KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES SUNIL K. CHEBOLU Abstract.Following Krause [Kra99], we prove Krull-Schmidt theorems for thick subcategories of various triangulated categories: derived categori* *es of rings, noetherian stable homotopy categories, stable module categories o* *ver Hopf algebras, and the stable homotopy category of spectra. In all these* * cat- egories, it is shown that the thick ideals of small objects decompose un* *iquely into indecomposable thick ideals. Some consequences of these decompositi* *on results are also discussed. In particular, it is shown that all these de* *composi- tions respect K-theory. 1.Introduction Using ideas from modular representation theory of finite groups, Krause [Kra9* *9] proved a Krull-Schmidt theorem for thick subcategories of the stable module cat- egory. More precisely, he showed that the thick ideals of stmod(KG) decompose uniquely into indecomposable thick ideals. In this paper, we show that such de- compositions exist in various other stable homotopy categories like the derived categories of commutative rings and the stable homotopy category of spectra. To this end, we first generalise Krause's definition to arbitrary triangulated cat* *egories. Definition 1.1. [Kra99] Let T denote a tensor triangulated category. A thick subcategory A of T is a thick ideal if X ^ Y belongs to A for all X 2 A and all Y 2 T . If A is a thick (ideal) subcategory of T , a family of thick (ide* *al) subcategories (Ai)i2Iis a decomposition of A if (1) theTobjects of A are the finite coproducts of objects from the Ai, and (2) Ai Aj = 0 for all i 6= j. A decomposition (Ai)i2Iof A is denoted by A = qi2IAi, and we say that that A is indecomposable if A 6= 0 and any decomposition A = C q D implies that C = 0 or D = 0. A Krull-Schmidt decomposition of A is a unique decomposition A = qi2IAi where all the Ai are indecomposable. The above definition reminds one of the classical Krull-Schmidt theorem which states that any finite length module over an Artin ring admits a unique direct * *sum decomposition into indecomposable modules. Since all thick subcategories studied in this paper consist of finite objects in some triangulated category, it is re* *asonable to call these decompositions as Krull-Schmidt decompositions. ____________ Date: July 8, 2005. 2000 Mathematics Subject Classification. Primary:55p42; Secondary:18E30. Key words and phrases. Krull-Schmidt, tensor product property, stable homoto* *py theory, sta- ble module category, thick subcategory, derived category, Grothendieck groups. 1 2 SUNIL K. CHEBOLU We now discuss the main results in this paper. The first one is about the eff* *ect of K-theory of a thick subcategory under a Krull-Schmidt decomposition. Theorem 1.2. Let T denote a triangulated category and let C be a thick subcate- gory of T . If qi2ICiis a Krull-Schmidt decomposition of C, then under some mild conditions (see theorem 2.4), there is a natural isomorphism in K-theory, M K0(Ci)~= K0(C). i2I Our motivation for studying K-theory for thick subcategories comes from the problem of classifying the triangulated subcategories in a given triangulated c* *ate- gory. Thomason [Tho97 ] proved that if T is any triangulated category, then the subgroups of T are in bijection with the dense triangulated subcategories of T * *. (A triangulated subcategory A is dense in T if T is the smallest thick subcategory* * that contains A.) Since every triangulated subcategory A of T is dense in a unique t* *hick subcategory (namely, the one obtained by taking the intersection of all the thi* *ck subcategories of T that contain A), one can classify all the triangulated subca* *te- gories of T by computing the Grothendieck groups of all the thick subcategories of T . Towards this, the above theorem tells us that we can restrict ourselves* * to indecomposable thick subcategories. We use Landsburg's criterion (see lemma 2.1) and some general lemmas about triangulated categories which are developed in the Section 2 to prove the above theorem. It applies to all the decompositions we study in this paper. The next theorem deals with Krull-Schmidt decompositions in the aforemen- tioned categories. Theorem 1.3. The thick subcategories of finite spectra and those of perfect com- plexes in the derived category of a noetherian ring admit Krull-Schmidt decompo- sitions. Conversely,Tin both these cases, given any collection of thick subcate* *gories (Ci)i2Isuch that Ci Cj = 0 for all i 6= j, there exists a unique thick subcate* *gory C such that C = qi2ICi. In proving these decomposition theorems, we make good use of the thick sub- category theorems of Hopkins-Smith [HS98 ] and Hopkins-Neeman [Nee92]. In the derived category of a noetherian ring R, Hopkins-Neeman result states that the thick subcategories of perfect complexes are in bijection with the thick suppor* *ts (subset of Spec(R) that are a union of closed sets). Given a thick subcategory TS (corresponding to the thick support S under this bijection), we define a gra* *ph GS using the minimal prime ideals of S. It is shown that the indecomposable pieces that constitute TS in a Krull-Schmidt decomposition correspond precisely to the connected components of this graph. This decomposition theorem gives the following interesting algebraic result; see corollary 4.15 for a stronger resul* *t. Corollary 1.4. Let X be a perfect complex over a noetherian ring R. Then there exists a unique decomposition M X ~= Xi i2I such that the supports of the Xi are pairwise disjoint and indecomposable. We also generalise the above results on Krull-Schmidt decomposition to more general noetherian stable homotopy categories. We use the language and results KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES 3 from [HPS97 ] to achieve this generalisation. In particular, this generalisatio* *n gives Krull-Schmidt decompositions in some categories arising in modular representati* *on theory. We summaries these results in the next theorem. Theorem 1.5. Let B denote a finite dimensional graded co-commutative Hopf algebra satisfying