DECOMPOSING THICK SUBCATEGORIES OF THE STABLE
MODULE CATEGORY
HENNING KRAUSE
Abstract.Let mod_kG be the stable category of finitely generated modular*
* represen-
tations of a finite group G over a field k. We prove a Krull-Remak-Schmi*
*dt theorem for
thick subcategories of mod_kG. It is shown that every thick tensor-ideal*
* C of mod_kG
(i.e. a thick subcategory`which is a tensor ideal) has a (usually infini*
*te) unique de-
composition C = i2ICiinto indecomposable thick tensor-ideals. This deco*
*mposition
follows from a decomposition of the corresponding idempotent kG-module E*
*C into in-
decomposable modules. If C = CW is the thick tensor-ideal corresponding *
*to a closed
homogeneous subvariety W of the maximal ideal spectrum of the cohomology*
*Sring
H*(G; k), then the decomposition of C reflects the decomposition W = ni*
*=1Wiof W
into connected components.
Introduction
In modular representation theory of finite groups, one frequently passes to t*
*he stable
module category which is a triangulated category. Following ideas from stable h*
*omotopy
theory, Benson, Carlson, and Rickard studied in a number of papers the lattice *
*of thick
subcategories of the stable module category. In [4], these authors obtain for *
*p-groups
a description of this lattice which is an analogue of a result for the stable h*
*omotopy
category due to Devinatz, Hopkins, and Smith [9, 12]. In this paper we prove a *
*Krull-
Remak-Schmidt theorem for this lattice which we now explain.
Let = kG be the group algebra of a finite group G over a field k. The stable*
* category
mod_ of finitely generated -modules is a triangulated category and carries a te*
*nsor
product which is induced from the tensor product on mod . A full subcategory C*
* of
mod_ is called a thick tensor-ideal in mod_ if C is a thick subcategory and X *
*Y 2 C
for all X 2 C and Y 2 mod_. Given a thick tensor-ideal C, we say that a family *
*(Ci)i2I
of thick tensor-ideals in mod_ is a decomposition of C if
(1) the objects in C are the finite coproducts of objects from the Ci;
(2) Ci\ Cj = 0 for all i 6= j in I.
`
A decomposition (Ci)i2Iof C is denoted by C = ` i2ICi, and we say that C is ind*
*ecom-
posable if C 6= 0 and any decomposition C = C1 C2 implies C1 = 0 or C2 = 0.
Theorem`A. Every thick tensor-ideal C in mod_ has a unique decomposition C =
i2ICi into indecomposable thick tensor-ideals. Conversely, given any family (*
*Ci)i2Iof
thick tensor-ideals in mod_ satisfying`Ci\ Cj = 0 for all i 6= j in I, there ex*
*ists a thick
tensor-ideal C such that C = i2ICi.
Before we discuss this result in more detail let us mention that there seems *
*to be no
analogous statement for the stable homotopy category. Our analysis of thick sub*
*cate-
gories in mod_ is based on the analysis of certain objects in the`stable catego*
*ry Mod__
of all -modules.` In fact, we are able to compute the centre Z[C ] of the loca*
*lizing
subcategory C of Mod__ which is generated by a thick tensor-ideal C in mod_. R*
*ecall
that the centre of an additive category is the endomorphism ring of the identit*
*y functor.
1
2 HENNING KRAUSE
`
We prove that Z[C ] is a product of commutative local rings, and the correspon*
*ding
decomposition idC =`(ei)i2I of the identity functor into central idempotents gi*
*ves the
decomposition C = i2ICiif we define Ci= ei(C) for every i. Note that the cent*
*re of the
stable homotopy category is Z which is not a local ring. Therefore one cannot e*
*xpect
an analogue of Theorem A for the stable homotopy category because any Krull-Rem*
*ak-
Schmidt theorem requires local endomorphism rings for the indecomposable object*
*s.
In [19], Rickard introduced for every thick subcategory C of mod_ a particula*
*r -
module EC. It`is obtained from the`trivial representation k by applying a right*
* adjoint
e: Mod__ ! C of the inclusion C ! Mod__, i.e. EC = e(k). If C is a tensor-i*
*deal,
then the canonical map EC ! k induces an isomorphism EC EC ! EC in Mod__,
and therefore EC is called idempotent. In recent years, the study of such modul*
*es has
become one of the most successful methodologies in modular representation`theor*
*y of
finite groups. We obtain Theorem A from the following result since Z[C ] ' End*
*_ (EC).
