Note that Lm is the identity functor, since it is defined by inve*
*rting
the nonzero elements not in p, i.e., in Ext0B(k; k) = k. Hence K(m) = S=m. By t*
*he
proof of [HPS97 , Lemma 6.0.9], we know that every element of m acts nilpotently
on ss*S=m, and so ss*S=m is finite-dimensional as a vector space over k. Theref*
*ore
it has a finite Postnikov tower, and so **** **~~.
Note also that L . |__|
Now we deal with the difference between the singly graded category S0(B) and
the bigraded one S(B).
Corollary 6.13.Every finite-dimensional graded elementary Hopf algebra over an
algebraically closed field has the tensor product property.
Proof.Let U denote the forgetful functor from graded B-modules to B-modules; we
also use U for the induced functor U :S(B) ! S0(B), and other related functors.
For example, if p is a bihomogeneous prime in Ext**B(k; k), then we write Up for
the corresponding homogeneous prime in Ext*B(k; k).
Note that if p is a bihomogeneous ideal, then we may choose bihomogeneous
generators, so we may assume that U(S=p) = S=(Up): when taking the quotient by
p, it doesn't matter whether one pays attention to the grading or not. On the o*
*ther
hand, localization could be rather different: in S(B), the functor Lp inverts *
*all
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 25
bihomogeneous elements not in p, whereas in S0(B), Lp inverts more elements_all
homogeneous elements not in p. So UMp will in general be different from M0Up.
We claim, though, that if X is an object in S(B), then
VS(B)(X) = VS(B)(S) \ VS0(B)(X):
In other words, VS(B)(X) is the set of all bihomogeneous primes contained in
VS0(B)(X). Given this equality, the tensor product property for S0(B) implies
it for S(B).
Since
VS(B)(X) = {p | p is bihomogeneous; X ^ Mp 6= 0}
VS0(B)(UX) = {q | q is homogeneous; UX ^ M0q6= 0};
and since M0pinvolves inverting more elements than Mp, we clearly have the con-
tainment
VS(B)(X) VS(B)(S) \ VS0(B)(UX):
We want to know that if X is a graded object and p is bihomogeneous prime ideal
so that UX ^ M0Up= 0, then X ^ Mp = 0.
Since Mp is Bousfield equivalent to K(p), and since S=p makes sense indepen-
dently of the grading, we want to know that
LUp(UX ^ S=p) = 0 ) Lp(X ^ S=p) = 0:
Since the effect of Lp is to localize homotopy groups at p, we can state this p*
*urely
module-theoretically: we want to know that if R is a graded ring, p is a homoge*
*neous
prime ideal of R, and N is a graded R-module, then
UNUp = 0 ) Np = 0:
This is straightforward: if UNUp = 0, then every element x 2 UN is annihilated
by some a 62 Up. Hence every homogeneous element x 2 N is annihilated by some
a 62 Up. Write a as a sum of homogeneous elements: a = a1 + . .+.an with the ai
inPdistinct degrees. Then each aix is in a different degree, so must be zero. S*
*ince
ai= a is not in Up, then some aiis not in p. Hence every homogeneous element_
in n is annihilated by a homogeneous element not in p. |__|
7. Rank varieties-philosophy
In this short section, we try to motivate the rank varieties used in the prev*
*ious
section. To some extent, rank varieties are an ad hoc construction, but a stab*
*le
homotopy theoretic point of view helps to explain some of their usefulness.
Rank varieties are used to compute support varieties, via results like Theo-
rem 6.10; as we have noted, one of their main uses is to verify the tensor prod*
*uct
property for support varieties. So in our search for more precise motivation f*
*or
rank varieties, we examine the tensor product property. Ideally, we would like *
*to
prove it directly for support varieties, by imitating the proof for rank variet*
*ies in
Theorem 6.6.
Let C be a bigraded Noetherian stable homotopy category. We point out in
Section 3 that the tensor product property is equivalent to the following condi*
*tion
on the Mp, for objects X and Y of C:
o For all prime ideals p of ss**S, Mp^X ^Y = 0 if and only if either Mp^X = 0
or Mp ^ Y = 0.