the tensor product property (Examples: group algebras of fin* *ite groups and the finite dimensional sub-Hopf algebras of the mod 2 Steenrod algeb* *ra). Then, (1) Every thick ideal of small objects in K(Proj B) (the chain-homotopy cate- gory of graded projective B-modules) is indecomposable. (2) Every thick ideal of stmod(B) admits a Krull-Schmidt decomposition. As mentioned above, Krause [Kra99] proved that thick ideals of stmod(KG) (G a finite group) admit Krull-Schmidt decompositions. It is known [BCR96 ] that group algebras of finite groups satisfy the tensor product property and therefo* *re part 2 of the above theorem is a generalisation of Krause's result. The paper is organised as follows. In section 2, after developing necessary l* *emmas about triangulated categories and thick subcategories, we show that Krull-Schmi* *dt decompositions respect K-theory under some mild hypothesis. In the later sectio* *ns we study Krull-Schmidt decompositions in various stable homotopy categories: We review Krause's result for the stable module categories of group algebras in se* *ction 3, derived categories of rings in section 4, noetherian stable homotopy categor* *ies in section 5, stable module categories over Hopf algebras in section 6, and fin* *ally the stable homotopy category of spectra in section 7. We end the paper with a f* *ew questions which ask for further extensions of these decompositions. Acknowledgements: I would like to thank Henning Krause, Srikanth Iyengar, and John Palmieri for the many interesting and fruitful discussions I had with them on this subject. 2. K-theory for thick subcategories In this section we study how the K-theory of triangulated categories behaves * *un- der a Krull-Schmidt decomposition. We begin with some preliminaries on triangu- lated categories. Let T denote a triangulated category that is essentially smal* *l (i.e., a category that has only a set of isomorphism classes of objects). The Grothend* *ieck group K0(T ) is defined to be the free abelian group on the isomorphism classes* * of T modulo the Euler relations: [B] = [A] + [C], whenever A ! B ! C ! A is an exact triangle in T (here [X] denotes the element in the Grothendieck group tha* *t is represented by the isomorphism class of the object X). This is clearly an Abeli* *an group with [0] as the identity element and [ X] as the inverse of [X]. We always have the identity [A] + [B] = [A q B] in the Grothendieck group. Also note that any element of K0(T ) is of the form [X] for some X in T . Having defined K-theory, we now study how it behaves under a Krull-Schmidt de- composition. We begin with some lemmas that will be needed in studying K-theory. We will start with the following extremely useful Lemma due to Landsburg [Lan91* *]. This is a nice criterion for the equality of two classes in the Grothendieck gr* *oup. Lemma 2.1. [Lan91] Let T be an essentially small triangulated category. If X and Y are objects in T , then [X] = [Y ] in K0(T ) if and only if there are objects* * A, B, 4 SUNIL K. CHEBOLU C in T and maps such that there are exact triangles A -f!B q X -g! C -h! A, 0 g0 h0 A -f!B q Y -! C -! A. It is well-known that coproducts of exact triangles are exact in any triangul* *ated category; see [Mar83 , Appendix 2, Prop. 10]. Under some additional hypothesis, the next lemma gives a converse to this well-known fact. Lemma 2.2. Let T denote a triangulated category and let 0 A -f!B ! C ! A and A0-f! B0! C0! A0 0 be two sequences of maps in T such that their sum A q A0fqf-!B q B0! C q C0! (A A0) is an exact triangle. If we also have (1) Hom (Cone (f), C0) = 0 = Hom (Cone (f0), C) and (2) Hom (C0, Cone(f)) = 0 = Hom (C, Cone(f0)), then the given sequences of maps are exact triangles. Proof.Complete the maps f and f0 to exact triangles in T : 0 A -f!B ! Cone(f) ! A and A0-f! B0! Cone(f0) ! A0. Since the coproduct of exact triangles is exact, adding these two triangles giv* *es another triangle, 0 A q A0f-f!B q B0! Cone(f) q Cone(f0) ! (A A0). We know from the axioms for a triangulated category that there is a fill-in m* *ap H in the diagram below. fqf0 0 0 A q A0 ____//_B q B0___//Cone(f)_q Cone(f_)___// (A q A ) ___ | |=| =|| _H_____ |= fflffl|fqf0 fflffl| fflffl____ fflffl| A q A0 ____//_B q B0________//_C q C0_________// (A q A0) Note that three out of the four vertical maps in the above diagram are isomorph* *isms and therefore so is H; see [Mar83 , Appendix 2, Prop. 6]. Now the hypothesis Hom (Cone (f), C0) = Hom (Cone (f0), C) = 0 implies that H = h q h0. So we have h q h0: Cone(f) q Cone(f0) ! C q C0 is an isomorphism. The hypothesis Hom (C0, Cone(f)) = Hom (C, Cone(f0)) = 0 implies that the inverse G of this isomorphism is of the form g q g0. This forc* *es both h and h0 to be isomorphisms and hence C ~= Cone(f) and C0 ~=Cone(f0). Since exact triangles in T form a replete class, the two sequences of maps und* *er consideration are exact triangles. Lemma 2.3. Let C and D be thick ideals in T . If C \ D = 0, then C ^ D = 0. (C ^D is the full subcategory of objects of the form X ^Y where X 2 C and Y 2 D* *.) Proof.Clear, since C ^ D C \ D = 0. KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES 5 We are now ready to state and prove our main theorem on K-theory for thick subcategories. ` Theorem 2.4. Let A = Ai be a Krull-Schmidt decomposition of a thick (ideal) i2I subcategory A in a triangulated category T . If A is thick, assume that (1) hol* *ds; if A is a thick ideal, assume any one of the following three conditions. (1) Hom (Ai, Aj) = 0 for all i 6= j. (2) There exist an object Si in Ai such that Si^ X = X for all X 2 Ai. (3) There exists an object Si in T such that Si^ X = X for all X 2 Ai, and whenever j 6= i, Si^ X = 0 for all X 2 Aj. Then the inclusion functors Ai,! A give rise to an isomorphism M : K0(Ai) ~=K0(A). i2I Proof.The inclusion functors Ai ,! A induce mapsLon the Grothendieck groups which can be assembled to obtain the map : K0(Ai) ! K0(A) that we want i2I to show is an isomorphism. Showing that is surjective is easy: note that every element in K0(A) is of the form [X] for some X 2 A. Since the family of thick ideals`(Ai)i2I is a decomposition for A,`we can express X as a finite coproduct; X = Xi with Xi2 Ai.LThis gives [X] = [ Xi] = i2I[Xi]. The last quantity is clearly the image of [Xi] under the map and therefore is surjective. L To show that is injective,Lwe use Landsburg's`criterion 2.1.