Theorem`B. The -module EC has, up to isomorphism, a unique decomposition EC =
i2IEi into indecomposable -modules with local endomorphism ring. Moreover, t*
*he
endomorphism ring of EC has the following properties:
T
(1) n2Nrn = 0 for the Jacobson radical r ofQEnd_ (EC);
(2) if C is a tensor-ideal, then End_ (EC) ' i2IEnd_(Ei)
The decomposition of the module EC is the consequence of a more general resul*
*t.
In fact, we introduce the concept of an endofinite object in a compactly genera*
*ted
triangulated category and prove that any endofinite object decomposes uniquely *
*into
indecomposable objects. Theorem B then follows since EC is an endofinite object*
* in the
localizing subcategory of Mod__ which is generated by C.
Interesting examples of endofinite objects also arise in stable homotopy theo*
*ry. For
instance, the classifying space BG of G is endofinite in the category of spectr*
*a [16].
Therefore we obtain a new proof for the existence of a stable splitting of BG w*
*hich does
not involve Segal's conjecture; see Benson's survey [2] for more details.
Let us end this introduction with a geometric interpretation of Theorem A in *
*terms
of the projective prime ideal spectrum of the cohomology ring H*(G; k). More pr*
*ecisely,
we denote by VG (k) the set of closed homogeneous irreducible non-zero subvarie*
*ties of
the maximal ideal spectrum VG (k) of H*(G; k). Generalizing the definition of a*
* closed
homogeneous subvariety of VG (k), Benson, Carlson, and Rickard consider in [3] *
*subsets
W of VG (k) which are closed under specialization, i.e. W 2 W and V W implies
V 2 W. For instance, any closed homogeneous subvariety W of VG (k) is determine*
*d by
W = {V 2 VG (k) | V W }. Given any subset W of VG (k) closed under specializat*
*ion,
we say that W is projectively connected if any decomposition W = W1[ W2 into su*
*bsets
of VG (k) closed under specialization such that W1\W2 = ; implies W = W1 or W =*
* W2.
In [4], Benson, Carlson, and Rickard establish a bijection W 7! CW between the*
* subsets
of VG (k) which are closed under specialization`and the thick tensor-ideals in *
*mod_.
Under this bijection the decomposition C = i2ICiofSC = CW into indecomposabl*
*e thick
tensor-ideals corresponds to the decomposition W = i2IWi of W into projective*
*ly
connected non-empty components, i.e. Wi\ Wj = ; and Ci= CWi for all i 6= j in I*
*. We
have therefore the following consequence.
Theorem C. Let C be a thick tensor-ideal in mod_, and suppose that W is the c*
*or-
responding subset W of VG (k) closed under specialization such that C = CW . Th*
*en the
following conditions are equivalent:
(1) W is non-empty and projectively connected;
DECOMPOSING THICK SUBCATEGORIES 3
(2) C is an indecomposable thick tensor-ideal in mod_;
(3) EC is an indecomposable object in Mod__.
Similar results have been obtained independently by Daugulis in a recent pape*
*r [8].
For instance, using some different methods, he shows that the endomorphism ring*
* of
an idempotent module EC is local provided that C corresponds to a connected clo*
*sed
homogeneous subvariety of VG (k) (or to a collection of closed homogeneous subv*
*arieties
of VG (k) which is connected in a suitable sense).
1. Endofiniteness
Let T be a triangulated category [20] and suppose that arbitrary coproducts e*
*xist in
T . `An object X in T is called`compact if for every family (Yi)i2I in T the *
*canonical
map iHom (X; Yi) ! Hom (X; iYi) is an isomorphism. We denote by T0 the full
subcategory of compact objects in T and observe that T0 is a triangulated subca*
*tegory
of T . Following [18], the category T is called compactly generated provided *
*that the
isomorphism classes of objects in T0 form a set, and Hom (C; X) = 0 for all C i*
*n T0
implies X = 0 for every object X in T .