26 MARK HOVEY AND JOHN H. PALMIERI
As remarked in Section 3, this would follow if the Mp were (Bousfield equivalent
to) skew field objects. Let B be a Hopf algebra and let C = S(B). Here are the
steps involved in proving the tensor product theorem for S(B):
Step 1. The coproduct : B -! BB induces a map Ext**BB(k; k) ! Ext**B(k; k),
and hence a map on varieties: *: VB (S) -! VBB (S). One can show that
VBB (S) = VB (S) x VB (S), and one can compute the effect of the map *:
*: VB (S) -!VB (S) x VB (S)
p 7-! (p; p):
For support varieties, this is easy to check: first, there is a K"unneth isom*
*orphism
Ext**BB(k; k) ~=Ext**B(k; k) Ext**B(k; k);
which gives the identification of VBB (S). Second, it is well known that indu*
*ces
the usual Yoneda product on Ext, and it is easy to compute the effect of the pr*
*oduct
map on Spec.
For rank varieties, this is a bit harder. When B is elementary, we complete S*
*tep
1 in Lemma 6.8. When B is not elementary, though, we do not know how to verify
this; hence we need to assume it in Theorem 6.6.
Given Step 1, the proof of Theorem 6.6 reduces us to the following. (In that
proof, we apply this step to the Hopf algebra inclusion : B ! B B.)
Step 2. Let i: B1 ,! B2 be an inclusion of finite-dimensional Hopf algebras for
which the trivial module k is the only simple module, and assume that the induc*
*ed
map i*: SpecExt**B1(k; k) ! SpecExt**B2(k; k) is an inclusion. Then for any obj*
*ect
X in S(B2),
(7.1) i*VB1(resX) = VB2(X) \ i*VB1(S):
We state this for rank varieties in Lemma 6.7. Note also that in the graded
connected case, if B1 ,! B2 is an inclusion, then so is the induced map on Spec*
*, by
Lemma 8.1.
It turns out that to show the two inclusions and of Equation 7.1, one needs
to understand the properties of the objects Mp under restriction and induction,
respectively. In particular, if one assumes that
for all prime ideals q in Ext**B1(k; k), then the inclusion follows. (Since i**
* is an
inclusion, then Corollary 4.12 implies an even stronger result, that resMi*q= M*
*q.)
Here are the details: we assume that p 62 VB2(X) \ i*(VB1(S)). Note that if
p 62 i*(VB1(S)), then p is certainly not in i*VB1(resX), so we assume that p = *
*i*q
for some q 2 VB1(S). Since p 62 VB2(X), then Mp ^ X = 0. Applying res, we find
that 0 = res(Mp ^ X) = resMp ^ resX. We are assuming that resMp has a larger
Bousfield class than Mq; hence Mq ^ resX = 0. In other words, q 62 VB1(resX).
If one knows, in addition, that
;
for all prime ideals q in Ext**B1(k; k), then the inclusion of Equation (7.1)f*
*ollows.
(Unfortunately, we do not know how to prove this inequality of Bousfield classes
without using rank varieties. We discuss this below.) Here are the details of t*
*he
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 27
verification of the inclusion : assume that q 62 VB1(X), so that Mq ^ resX = 0.
Since = , then
0 = resMi*q^ resX = res(Mi*q^ X):
Applying indgives indres(Mi*q^ X). For modules, indresM ~=ind(k) M; we
apply the corresponding result for chain complexes (twice) to get
indres(Mi*q^ X) = ind(S) ^ Mi*q^ X = (indresMi*q) ^ X = indMq ^ X = 0:
Finally, since = , we see that
Mi*q^ X = 0;
so i*q 62 VB2(X).
(By the way, one can show, using the equalities resMi*q= Mq and indresX =
ind(S) ^ X, that
;
so one wants to know that these are in fact Bousfield equivalent.)