`Suppose ( [X* *i]) = 0. Then, as noted above, ( [Xi]) = [Xi] = [ Xi] and hence [ Xi] = [0]. This now gives, by Landsburg's criterion, two exact triangles in A: 0 1 a (3a) A ! B q @ XjA ! C ! A j2I (3b) A ! B ! C ! A L We want to show that [Xi] = 0 or equivalently [Xi] = 0 for all i 2 I.` First assume`that condition`(1) holds. Then consider the`decompositions A = Ai, B = Bi, and C = Ci (which exist because A = Ai ). Substituting these i2I decompositions in the above triangles (3a)(3b)gives a a a a Ai! (Biq Xi)! Ci! Ai, a a a a Ai! Bi! Ci! Ai. Now our assumption (1) together with lemma 2.2 will enable us to split these two triangles into exact triangles in Ai. So for each i 2 I, we get exact triangles Ai! Biq Xi! Ci! Ai, Ai! Bi! Ci! Ai. in Ai. This implies (by Landsburg's criterion) that [Xi] = 0 in K0(Ai). So is injective if condition (1) holds. 6 SUNIL K. CHEBOLU Now assume that A is a thick ideal and that for each fixed i 2 I either (2) o* *r (3) holds. Smash the above triangles (3a)(3b)with Sito get two exact triangles in T* * : 0 1 a A ^ Si! (B ^ Si) q @ (Xj^ Si)A ! C ^ Si! (A ^ Si) j2I A ^ Si! B ^ Si! C ^ Si! (A ^ Si) It is easily seen that these triangles are in fact triangles in Ai: This is tri* *vial if condition (2)`holds (because Ai are ideals and`Si 2 Ai). If condition (3) hold* *s, write A = Ai with Ai 2 Ai, then A ^ Si = (Ai^ Si) = Ai 2 Ai. Similarly B ^ Si and C ^ Si also`belong to Ai. Now we claim that (Xj ^ Si) = Xi. If (2) holds, then by the lemma 2.3, we j2I get Xj^ Si= 0 whenever i 6= j, and since Si is a unit for Ai, Xi^ Si= Xi. If (3) holds, this is obviously true. So in both cases (conditions (2) and (3)), the a* *bove triangles can be simplified to obtain the following triangles in Ai. A ^ Si! (B ^ Si) q Xi! C ^ Si! (A ^ Si), A ^ Si! B ^ Si! C ^ Si! (A ^ Si). This implies that [Xi] = 0 in K0(Ai). This show that is injective, completing* * the proof of the theorem. Here is another crucial lemma for studying Krull-Schmidt decompositions. Lemma 2.5. Let T be a triangulated category and let A and B be two thick (ideal) subcategories of T . If Hom (A, B) = 0 = Hom (B, A), then the full subcategory A q B, consisting of objects of the form A q B with A 2 A and B 2 B, is a thick (ideal) subcategory of T . Proof.The key observation here is that every map H : A q B ! A0q B0 in A q B is forced by the given hypothesis to be of the form f q g. It is very clear tha* *t A q B satisfies the ideal condition. To see that AqB satisfies the 2 out of 3 conditi* *on, start with a map H as above and complete it to a triangle. Then we have the following diagram where the rows are triangles in T . (The bottom row is the coproduct of two triangles in T .) fqg A q B ____//_A0q B0______//_Cone(f_q g)_____//_ (A q B) ___ | |=| |=| _H______ |= fflffl|fqg fflffl| fflffl___ fflffl| A q B ____//_A0q B0___//_Cone(f) q Cone(g)__//_ (A q A0) There exits a fill-in map H which turns out to be an isomorphism as before. The* *re- fore A q B is a triangulated subcategory. It remains to show thickness, i.e., A q B is closed under retractions. Consid* *er a retraction map e : A q B ! A q B (so e2 = e). Since e = a q b, the equation e2 * *= e implies (a q b)2 = a2 q b2 = a q b. This shows both a and b are retractions. So* * we are done. Having developed all the necessary tools, we now turn our attention to Krull- Schmidt decompositions for thick subcategories. We begin with Krause's decom- position result [Kra99] for the stable module category in the next section. KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES 7 3.Stable module categories over group algebras Consider the stable module category StMod (KG), where G is a finite group and K is some field. The objects of this category are the (right) KG-modules and morphisms are equivalence classes of KG-module homomorphisms where two homomorphisms are equivalent if and only if their difference factors through a projective module. This category is well-known to be a triangulated category and has a well-defined tensor product (ordinary tensor product of K-vector spaces w* *ith the diagonal G-action) which makes it into a tensor triangulated category. The * *full subcategory of small objects in StMod (KG) is equivalent to the full subcategory consisting of finitely generated KG-modules and is denoted by stmod(KG). The main theorem of [Kra99] then states: Theorem 3.1. [Kra99] Every thick ideal A in stmod(KG)`decomposes uniquely into indecomposable thick ideals (Ai)i2I; A = i2IAi. Conversely given thick ideals (Ai)i2Isuch`that Ai\ Aj = 0 for all i 6= j 2 I, there exists a thick ide* *al A such that A = i2IAi. We will briefly outline how Krause arrives at this decomposition. The key idea is to consider the Bousfield localisation with respect to the localising subcat* *egory Aq generated by A. The inclusion Aq ,! StMod (KG) has a right adjoint e : StMod (KG) ! Aq . In fact, e(M) is just the fibre of the Bousfield localisation map M ! MAq . Thus for each KG-module M, there is a natural triangle in StMod (KG) given by e(M) -ffl!M ! MAq ! e(M). Applying this right adjoint to the trivial module K gives a module e(K), which is denoted by EA . The KG- module EA associated to the thick ideal A in this way is an idempotent module (* *i.e., EA EA = EA ). (These modules were introduced by Rickard and they proved to be very useful objects in modular representation theory.) Krause then shows that this idempotent module is an endofinite object (see [Kra99, DefinitionL1.1]) in* * Aq and hence admits a splitting into indecomposable`modules: EA = i2IEi with Ei Ej = 0 for i 6= j. Let (ffli) : iEi! K be the decomposition of ffl : EA ! * *K and define Aito be the collection of all modules`X in stmod(KG) such that ffli X is* * an isomorphism. It then follows that A = iAi is the