In [7], Crawley-Boevey introduced the concept of endofiniteness for locally f*
*initely
presented categories. We make the following analogous definition for a compactl*
*y gen-
erated triangulated category T .
Definition 1.1. An object X in T is called endofinite if for every compact obje*
*ct C in
T the End (X)-module Hom (C; X) has finite composition length.
The following result collects some useful properties of endofinite objects.
Theorem 1.2. An endofinite object X has the following properties:
`
(1) there exists, up to isomorphism, a unique decomposition X = i2IXi into *
*inde-
composableTobjects with local endomorphism ring;
(2) n2Nrn = 0 for the Jacobson radical r of End (X);
(3) if OE: Y ! X is a phantom map, i.e. the induced map Hom (C; Y ) ! Hom (C;*
* X)
is zero for all compact C, then OE = 0.
Proof.The proof uses the fact that the category T can be embedded (modulo the
phantom maps in T ) into an abelian Grothendieck category. We recall briefly t*
*his
construction and refer to [15] for more details. Let Mod T0 be the category of *
*additive
functors T0op! Ab into the category of abelian groups, and denote by mod T0 the*
* full
subcategory of finitely presented functors. We shall use the restricted Yoneda *
*functor
h: T - ! Mod T0; X 7! HX = Hom (-; X)|T0:
If X is endofinite, then Hom (F; HX ) is a End (HX )-module of finite length fo*
*r all F
in mod T0, and therefore HX is an endofinite object of Mod T0 in the sense of *
*[7]. We
claim that HX is an injective object. To this end let ": 0 ! HX ! M ! N ! 0
be an exact sequence. Every map F ! N with F in mod T0 factors through M ! N
since HX is fp-injective, i.e. Ext 1(-; HX ) vanishes on finitely presented ob*
*jects, by
[15, Lemma 1.6]. Thus " is a pure-exact sequence which is split exact since ev*
*ery
endofinite object is pure-injective [7, 3.6]. The injectivity of HX implies tha*
*t h induces
an isomorphism End (X) ! End (HX ) by [15, Lemma 1.7]. Any endofinite object in
Mod T0 has a decomposition into indecomposable objects with local endomorphism *
*ring
by [7, Theorem 3.5.2], and this gives the desired decomposition of X in T .
Let us give an alternative proof for (1). To this end consider the localizing*
* subcategory
L of Mod T0 which is generated by all F in mod T0 with Hom (F; HX ) = 0. We c*
*an
4 HENNING KRAUSE
form the quotient category Mod T0=L in the sense of [10], and obtain a locally*
* finite
Grothendieck category since the objects in mod T0 form a generating set of fini*
*te length
objects. Moreover, the quotient functor q :Mod T0 ! Mod T0=L identifies HX wit*
*h an
injective object in Mod T0=L. More precisely, q(HX ) is fp-injective in Mod T0=*
*L since HX
is fp-injective in Mod T0 by [15, Lemma 1.6]. However, any fp-injective object *
*in a locally
noetherian category is injective by an analogue of Baer's criterion for Grothen*
*dieck
categories. It follows from [13, Corollary 2.11] that HX is L-closed, i.e. Hom *
*(L; HX ) =
0 = Ext1(L; HX ), since L is generated by finitely presented objects. Therefor*
*e HX is
an injective object in Mod T0 since q(HX ) is injective, and End (HX ) ' End (*
*q(HX ));
see [10]. Any injective object in a locally finite category decomposes uniquel*
*y into
indecomposables with local endomorphism ring [10, IV.2], and this gives an alte*
*rnative
proof for (1) since
End(X) ' End (HX ) ' End (q(HX )):
T n
Part (2) follows from the general fact that n2Nrad End(Q) = 0 for every inje*
*ctive
object Q in a locally finite category [10, IV.4]. To prove (3) consider a phant*
*om map
OE: Y ! X. The functor h induces an isomorphism Hom (Y; X) ! Hom (HY ; HX ), b*
*y __
[15, Lemma 1.7], since HX is injective, and therefore OE = 0. This finishes the*
* proof. |__|
We include for later reference the following lemma.
Lemma 1.3. Let S be a localizing subcategory of T which is generated by comp*
*act ob-
jects from T , and denote by e: T ! S a right adjoint of the inclusion S ! T .