The point of using rank varieties is that "they are well-behaved under induct*
*ion".
For each x in the rank variety VrB(k), we construct an object Nx: first, consid*
*er the
B K-module H(x) = (B K) alg(x)K. This has Ext groups
ExtBK (H(x); K) ~=Extalg(x)(K; K) ~=K[y]
for some y in Ext1. Nx is the chain complex obtained from H(x) by inverting y, *
*so
its homotopy groups are K[y; y-1]. Note that Nx need not be a ring object, even
though its homotopy groups make it look like it is trying to be a skew field in*
* the
category S(B).
It is easy to check that the objects Nx are well-behaved under induction: if *
*B1 is
a sub-Hopf algebra of B2 so that VB1(S) includes into VB2(S), then since support
varieties are the same as rank varieties by Corollary 6.5, we get a similar inc*
*lusion
for rank varieties. Hence for x 2 VrB1(k), we have __x, the image of x in VrB2(*
*k). By
construction, indH(x) = H(__x); inverting y in both of these B2-chain complexes
yields indNx = N_x, as desired.
Corollary 6.5 gives a bijection VStMod0(B)(k) -~!VrB(k). Using all of the res*
*ults
about both kinds of varieties, one can do better: one sees that if p 2 VStMod0(*
*B)(k)
corresponds to x 2 VrB(k), then Mp and Nx are Bousfield equivalent. Hence, one
can complete Step 2 for support varieties, albeit in a roundabout fashion.
8.Examples
In this section, we apply our results to some particular Hopf algebras B_those
whose quasi-elementary sub-Hopf algebras are elementary.
The class of quasi-elementary Hopf algebras, even the definition thereof [Pal*
*97],
is quite complicated. They are well understood in two cases: the group algebra
kG of a p-group G is quasi-elementary if and only if G is elementary abelian, by
Serre's theorem about products of Bocksteins of elements in group cohomology
[Ser65]. Of course in this setting, we know by the results in [BCR96 ] that kG *
*has
all of the desired properties_Avrunin-Scott stratification and the tensor produ*
*ct
property_and hence Corollary 3.7 holds for the group algebras of finite p-group*
*s.
We are most interested in the second case, the finite sub-Hopf algebras of the
mod 2 Steenrod algebra A, as these are useful in algebraic topology. Note that
A is defined over the Galois field F2, but we can consider A over any field of
28 MARK HOVEY AND JOHN H. PALMIERI
characteristic 2 by a central extension of scalars. The quasi-elementary finite*
* sub-
Hopf algebras of A were essentially classified in [Wil81, Theorem 6.4] (see also
[NP98 , Proposition 5.2]); they are all elementary Hopf algebras. Therefore, by
Corollary 6.13, they all have the tensor product property. We will show that, s*
*ince
they are connected, this implies that they satisfy Avrunin-Scott stratification*
*, so
Corollary 5.10 implies that Corollary 3.7 holds in this case also_see Corollary*
* 8.5
below.
Note that when p > 2, the quasi-elementary sub-Hopf algebras of the mod p
Steenrod algebra are classified in [NP98 ]; however, the proof given there has *
*a gap
which these authors do not know how to fill.
We start with the following observation.
Lemma 8.1. Suppose that B is a finite-dimensional graded connected cocommuta-
tive Hopf algebra over a field k of characteristic p, and let Q be a sub-Hopf a*
*lgebra
of B. Then the induced map
res*Q:SpecExt**Q(k; k) -!Spec Ext**B(k; k)
is injective.
Proof.This follows from [HS98 , Theorem 4.13],nwhich says that there is an n so
that for every element y 2 Ext**Q(k; k), yp is in the image of
resQ:Ext**B(k; k) ! Ext**Q(k; k):
For a prime ideal p Ext**Q(k; k), res*Q(p) is defined to be res-1Q(p). Assume *
*that
res*Q(p) = res*Q(q) for prime ideals p and q. For y 2 p, then ypn = resQ(x) for*
* some
x, so x 2 res*Q(p) = res*Q(q), so resQ(x) = ypn is in q. Since q is prime,_we_c*
*onclude
that y 2 q. Therefore p q. By the same argument, q p. |__|
Whenever the maps induced on Specby inclusions of quasi-elementary sub-Hopf
algebras are injective, Avrunin-Scott stratification follows from the tensor pr*
*oduct
property. To see this, we calculate the acyclics for the restriction functor.