Krull-Schmidt decomposition for A. See [Kra99] for more details. We draw the following corollary by applying theorem 2.4 to the above decompo- sition. Proposition 3.2. Let A be a thick ideal of stmod(KG) and let qAi be the Krull- Schmidt decomposition of A. Then, M K0(A) ~= K0(Ai). i2I Proof.We show that condition (1) of our main theorem is satisfied. This is done in [Kra99, Lemma 2.4] but we include it here for the reader's convenience. For * *any two thick ideals A1 and A2 that occur in the decomposition for A, we want to sh* *ow that Hom (A1, A2) = 0. Towards this, consider objects X 2 A1 and Y 2 A2 and note that every map X ! Y in stmod(KG) factors through X Hom K(X, Y ) in the obvious way. So we will be done if we can show that Hom K(X, Y ) is a proje* *ctive KG-module or equivalently a trivial module in stmod(KG). To see this, first note that Hom K (X, Y ) ~=X* Y , where X* denotes the K-dual of X. But X* is a retract of X* X X* (see [HPS97 , Lemma A.2.6]) and therefore belongs to A1. 8 SUNIL K. CHEBOLU This implies that X* Y 2 A1 A2. The latter is zero by lemma 2.3 and therefore Hom K(X, Y ) = X* Y is zero in stmod(KG). 4. Derived categories of rings In this section we will prove a Krull-Schmidt theorem for the thick subcatego* *ries of perfect complexes over a noetherian ring. We work in some level of formality here so that we can generalise our results easily to noetherian stable homotopy categories in the sense of Hovey-Palmieri-Strickland [HPS97 ]. Recall that a subset S of Spec(R) is a thick support if it is a union of (Zar* *iski) closed sets Sffsuch that Spec(R) - Sffis a quasi-compact set. It is an exercise [AM69 , Page 12, Exc.17(vii)] to show that Spec(R)-Sffis quasi-compact if and o* *nly if Sff= V (Iff) for some finitely generated ideal Iff. (V (I) denotes the colle* *ction of all primes that contain I.) Since all ideals in a noetherian ring are finitely gene* *rated, it follows that the thick supports for noetherian rings are precisely subsets of S* *pec(R) that are a union of closed sets (also known as specialisation closed subsets). * *It is a celebrated theorem of Thomason that the thick subcategories of perfect complexes are completely determined by these thick supports. More precisely: Theorem 4.1. [Tho97 ] The lattice of thick subcategories in Db(projR) is isomor- phic to the lattice of thick supports of Spec(R). Under this isomorphism, a thi* *ck support S corresponds to the thick subcategory TS consisting of all complexes X suchSthat Supp(X) S, and a thick subcategory C corresponds to the thick suppo* *rt X2C Supp(x). Using this theorem, we now work our way to the Krull-Schmidt theorem for thick subcategories of perfect complexes. We begin with some lemmas. Lemma 4.2. For X in Db(projR), let DX = RHom (X, R) denote its Spanier- Whitehead dual. Then Supp(X) = Supp(DX). Proof.X is a retract of X DX X [HPS97 , Lemma A.2.6], and therefore DX is a retract of DX X DX. Now it is clear that both X and DX have the same support. Following [HPS97 ], we will denote Hom Db(projR)( *X, Y ) by [X, Y ]* and the internal function spectrum RHom (X, Y ) by F (X, Y ). With these notations, we have the following natural isomorphism [HPS97 ] [X A, B]* ~=[X, F (A, B)]*. This isomorphism gives the following useful lemma. Lemma 4.3. If X and Y are perfect complexes such that Supp(X) \ Supp(Y ) = ;, then [X, Y ]* = 0. In particular if A and B are any two disjoint thick supports* * of Spec(R), then [TA , TB ]* = 0. Proof.By Spanier-Whitehead duality, [X, Y ]* = [R X, Y*] = [R, F (X, Y )]*. Therefore [X, Y ]* = 0 if and only if F (X, Y ) = 0. But since X is a small ob- ject in D(R), F (X, Y ) = DX Y [HPS97 , Appendix A.2]. So we have to show that DX Y = 0. Since Supp(DX) = Supp(X) is disjoint with Supp(Y ), given any prime p, either p is not in Supp(DX) or it is not in Supp(Y ). In the former ca* *se, Rp DX = 0, and in the latter, Rp Y = 0. In either case, we get DX Y Rp = 0. Since p is an arbitrary prime, we get DX Y = 0. KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES 9 BySa thick decomposition of a thick support S, we will mean a decomposition S = Siinto non-empty thick supports, where Si\Sj = ; if i 6= j. A thick decom- position is Krull-Schmidt if the Si do not admit nontrivial thick decomposition* *s. Here are a few examples. Example 4.4. All rings considered are commutative. (1) If R is a PID, then every non-zero prime ideal is a maximal ideal and so* * the thick supports are: Spec(R), and all subsets of the set of maximal ideal* *s. In particular, Spec(R) is an indecomposable thick support. (2) If R is an Artinian ring, it has only finitely many prime ideals and eve* *ry prime ideal is also a maximal ideal. So everySnon-empty subset S of Spec* *(R) is a thick support. It is then clear that p2S{p} is the Krull-Schmidt decomposition for S. Note: The components of the Krull-Schmidt decomposition of a thick support S are not (in general) the connected components of S. For example, Spec(Z) - {0} is a connected subset of Spec(Z); however, it is not indecomposable as a thick suppo* *rt. In fact, [ Spec(Z) - {0} = {(p)} p prime is its Krull-Schmidt decomposition. Now we establish the strong connection between the decompositions of thick supports and the decompositions of thick subcategories. Proposition 4.5. Let C = A[B be a thick decomposition of a thick support. Then this induces a decomposition of the associated thick subcategories: TC = TA q TB . Proof.Clearly TA \ TB = 0 because A and B are disjoint by definition. We have to show that the objects of TC are coproducts of objects in TA and TB . We give* * an indirect proof of this statement using the thick subcategory theorem. Lemma 4.3 and lemma 2.5 tell us that TA q TB is a thick subcategory. So we will be done i* *f we can show that the thick support correspondingSto TA q TB is C. The thick support corresponding to TA q TB is given by Supp (x). This clearly contains both x2TAqTB A and B and hence their union (= C). To see the other inclusion, just note that Supp(a b) = Supp(a) [ Supp(b) A [ B = C. Now we show that decompositions of