(1) S is a compactly generated triangulated category.
(2) If X is an endofinite object in T , the e(X) is endofinite in S.
Proof.Suppose that S is generated by a class D of objects in T0 and denote by C*
* the
thick subcategory of T0 which is generated by D. Clearly, C S0. Suppose now th*
*at
Hom (S0; X) = 0 for some X in S, and denote by R the localizing subcategory of *
*all Y
in T such that Hom (Y; X[n]) = 0 for all n 2 Z. Clearly, R contains S and the*
*refore
X = 0. It is not hard to check that S0 = C (e.g., see [17, Lemma 2.2]), and we *
*conclude
that S is compactly generated.
Now let X be an endofinite object in T and fix C in S0. The functorial isomor*
*phism
Hom (C; X) ' Hom (C; e(X)) shows that Hom (C; e(X)) has finite length over End(*
*e(X)) __
since C is also compact in T . Therefore e(X) is endofinite in S. *
* |__|
Example 1.4. (1) A spectrum X is endofinite in the stable homotopy category if*
* and
only if the stable homotopy groups sssn(X) have finite length over the endomorp*
*hism
ring [X; X] of X for all n 2 Z; see [16].
(2) A -module X is an endofinite object in the stable module category Mod__ i*
*f and
only if the endolength of X (i.e. the length of X as module over End (X)) is f*
*inite; see
[14, Corollary 3.3].
2. Decomposing thick subcategories
Let = kG be the group algebra of a finite group G over a field k. We consid*
*er
the category Mod of (right) -modules, and mod denotes the full subcategory of
all finitely generated -modules. The algebra is quasi-Frobenius, i.e. projecti*
*ve and
injective -modules coincide. Therefore the stable category Mod__ is triangulate*
*d; e.g.,
see [11]. Recall that the objects in Mod__ are those of Mod , and for two -modu*
*les
X; Y one defines Hom__(X; Y ) to be Hom (X; Y ) modulo the subgroup of maps wh*
*ich
factor through a projective -module. Note that the projection functor Mod ! Mo*
*d__
DECOMPOSING THICK SUBCATEGORIES 5
preserves coproducts. Thus Mod__ has arbitrary coproducts, and it is not diffi*
*cult to
check that an object X in Mod__ is compact if and only if X ' Y in Mod__ for so*
*me
finitely generated -module Y . Therefore we shall not distinguish between (Mod*
*__)0
and mod_. The description of the compact objects shows that Mod__ is compactly
generated.
2.1. Thick subcategories. Let C be a thick subcategory of mod_, i.e. C is a ful*
*l trian-
gulated`subcategory of mod_ which is closed under taking direct summands. We de*
*note
by C the localizing subcategory of Mod__ which is generated by C. Recall that *
*a full
triangulated subcategory`of Mod__ is localizing if it is closed`under taking co*
*products.
The inclusion C ! Mod__ has a right adjoint e: Mod__ ! C (e.g., see [17, 19])*
*, and
we denote by ": EC = e(k) ! k the corresponding map for the trivial representat*
*ion
k. Using the results about endofinite objects we can now prove the first porti*
*on of
Theorem B.
Theorem`2.1. The -module EC has, up to isomorphism, a unique decomposition
ECT= i2IEi into indecomposable -modules with local endomorphism ring. Moreove*
*r,
n
n2Nr = 0 for the Jacobson radical r of End_ (EC).
Proof.The trivial representation`k is certainly an endofinite object in Mod__kG*
*, and
therefore EC is endofinite`in C by Lemma 1.3. We can apply Theorem 1.2 and obt*
*ain
a decomposition EC = i2IEi into indecomposables in Mod__ plus the assertion a*
*bout
the Jacobson radical`of End_ (EC). It is well-known that any -module X has a de*
*com-
position X = XP PX such that XP has no non-zero projective direct sumand and *
*PX
is projective [14, Proposition 5.10]. Clearly, any projective -module is a copr*
*oduct of
indecomposable modules since is artinian. On the other hand, the endomorphisms
of XP which factor through projectives belong to the Jacobson radical of End (*
*XP ),
and therefore the decomposition of EC in Mod__ gives the desired decomposition_*
*in
Mod . |__|
T
Remark 2.2. Let s be the Jacobson radical of End (EC). Then ( n2N sn)l= 0 w*
*here
l denotes the Loewy length of .