Proposition 8.2.Suppose that B is a finite-dimensional graded cocommutative
Hopf algebra which has the tensor product property. Suppose as well that Q is a
sub-Hopf algebra of B, and X 2 S(B). Then resX = 0 if and only if V(X) \
res*(SpecExt **Q(k; k)) = ;.
Proof.Note that resX = 0 if and only if S(Q)(S; resX)** = 0. By adjointness,
this holds if and only if S(B)(indS; X)**= 0. Now indS is a finite object of S(*
*B),
since B is finite-dimensional, so we can use Spanier-Whitehead duality (see [HP*
*S97 ,
Section 2.1]) to conclude that this holds if and only if D(indS)^X = 0. Therefo*
*re,
the acyclics for the restriction functor form the Bousfield class of D(indS). B*
*ut,
since B has the tensor product property, any Bousfield class is completely dete*
*r-
mined by the Mp which belong to it by Corollary 3.4. But Corollary 4.12 implies*
*__
that resMp = 0 if and only if p 62 res*SpecExt**Q(k; k). |*
*__|
This proposition yields the following theorem.
Theorem 8.3. Suppose B is a finite-dimensional graded cocommutative Hopf al-
gebra which has the tensor product property. Suppose as well that, for every qu*
*asi-
elementary sub-Hopf algebra Q of B, the induced map
res*Q:SpecExt**Q(k; k) -!Spec Ext**B(k; k)
STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS 29
is injective. Then B satisfies Avrunin-Scott stratification.
Proof.Suppose X 2 S(B), Q is a quasi-elementary sub-Hopf algebra of B, and
q 2 (res*Q)-1VB (X). Let p = res*Qq. Then X ^ Mp is nonzero by hypothesis.
Proposition 8.2 implies that resQ(X ^ Mp) is also nonzero. Since res*Qis inject*
*ive,_
Theorem 4.11 then implies that resQX ^ Mq is nonzero, as required. |__|
Hence we have the following corollaries.
Corollary 8.4.Suppose Q is a graded connected elementary Hopf algebra over an
algebraically closed field. Then Q satisfies Avrunin-Scott stratification.
Proof.This follows from Corollary 6.13, Lemma 8.1, and Theorem 8.3. |___|
Corollary 8.5.If B is a finite-dimensional graded connected cocommutative Hopf
algebra over an algebraically closed field, and if every quasi-elementary sub-H*
*opf
algebra of B is elementary, then B has the tensor product property. Hence Corol-
lary 3.7 applies.
Proof.This follows from Corollaries 5.10, 6.13, and 8.4. |_*
*__|
Finally, since every quasi-elementary sub-Hopf algebra of the mod 2 Steenrod
algebra is elementary, we have the following.
__
Corollary 8.6.Let A be the mod 2 Steenrod algebra, and let F2 be_the_algebraic
closure of F2. If B is a finite-dimensional sub-Hopf algebra of F2 F2 A, then B
has the tensor product property. Hence Corollary 3.7 applies.
We point out that the usual classification_of sub-Hopf algebras of A [AD73 ]
also classifies sub-Hopf_algebras of F2F2A;_in_particular, every finite-dimensi*
*onal
sub-Hopf algebra of F2 F2 A is of the form F2 F2 B for some finite-dimensional
sub-Hopf algebra B of A.
Also note that in [HP99 ], we have removed the requirement from Corollaries 8*
*.5
and 8.6 that the field be algebraically closed.