thick subcategories gives rise to thick de- compositions of the corresponding thick supports. Proposition 4.6. Let C = A q B be a decomposition of a thick subcategory C of Db(projR) and let A, B, and C denote the corresponding thick supports of these thick subcategories. Then C = A [ B is a thick decomposition of C. Proof.We first observe that the intersection of thick supports isSagain a thick support:SLet S and T be two thick supports of Spec(R). Then S = ffV (Iff) and T = fiV (Jfi) for some finitely generated ideals Iffand Jfiin R. Now, [ [ S \ T = (V (Iff) \ V (Jfi))= V (Iff+ Jfi). ff,fi ff,fi 10 SUNIL K. CHEBOLU Since the sum of two finitely generated ideals is finitely generated, we conclu* *de that S \ T is a thick support. We now argue that A and B are disjoint. If A and B are not disjoint, then the* *ir intersection being a thick support corresponds to a non-zero thick subcategory * *that is contained in both A and B. This contradicts the fact that A \ B = 0, so A and B have to be disjoint. It remains to show that A [ B is C. For this, note that * *every complex c 2 C splits as c = a q b and recall that Supp(c) = Supp(a) [ Supp(b). * *It follows that C = A [ B. Remark 4.7. It can be easily seen that the last proposition, together with lemm* *a 4.3, implies that the cancellation property holds for thick subcategories of Db(proj* *R). Combining proposition 4.5 and 4.6 we get the following decomposition result. Theorem 4.8. Let R be a commutative ring and let TS be the thick subcategory of Db(projR) corresponding to a thick support S. Then TS admits a Krull-Schmidt decomposition if and only if S admits one. Applying our main theorem 2.4 to the above decomposition gives, ` Corollary 4.9. LetSTS = i2ITSi be a decomposition corresponding to a thick decomposition S = i2ISi. Then, M K0(TS) ~= K0(TSi). i2I Proof.Condition (1) of our main theorem holds here by lemma 4.3. The question that remains to be addressed is the following. When do thick supports of Spec(R) admit Krull-Schmidt decompositions? We show that this is always possible if R is noetherian. Proposition 4.10. Let R be a noetherian ring and let S be a thickSsupport of Spec(R). Then there exists a unique Krull-Schmidt decomposition Si for S. Proof.It is well-known that the set of prime ideals in a noetherian ring satisf* *ies the descending chain condition [AM69 , Corollary 11.12]. (This condition is equival* *ent to saying that any non-empty collection of primes ideals has a minimal element.) To start, let S be a thick support in Spec(R) and let (pi)i2Ibe the collection * *of all minimal elements in S - i.e., primes p 2 S which do not contain any other prime* * in S. It is now clear (using the above fact about noetherianSrings) that every pri* *me p 2 S contains a minimal element pi2 S, therefore S = i2IV (pi). (Also note th* *at each V (pi) is a closed subset of Spec(R) and hence a thick support.) Now define the thick graph GS of S as follows: The vertices are the minimal primes (pi)i2I* *in S, and two vertices pi and pj are adjacent if and only if V (pi) \ V (pj) 6= ;.* * Let (Ck)k2K be the connected components of this graph and for each Ck define a thick support [ Sk := V (pi). pi2CkS By construction it is clear that Sk is a thick decomposition of S. It is not * *hard to see that each Sk is indecomposable. This is done by showing that any thick decomposition of Sk disconnects the connectedScomponent Ck of the thick graph of S. Finally the uniqueness part: let Tk be another Krull-Schmidt decomposition KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES 11 of S. It can be easily verified that the minimal primes in Tk are precisely the* * min- imalSprimes of S that are contained in Tk. Thus the Krull-Schmidt decomposition Tk gives a partition of the set of minimal primes in S. This partition induc* *es a decomposition of the thick graph of S into its connected (since each Tk is in* *de- composable) components. Since the decomposition of a graph into its connected components is unique, the uniqueness of Krull-Schmidt decomposition follows. Remark 4.11. The careful reader will perhaps note that the proof of proposition 4.10 makes use of only the following two conditions on R. (1) Every open subset of Spec(R) is compact. (2) Spec(R) satisfies the descending chain condition. Therefore the proof generalises to any R which satisfies these two properties. * *As mentioned above, it is well-known that noetherian rings satisfy these two prope* *rties. Moreover, any ring which has finitely many prime ideals automatically satisfies these properties. Note that there are non-noetherian rings which have finitely many primes. For example, the ring R = C[x2, x3, x4, . .].=(x22, x33, x44, . .). is a non-noetherian ring with only one prime ideal. So everything that we are g* *oing to state for the remainder of this section will work for rings which have these* * two properties, but we state our results only for noetherian rings for obvious cogn* *itive reasons. After all that work, the following theorem is now obvious. Theorem 4.12. If R is a noetherian ring, then every thick subcategory of Db(pro* *jR) admits a unique Krull-Schmidt decomposition. Conversely, given any collection of thick subcategories (Ci)i2Iin Db(projR) such`that Ci\ Cj = 0 for all i 6= j, th* *ere exists a thick subcategory C such that C = i2ICi. Proof.The first part follows from theorem 4.8 and proposition 4.10. For the sec* *ond part, define C be the full subcategory of all finite coproducts of objects from* * the Ci. Thickness of C follows by combining theorem 4.8, lemma 4.3, and lemma 2.5. Corollary 4.13. A noetherian ring R is local if and only if every thick subcate* *gory of Db(projR) is indecomposable. Proof.If R is local, then it is clear that every thick support of Spec(R) conta* *ins the unique maximal ideal. In particular, we cannot have two non-empty disjoint thick supports and therefore every thick subcategory of Db(projR) is indecomposable. Conversely, if R is not local, it is clear that the thick subcategory supported* * on the set of maximal ideals is decomposable. Theorem 4.12 also gives the following interesting algebraic results. Corollary 4.14. Let X be a perfect complex over a noetherian ring R. Then X admits a unique splitting into perfect complexes, M X ~= Xi i2I such that the supports of the Xi are pairwise disjoint and indecomposable. 