2.2. Thick tensor-ideals. Let C be a thick subcategory of mod_ and assume in ad*
*di-
tion that X Y 2 C for all X 2 C and Y 2 mod_. We call a subcategory C with the*
*se
properties a`thick tensor-ideal in mod_. We shall use the following description*
* of the
objects in C which is due to Rickard.
Proposition 2.3. Suppose that C is a thick tensor-ideal in mod_. Then the foll*
*owing
are equivalent for an object X in Mod__:
`
(1) X belongs to C ;
(2) " X :EC X ! k X ' X is an isomorphism.
*
* __
Proof.Combine Proposition 5.6 and Proposition 5.13 from [19]. *
* |__|
The following lemma will be useful for decomposing a thick tensor-ideal.
Lemma 2.4. Let C and D be thick tensor-ideals in mod_ . Then the following a*
*re
equivalent:
(1) C \ D = 0;
(2) C D = 0, i.e. X Y = 0 for all X 2 C and Y 2 D;
(3) Hom__(C;`D) =`0, i.e. Hom__(X; Y ) = 0 for all X 2 C and`Y 2 D; `
(4) Hom__(C ; D ) = 0, i.e. Hom__(X; Y ) = 0 for all X 2 C and Y 2 D ;
6 HENNING KRAUSE
` ` ` `
(5) C D = 0, i.e. X Y = 0 for all X 2 C and Y 2 D ;
(6) EC` ED`= 0.
(7) C \ D = 0;
Proof.(1) ) (2) Clear, since C D C \ D.
(2) ) (3) Let X 2 C and Y 2 D. Then the k-dual X* is a direct summand of X*
X X* and belongs therefore to C . It follows that Hom k(X; Y ) ' X*Y is a proje*
*ctive
-module since C D = 0. Any -map X ! Y factors through X Hom k(X; Y ), and
therefore Hom__(X; Y ) = 0.
(3) ) (4) Let`D0be the localizing subcategory of all X in Mod__ such that Hom*
*__(C; X) =
0. Clearly, D D0 since D D0. Now let C0 be the`localizing subcategory of all*
* X
in Mod__`such`that Hom__ (X; D0) = 0. We have C C0 since C C0, and therefore
Hom__ (C ; D ) = 0. ` ` ` ` ` ` ` `
(4) ) (5) If Hom__ (C ; D ) = 0, then C D = 0 since C D C \ D by
Proposition 2.3.
(5) ) (6) Clear. ` `
(6) ) (7) Clear, since EC ED X ' X for every X 2 C \ D by Proposition 2.3*
*._
(7) ) (1) Clear. |*
*__|
We continue with two results which establish the strong connection between de*
*com-
positions of C and EC. To this end the following definition is needed. Let ": E*
* ! k be a
map in Mod__. We denote by L" the localizing subcategory of all X in Mod__ such*
* that
" X is an isomorphism. The intersection C" = L" \ mod_ gives a thick subcatego*
*ry
of mod_, and we observe that C" is a thick tensor-ideal if " E is an isomorphi*
*sm.
Proposition`2.5. Let C be a thick tensor-ideal in mod_. Every decomposition`EC*
* =
iEi satisfying Ei Ej = 0 for all i 6= j, induces a decomposition C = iCi of*
* C such
that Ei' ECi for all i.
`
Proof.Let ("i): iEi ! k be the decomposition of ": EC ! k. The assumption
Ei Ej = 0 for all i 6= j`implies that "i Ei is an isomorphism for all i. Theref*
*ore we
get a decomposition C = iCi of C if we define Ci = C"ifor all i. It remains t*
*o show
that Ei ' ECi for all i. Observe first`that Ei ECj = 0 for all i 6= j since Ei *
*Cj = 0.
Now consider the canonical map OE: jECj ! k. We`have ECi ECj = 0 for all i 6=*
* j
by Lemma 2.4. Therefore C LOEand consequently C LOE. It follows that
a a
Ei' Ei ( ECj) ' (Ei ECj) ' Ei ECi:
j j
` __
On the other hand, Ei ECi' ECisince Ci L"i. Thus Ei' ECifor all i. |__|
Proposition`2.6. Let C be a thick tensor-ideal`in mod_. Every decomposition C*
* =
iCi of C induces a decomposition EC = iEi such that Ei Ej = 0 for all i 6= *
*j and
ECi' Ei for all i.