As examples, we consider the sub-Hopf algebras_A(n) of the mod 2 Steenrod
algebra, after extending scalars from F2 to F2. Recall that A is generated_as an
algebra by the elements {Sqk | k 1}, and A(n) is the subalgebra of F2 F2 A
generated by {Sqk | 1 k 2n}; it turns out that this is a finite-dimensional
sub-Hopf algebra._
Let k = F 2. When n = 0, then A(0) is isomorphic to the exterior algebra
k[Sq1]=(Sq1)2, with Sq1primitive. It is easy to compute Ext: ExtA(0)(k; k) = k[*
*h0],
where h0 is in bidegree (1; 1). The only bihomogeneous prime ideals are (0) and
(h0), and one can describe M(0)and M(h0): first, M(0)= L(0)S, and this is a
bigraded field object with ss**M(0)= k[h0; h-10]. Second, ss**M(h0)is isomorphi*
*c to
a shifted copy of k[h0; h-10]=k[h0]. (By the way, in [HPS97 , Section 6], the a*
*uthors
construct an object K(p) which is Bousfield equivalent to Mp for each prime p.
In the case at hand, K(h0) is, by definition, the cofiber S=h0 of h0, which is *
*the
chain complex consisting of the module A(0) concentrated in degree 0. This obje*
*ct
represents ordinary homology of chain complexes in the category S(A(0)).)
Since there are three subsets of Spec which are closed under specialization_;,
{(h0)}, {(0); (h0)}_then Corollary 3.7 tells us that there are three nonempty t*
*hick
subcategories of finite objects in S(A(0)): respectively, the thick subcategory*
* con-
sisting of the zero chain complex, the thick subcategory generated by A(0), and
30 MARK HOVEY AND JOHN H. PALMIERI
the thick subcategory of all finite objects. Similarly, since there are two bih*
*omo-
geneous prime ideals, the Bousfield lattice has four elements: <0>, , ,
and ~~~~ = .
Next, we consider A(1), the sub-Hopf algebra of k F2 A generated by Sq1 and
Sq2. This is 8-dimensional as a vector space. We have the following computation
of Ext (as given in, say, [Wil81, p. 142]):
Ext**A(1)(k; k) ~=k[h0; h1; w; v]=(h0h1; h31; h1w; w2 - h20v);
where h0 has bidegree (1; 1), h1 has bidegree (1; 2), w has bidegree (3; 7), and
v has bidegree (4; 12). Note that there are four bihomogeneous prime ideals in
Ext**A(1)(k; k); the nilradical (h1), the ideals p0 = (h1; w; h0) and p1 = (h1;*
* w; v),
and the maximal ideal (h1; w; h0; v). Hence there are six nonempty thick subcat*
*e-
gories of finites and sixteen Bousfield classes in S(A(1)). Write M0 for a proj*
*ective
resolution of A(1)=A(1) Sq1, and write M1 for a projective resolution of A(0) w*
*ith
the apparent A(1)-module structure. One can check that the support variety of M0
is the closure of {p0}, and that the support variety of M1 is the closure of {p*
*1}. The
thick subcategories in S(A(1)) are those generated by 0, A(1), M0, M1, M0 _ M1,
and S.
The situation gets much more complicated with A(n) for n 2. For example,
from the results of [Pal97] and [NP98 ], one knows that Ext**A(2)(k; k) is F -i*
*somorphic
to
k[h0; v1; v2; h21]=(h0h21);
where |h0| = (1; 1), |v1| = (1; 3), |v2| = (1; 7), and |h21| = (1; 6). Since a*
*n F -
isomorphism of algebras induces a poset isomorphism on Spec, one concludes that
there are infinitely many prime ideals in Ext**A(2)(k; k); for instance, one ha*
*s the
ideals (ffv31+ fiv20v2), for any scalars ff and fi. So there are infinitely man*
*y thick
subcategories, and infinitely many Bousfield classes.
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Department of Mathematics, Wesleyan University, Middletown, CT 06459
E-mail address: hovey@member.ams.org
Department of Mathematics, University of Washington, Seattle, WA 98195
E-mail address: palmieri@member.ams.org
~~