12 SUNIL K. CHEBOLU S Proof.Let Si be the Krull-Schmidt decomposition of Supp(X) (which exists by proposition 4.10). It is clear from theorem 4.12 that X admits a splitting (X * *~= Xi) where Supp(Xi) Si. To see that we have equality in this inclusion, obser* *ve that _ ! M [ [ Supp(X) = Supp Xi = Supp (Xi) Si= Supp(X). i2I i2I i2I Uniqueness: If X admitsSanother decompositionS Yias above, then by proposition 4.10 we know that both Supp(Xi) and Supp(Yi) are the same decompositions of Supp(X). Lemma 4.3 then implies that Xi~= Yi for all i. So the decomposition is unique. Srikanth Iyengar pointed out that this splitting holds in a much more general* *ity. In fact it holds for all complexes in the derived category which have bounded a* *nd finite homology (i.e. complexes whose homology groups Hi(-) are finitely genera* *ted and are zero for all but finitely many i.) The full subcategory of such complex* *es will be denoted by Df(R). Corollary 4.15. Let R be a noetherian ring. Every complex X in Df(R) admits a splitting M X ~= Xi i2I such that the supports of the Xi are pairwise disjoint and indecomposable. Proof.Since X has bounded and finite homology, Supp (X) is a closed set and therefore proposition 4.10 applies and we get a decomposition [ Supp(X) = Si. i2I We now use a result of Neeman [Nee92] which states that the lattice of localisi* *ng subcategories (thick subcategories that are closed under arbitrary coproducts) * *is isomorphic to the lattice of all subsets of Spec(R). If L denotes the localisi* *ng` subcategory generated by X, then we get a Krull-Schmidt decomposition L ~= Li that corresponds to the above decomposition of the support of X. (This follows exactly as in the case of thick subcategories.) The splitting of X as stated in* * the corollary is now clear. Example 4.16. We illustrate the last two corollaries with some examples. (1) For every integer n > 1, let M(n) denote the Moore complex 0 ! Z -n! Z ! 0 and letSpi11pi22. .p.ikkbe the unique prime factorisation of n. It* * is easy to see that kt=1{(pt)} is the Krull-Schmidt decomposition of Supp(M(n)). Then it follows that Mk M(n) ~= M(pitt) t=1 is the splitting of M(n) corresponding to this Krull-Schmidt decompositi* *on. (2) Let R be a self-injective noetherian ring (hence also an ArtinianSring).* * Then it follows (because every prime ideal is also maximal) that p2Spec(R){* *p} KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES 13 is the Krull-Schmidt decomposition for Supp(R). It can be shown that M R ~= E(R=p) p2Spec(R) is the splitting of R corresponding to this Krull-Schmidt decomposition. (Here E(R=p) denotes the injective hull of R=p.) 5.Noetherian stable homotopy categories. Motivated by the work in the previous section, we now state a Krull-Schmidt theorem for noetherian stable homotopy categories. We use the language and resu* *lts of [HPS97 ] freely. We explain how the proofs of the previous section generali* *se to give us a Krull-Schmidt theorem in this more general setting by invoking the appropriate results from [HPS97 ]. We begin with some definitions and prelimina* *ries from Axiomatic Stable Homotopy Theory [HPS97 ]. Definition 5.1. [HP01 ] A unital algebraic stable homotopy category is a tensor triangulated category C with the following properties. (1) Arbitrary products and coproducts of objects in C exist. (2) C has a finite set G of weak generators: i.e., X ~=0 if and only if [A, * *X]* = 0 for all A 2 G. (3) The unit object S and the objects of G are small. C is a noetherian stable homotopy category if in addition the following conditi* *ons are satisfied, o ss*(S) := [S, S]* is commutative and noetherian as a bigraded ring. o For small objects Y and Z of C, [Y, Z]* is a finitely generated module o* *ver [S, S]*. The stable homotopy category of spectra is unital and algebraic but not noe- therian. The derived category D(R) of a commutative ring R is a noetherian stab* *le homotopy category if and only if R is noetherian (since [R, R]* = R). Spec(ss*S) will stand for the collection of all homogeneous prime ideals of ss*S with the * *Zariski topology. Henceforth C will denote a bigraded noetherian stable homotopy category. For each thick ideal A of finite objects in C, there is a finite localisation funct* *or LA (also denoted LfA) on C whose finite acyclics are precisely the objects of A; s* *ee [HP01 , Theorem 2.3]. For each bihomogeneous prime ideal p in ss*(S), there are finite localisation functors Lp and L

= . p2Spec(ss*S) This implies (DX ^ Y ) ^ S = DX ^ Y = 0. Proposition 4.5: This goes through verbatim without any changes. Proposition 4.6: This is actually much easier. Thick supports of Spec(ss*S) a* *re just unions of Zariski-closed sets and therefore it is obvious that intersectio* *n of thick supports is thick. The rest of the proof of this proposition follows exac* *tly the same way. Using these lemmas and proposition, theorem 4.8 and corollary 4.9 follow. Pro* *po- sition 4.10 also holds in the graded noetherian case; see lemma 5.3. Thus the f* *irst part of theorem follows by combining all these lemmas and propositions. The converse follows as before by combining theorem 4.8, lemma 4.3, and lemma 2.5. Corollary 5.5. Let C be a noetherian stable homotopy category satisfying the te* *nsor product property. Then every object X in C admits a unique decomposition a X ~= Xi, i2I such that the support varieties of the Xi are pairwise disjoint and indecomposa* *ble. Remark 5.6. The derived category D(R) of a noetherian ring R is a noetherian stable homotopy category which satisfies the tensor product property. (Note that ss*(S) = [R, R]* = R.) So theorem 5.2 recovers our Krull-Schmidt theorem for derived categories of noetherian rings. We now give another example of a noetherian stable homotopy category for which theorem 