Proof.We define Ei = ECi for all`i and observe that Ei Ej = 0 for all i 6= j by
Lemma`2.4. We claim that EC`' iEi. To this end consider the canonical`map
OE: iEi! k. Observe that C LOEsince C LOE. Therefore EC ' ( iEi) EC. On
the other hand, a a a
( Ei) EC ' (Ei EC) ' Ei
i ` i i
*
* __
since Ci C for all i. Thus EC ' iEi. *
*|__|
DECOMPOSING THICK SUBCATEGORIES 7
We are now in a position to prove the decomposition theorem for thick tensor-*
*ideals.
Proof`of Theorem A. Let C be a thick tensor-ideal in mod_ and fix the decomposi*
*tion
EC = i2IEi in Mod__ into indecomposable objects which exists by Theorem 2.1. *
*We
claim that EiEj = 0 for all i 6= j in I. In fact, for every i 2 I we obtain a d*
*ecomposition
a a
Ei' Ei ( Es) ' (Ei Es);
s2I s2I
and we have Ei ' Ei Et for precisely one t 2 I since Ei is indecomposable. The
same argument gives Et ' Et Ei, and therefore`i = t. Thus Ei Ej = 0 for all
i 6= j in I. We obtain a decomposition C = i2ICi from Proposition 2.5, and it*
* follows
from Proposition 2.6`that every Ci is indecomposable. Moreover, the uniqueness *
*of the
decomposition C = i2ICi(up to permutation`of the Ci) follows from the corresp*
*onding
uniqueness of the decomposition EC = i2IEi.
Suppose now that (Ci)i2Iis a family of thick tensor ideals in mod_ with Ci\ C*
*j = 0
for all i 6= j in I. We define C to be the full subcategory of all finite copr*
*oducts of
objects from the Ci. Clearly, C is a tensor-ideal. By Lemma 2.4, Hom__ (Ci; Cj)*
* = 0 for
all i 6= j. Therefore C is a`thick subcategory`of mod_ since any map OE: X ! Y *
*in C
has a decomposition (OEi): `iXi ! iYi with OEi 2 Ci for all i. It follows th*
*at C is_a
thick tensor-ideal with C = i2ICi. This finishes the proof of Theorem A. *
* |__|
` The following corollary summarizes the basic properties of a decomposition C *
* =
i2ICi.
`
Corollary 2.7.`Let C = i2ICi be a decomposition of a thick tensor-ideal in mo*
*d_.
Then EC ' i2IECi andQC is indecomposable if and only if EC is indecomposable.
Moreover, End_ (EC) ' i2IEnd_(ECi).
2.3. The centre of a localizing subcategory. Given an additive category A, we
denote by Z[A] the centre of A which is the endomorphism ring of the identity f*
*unctor
idA. More precisely, Z[A] is the ring of all natural transformation idA ! idA.*
* Note
that Z[A]`is a commutative ring. Suppose now that C is a thick tensor-ideal in *
*mod_
and let C be the localizing`subcategory which is generated by C. Then we hav*
*e the
following description of Z[C ].
`
Proposition 2.8. The map Z[C ] ! End_ (EC), OE 7! OEEC, is a ring isomorphism*
*. In
particular, End_ (EC) is a commutative ring.
`
Proof.We construct an inverse for Z[C ] ! End_ (EC). To this end consider`the *
*canon-
ical map ": EC ! k and recall from Proposition 2.3 that for every X in C the *
*map
"X = " X :E X ! X is an`isomorphism. If is an endomorphism of EC, then we
obtain for every X in C an endomorphism "X O( X) O"-1Xof X. In fact, the map
` -1
End_ (EC) -! Z[C ]; 7! ("X O( X) O"X )X
` *
* __
gives a ring homomorphism which is an inverse for Z[C ] ! End_ (EC). *
* |__|
`
Corollary 2.9. Z[C ] is a product of commutative local rings.