5.4 applies. Example 5.7. [HP01 , HP00] Let B denote a finite dimensional graded co-commutat* *ive Hopf algebra over a field k (char(k) > 0). Then the category K(Proj B), whose objects are unbounded chain complexes of projective B-modules and whose mor- phisms are chain homotopy classes of chain maps, is a noetherian stable homotopy category. In this category the unit object is a projective resolution of k by * *B- modules and therefore ss*(S) is isomorphic to Ext*,*B(k, k). It is well-known * *that this ext algebra is a bigraded noetherian algebra; see [FS97 ] and [Wil81]. In * *partic- ular, if B is a finite dimensional sub-Hopf algebra of the mod 2 Steenrod algeb* *ra, it is a deep result [HP01 , Corollary 8.6] that the category K(Proj B) satisfie* *s the tensor product property. So theorem 5.4 applies and we get the following result. Theorem 5.8. Let B denote a finite dimensional graded co-commutative Hopf algebra satisfying the tensor product property. Then every thick subcategory of* * small objects in K(Proj B) is indecomposable. Proof.The augmentation ideal of Ext*,*B(k, k) which consists of all elements in positive degrees is the unique maximal homogeneous ideal and therefore is conta* *ined in every thick support of SpecExt*,*B(k, k). Now by theorem 5.4 it follows that* * every thick subcategory of small objects in this category is indecomposable. 16 SUNIL K. CHEBOLU 6. Stable module categories over Hopf algebras In this section we give a generalised version of Krause's result [Kra99, The- orem A]. To this end, let B denote a finite dimensional graded co-commutative Hopf algebra over a field k. Following [HP01 ], we say that B satisfies the ten* *sor product if the category K(Proj B) does. We now prove a Krull-Schmidt theorem for stmod(B), whenever B satisfies the tensor product property. The category StMod (B) fails to be a noetherian stable homotopy category in general. (In fac* *t, ss*(S) = Hom stmod(B)(k, k) is isomorphic to the Tate cohomology of B which is * *not noetherian in general.) So we take the standard route of using a finite localis* *ation functor [HPS97 , Section 9] to get around this problem. We now review some preliminaries from [HPS97 , Section 9]. There is a finite localisation functor LfB: K(Proj B) ! K(Proj B), whose finite acyclics are pre- cisely the objects in the thick subcategory generated by B. It has been shown [HPS97 ] that the category StMod (B) is equivalent to the full subcategory of L* *fB- local objects. This finite localisation functor also establishes a bijection b* *etween the nonzero thick ideals of small objects in K(Proj B) and the thick ideals of stmod(B). In fact, the thick ideals of stmod(B) are in poset bijection with the non-empty specialisation closed subsets of Spec Ext*,*B(k, k). (Note that for * *the category K(Proj B), the empty set of Spec Ext*,*B(k, k) corresponds to the thick subcategory consisting of the zero chain complex.) Finally let m denote the uni* *que bihomogeneous maximal ideal of Ext*,*B(k, k) which corresponds to the trivial t* *hick subcategory in stmod(B). Then here is the main result of this section. Theorem 6.1. Let B denote a finite dimensional graded co-commutative Hopf algebra satisfying the tensor product property. Then every thick ideal of stmod* *(B) admits a unique Krull-Schmidt decomposition. Conversely, given thick ideals Ti * *in stmod(B)`satisfying Ti\ Tj = 0 for i 6= j, there exists a thick ideal T such t* *hat T ~= Ti. Moreover, M K0(T ) ~= K0(Ti). i2I Proof.Recall once again that the thick ideals of the stable module category are in poset bijection with the non-empty subsets of Spec Ext*,*B(k, k) that are cl* *osed under specialisation. To begin, let T be a thick ideal of stmod(B) and let S denote the corresponding specialisation closed subset. Since Ext*,*B(k, k)Sis a* * graded noetherian algebra, it follows that S admits a unique decomposition S = iSisuch that Si\ Sj = m for all i 6= j and Siindecomposable (the proof of Proposition 4* *.10 can be`adapted to the set S - m by making use of lemma 5.3). We now claim that T = TSi, where TSi denotes the thick ideal of stmod(B) corresponding to Si. The first thing to note is that for i 6= j, TSi\ TSj = 0: Suppose not, then t* *he non-zero thick ideal TSi\ TSj corresponds to a non-empty subset of Si\ Sj - m which is impossible. Therefore it follows that TSi\ TSj = 0. The next step is to show that the objects in T are the finite coproducts`of t* *he objects in TSi. We give an indirect proof (as before) by showing that TSi is a thick ideal.`Then the theoremSfollows because the specialisation`closed subset * *corre- sponding to TSiis clearly Si= S. To show that TSiis thick (ideal condition holds trivially), by lemma 2.5, all we need to show is that Hom stmod(A, B) = 0, whenever A 2 TSi and B 2 TSj (i 6= j). This was already done for group algebras KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES 17 in proposition 3.2. The proof given there can be easily generalised to finite d* *imen- sional co-commutative Hopf algebras. So that completes the proof of the existen* *ce of Krull-Schmidt decompositions. For the converse, define T be the full subcategory of all finite coproducts * *of objects in (Ti)i2Iand use the fact that Hom stmod(Ti, Tj) = 0 for i 6= j, to co* *mplete the argument. The statement about K0 groups follows from theorem 2.4. Remark 6.2. It is worth pointing out the difference between the decompositions in theorem 5.8 and those in theorem 6.1: Every thick ideal of small objects in K(Proj B) is indecomposable; on the other hand, thick ideals of small objects in StMod (B) admit Krull-Schmidt decompositions. This striking difference is re- ally the effect of Bousfield localisation (StMod (B) is a Bousfield localisatio* *n of Stable(B)). 