3. Decomposing Varieties
3.1. Varieties for finitely generated modules. Let H*(G; k) = Ext*(k; k) be the
cohomology ring of G over the field k, and we assume for simplicity that k is a*
*lge-
braically closed. We denote by VG (k) the maximal ideal spectrum of H*(G; k) a*
*nd
for every finitely generated -module X we denote by V (X) the corresponding clo*
*sed
8 HENNING KRAUSE
homogeneous subvariety of VG (k) (see [1, Chapter 5] for precise definitions). *
* Given a
closed homogeneous subvariety W of VG (k), the finitely generated -modules whose
variety is contained`in W form a thick tensor-ideal CW in mod_. We shall see t*
*hat the
decomposition CW =S iCi into indecomposable thick tensor-ideals corresponds t*
*o the
decomposition W = iWi of W into connected components, i.e. Ci= CWi for all i.*
* We
proceed in two steps.
S n
Proposition 3.1. Let W = i=1Wi be a union of closed homogeneous`subvarieties*
* of
VG (k) such that Wi\ Wj = {0} for all i 6= j. Then CW = ni=1CWi.
Proof.Clearly, CWi CW and CWi \ CWj = 0 for all i 6= j.SSuppose now that X 2 *
*CW
and we may assume that X is indecomposable. Then V (X) = ni=1(V (X) \ Wi) and
therefore V (X) = V (X) \ Wi for some i since the variety`of an indecomposable *
*module_
is connected [6]. Thus X 2 CWi and therefore CW = ni=1CWi. *
* |__|
Proposition 3.2.` Let W be a closed homogeneous subvariety of VG (k). EverySdec*
*om-
position CW = iCi into thick tensor-ideals induces a decomposition W = iWi*
* into
closed homogeneous subvarieties such that Wi\ Wj = {0} for all i 6= j and CWi =*
* Ci for
all i.
Proof.Assume first that Ciis indecomposable for all i. We can choose a finitely*
* generated
-module X such that W = V (X) (e.g., see [1, Corollary 5.9.2]) and we assume in
addition that the number`of indecomposable direct summandsSof X is minimal. The
decomposition CW = ` iCi gives a decomposition W = iWi if we define Wi = V (*
*Xi)
for all i where X`= iXiSdenotes the decomposition of X such that Xi 2 Ci for *
*all i.
In fact, W = V ( iXi) = iV (Xi) (e.g., see [1, Proposition 5.7.5]), and Wi\ *
*Wj = {0}
for all i 6= j since V (Xi) \ V (Xj) = V (Xi Xj) (e.g., see [1, Theorem 5.7.11]*
*) and
V (P ) = {0} for any projective -module P (e.g., see [1, Proposition 5.7.2]). N*
*ote that
CWi \ Ci 6= 0 for all`i, since 0 6= Xi 2 CWi \ Ci. We obtain from Proposition`*
*3.1 a
decomposition CW = iCWi, and the uniqueness of the decomposition CW = iCi
into indecomposable`thick tensor-ideals from Theorem A implies CWi = Ci for all*
* i. If
CW` =` iCi is an arbitrary`decomposition, then we can pass to a refinement CW *
* =
i( j2IiCij) with Ci= j2IiCijand Cijindecomposable for all i and j, by Theo*
*rem A.
UsingStheSassertion for indecomposable decompositions, one obtains a decomposit*
*ionS
W = i( j2IiWij) such that CWij = Cijfor all i; j. If we define Wi = j2IiW*
*ijfor
every i, then Proposition 3.1 gives
a a
CWi = CWij = Cij= Ci:
j2Ii j2Ii
*
* __
This finishes the proof. *
* |__|
Recall that a closed homogeneous subvariety W of VG (k) is projectively conne*
*cted if
any decomposition W = W1[ W2 into closed homogeneous subvarieties with W1\ W2 =
{0} implies W = W1 or W = W2. We obtain the following immediate consequences of
Proposition 3.1 and Proposition 3.2.
Corollary 3.3. For a closed homogeneous subvariety W of VG (k) the following co*
*ndi-
tions are equivalent:
(1) W is non-zero and projectively connected;
(2) CW is an indecomposable thick tensor-ideal in mod_;
(3) ECW is an indecomposable object in Mod__.
DECOMPOSING THICK SUBCATEGORIES 9
CorollaryS3.4. Let W be a closed homogeneous subvariety of VG (k), and let W =
n
i=1Wi be the`decompositionnof W into projectively connected non-zero componen*
*ts.
Then CW = i=1CWi is the decomposition of CW into indecomposable thick tenso*
*r ideals.
Moreover, the corresponding idempotent -module ECW has precisely n non-project*
*ive
indecomposable direct summands.
` We give now an example of a thick tensor-ideal C such that the decomposition *
*C =
i2ICi into indecomposable thick tensor-ideals is infinite. However, it is cl*
*ear that
cardI cardS whenever there is a set S of indecomposable -modules such that C is
the smallest thick tensor-ideal containing S.
Example 3.5. Let G = Z2 x Z2 be the Klein four group and chark = 2. The finite*
*ly
generated`-modules of complexity at most 1 form a thick tensor-ideal C, and C =
1
2P1(k)C where {C | 2 P (k)} is the set of indecomposable thick tensor-idea*
*l of
the form CVG*for some 0 6= i 2 H1(G; k), and VG ** denotes the closed homoge*
*nous
subvariety of maximal ideals containing i.
In [5], Benson and Gnacadja conjecture that for every closed homogeneous subv*
*ariety
W of VG (k) the module ECW is pure-injective. This conjecture becomes true pro*
*vided
one works in the appropriate category. Recall from [15] the following concept o*
*f purity
for a compactly generated triangulated category T . A map X ! Y in T is a pu*
*re
monomorphism if the induced map Hom (C; X) ! Hom (C; Y ) is a monomorphism for
every compact object C in T , and X is pure-injective if every pure monomorphism
X ! Y splits. For example, a -module X is pure-injective in Mod if and only if*
* X
is a pure-injective object in Mod__. The statement in part (3) of Theorem 1.2 *
*shows
that every endofinite object X in T is pure-injective. We conclude that for eve*
*ry thick
subcategory C of mod_ the corresponding -module EC is a pure-injective object i*
*n the
localizing subcategory of Mod__ which is generated by C.
3.2. Varieties for infinitely generated modules. In [3], Benson, Carlson, and R*
*ickard
generalize the definition of the variety V (X) of a finitely generated -module *
*X. To this
end they consider the set VG (k) of closed homogeneous irreducible non-zero sub*
*varieties
of the maximal ideal spectrum VG (k) of H*(G; k). Note that VG (k) can be iden*
*tified
with the projective prime ideal spectrum ProjH*(G; k). Under this bijection a u*
*nion of
Zariski-closed subsets of ProjH*(G; k) corresponds to a subset of VG (k) which *
*is closed
under specialization. Recall that a subset W of VG (k) is closed under speciali*
*zation if
W 2 W and V W implies V 2 W. It is clear that the subsets of VG (k) which are
closed under specialization form the closed subsets of a topology on VG (k). Gi*
*ven any
-module X, there is associated a subset V(X) of VG (k) which is closed under sp*
*ecial-
ization [3, Definition 10.2]. For example, V(X) = {V 2 VG (k) | V V (X)} if*
* X is
finitely generated. The following result is due to Benson, Carlson, and Rickard*
* [4]. It is
an analogue of Hopkins' classification of thick subcategories of the homotopy c*
*ategory
of unbounded complexes of finitely generated projective modules over a commutat*
*ive
noetherian ring [12].
Proposition 3.6. The map W 7! CW = {X 2 mod_ | V(X) W} defines a bijective
correspondence between the subsets of VG (k) which are closed under specializat*
*ion and
the thick tensor-ideals in mod_.
*
* __
Proof.See Theorem 3.4 in [4]. *
*|__|
Clearly, the map W 7! CW is inclusion preserving and identifies the project*
*ively
connected non-empty subsets of VG (k) with the indecomposable thick tensor-idea*
*ls in
10 HENNING KRAUSE
mod_. Therefore Theorem C is a direct consequence of Corollary 2.7 and the prec*
*eding
proposition.
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Fakult"at f"ur Mathematik, Universit"at Bielefeld, 33501 Bielefeld, Germany
E-mail address: henning@mathematik.uni-bielefeld.de
*