6.1. Sub-Hopf algebras of the mod-2 Steenrod algebra. It has been shown in [HP01 ] that every finite dimensional sub-Hopf algebra of A (the mod 2 Steen* *rod algebra) satisfies the tensor product property. So theorem 6.1 applies and we g* *et interesting Krull-Schmidt decompositions in the stable module categories over t* *hese sub-Hopf algebras. We now look at some standard sub-Hopf algebras of A and analyse decompositions in their stable module categories. For every non-negative integer n, denote by A(n) the finite dimensional sub-Hopf algebra of A generated by {Sqk | 1 k 2n}. A(0) - This is the sub-Hopf algebra of A generated by Sq1 and therefore it is i* *so- morphic to the exterior algebra F2[Sq1]=(Sq1)2 on Sq1. It can be easily shown t* *hat Ext*,*A(0)(F2, F2) ~=F2[h0] with |h0| = (1, 1). Therefore the bihomogeneous pr* *ime spectrum of Ext*,*A(0)(F2, F2) is {(0), (h0)}. It is now clear from the thick s* *ubcat- egory theorem that there are only two thick subcategories of stmod(A(0)); the trivial category which corresponds to maximal prime (h0) and the entire category stmod(A(0)) which corresponds to {(0), (h0)}. These subsets of prime ideals are clearly indecomposable closed subsets and therefore we conclude by theorem 6.1 that all thick subcategories of stmod(A(0)) are indecomposable. A(1) - This is the sub-Hopf algebra of A generated by Sq1and Sq2. This algebra * *can be shown (using the Adem relations) to be isomorphic to the free non-commutative algebra (over F2) generated by symbols Sq1and Sq2modulo the two sided ideal (Sq1 Sq1, Sq2Sq2+ Sq1Sq2Sq1). The cohomology algebra of A(1) has been computed in [Wil81]: Ext*,*A(1)(F2, F2) ~=F2[h0, h1, w, v]=(h0h1, h31, h1w, w2 - h20v), where the bi-degrees are: |h0| = (1, 1), |h1| = (1, 2), |w| = (3, 7), and |v| =* * (4, 12). The bihomogeneous prime lattice of the above cohomology ring consists of the minimal prime (h1); primes p0 = (h1, w, h0) and p2 = (h1, w, v) of height one; * *and the maximal prime m = (h1, w, h0, v). The thick subcategory theorem now tells us that there are five thick subcate- gories in stmod(A(1)) which correspond to the thick supports {m}, {p0, m}, {p1,* * m}, 18 SUNIL K. CHEBOLU m E yy EEE yyy EEE yyy E p0 p1 CC -- CCC --- CCC --- (h1) Figure 1. Bihomogeneous prime lattice of Ext*,*A(1)(F2, F2) {p0, p1, m}, and {(h1), p0, p1, m}. It is clear that the only decomposable thic* *k sup- port in this list is {p0, p1, m}; in fact, {p0, p1, m} = {p0, m} [ {p1, m} is the desired decomposition. This induces a Krull-Schmidt decomposition of thi* *ck subcategories supported on these thick supports: T{p0,p1,m}~=T{p0,m}q T{p1,m}. Note that the remaining four thick subcategories in stmod(A(1)) are indecompos- able. Decompositions in stable module categories over A(2) and higher are much more complicated. 7. Finite spectra In this section we work in the triangulated category F of finite spectra. Rec* *all that a spectrum X is torsion if ss*(X) is a torsion group. In particular, a fi* *nite spectrum X is torsion (p-torsion) if ss*(X) consists of finite abelian groups (* *p- groups) in every degree. We will now show that the thick subcategories of finite spectra admit Krull-Schmidt decompositions. Ftorwill denote the category of finite torsion spectra, and Cn,pwill denote t* *he thick subcategory of finite p-torsion spectra consisting of all K(n - 1)* acycl* *ics Lemma 7.1. If A is a thick subcategory of F, then either A = F or A Ftor. Proof.It suffices to show that if X is any non-torsion finite spectrum, then th* *e inte- gral sphere spectrum S belongs to the thick subcategory generated by X. Consider the cofibre sequence X -p! X ! X ^ M(p) ! X. It is easy to see that X ^M(p) is a type-1 spectrum, therefore the thick subcat* *egory generated by X contains all finite torsion spectra. The next step is to show th* *at there is a cofibre sequence a a Sr ! X ! T ! Sr+1, where T is a finite torsion spectrum; see [Mar83 , Proposition 8.9]. It is clea* *r from these exact triangles that S belongs to the thick subcategory generated by X. We now start analysing decompositions for thick subcategories of finite spect* *ra. KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES 19 Proposition 7.2. The category F is indecomposable. Proof.If F = A q B is any decomposition of F, then that would giving a splitting A _ B of the sphere spectrum (S). Since S is an indecomposable spectrum, either A or B must be the sphere spectrum, which implies that either A or B is equal to F. We now show that the thick subcategories of finite torsion spectra admit non- trivial Krull-Schmidt decompositions. Theorem 7.3. Every thick subcategory D of Ftoradmits a unique Krull-Schmidt decomposition. Conversely, given any collection of thick subcategories Di in Ft* *or` such that Di\ Dj = 0 for all i 6= j, there exits a thick subcategory D such tha* *t Di L * * i is a Krull-Schmidt decomposition for D. Moreover K0(D) ~= iK0(Di). Proof.For every prime number p, define Dp := D \ C1,p.WBy [Mar83 , 8.2, Theorem 20] every torsion spectrum X can be written as X = pX(p)where X(p)is the p-localisation of X. If X is a finite torsion spectrum, it is easy to see that* * X(p) is a finite p-torsion spectrum and therefore belongs to C1,p( Ftor). Since D i* *s a thick subcategory of Ftor, all the summands X(p)also belong to D. Therefore X(p) belongs to C1,p\ D = Dp. Also since X is a finite spectrum, the above wedge runs over only a finite set of primes. This implies that the spectra in D are the fi* *nite coproducts of spectra in (Dp). Dp is a thick subcategory in Ftor: Showing the C1,pis a thick subcategory of * *Ftor is a routine verification. Since intersection of thick subcategories is thick, * *D \ C1,p is thick in Ftor. Dp\ Dq = 0 : It is clear that if a finite spectrum is both p-torsion and q-to* *rsion, for primes p 6= q, then it is trivial - i.e., C1,p\ C1,q= 0. But Dp \ Dq C1,p* *\ C1,q, therefore Dp \ Dq = 0. Dp is indecomposable: Dp is by definition a thick subcategory of C1,p. The th* *ick subcategories of C1,pare well-known to be nested [HS98 ]. In particular they ar* *e all indecomposable. This completes the proof of the existence of a Krull-Schmidt decomposition for any thick subcategory`D in Ftor. Uniqueness: Let Ek be another Krull-Schmidt decomposition for D. Then since each Ek is an indecomposable thick subcategory of Ftor, we conclude that Ek = Cnk,pkfor some integer nk and some prime pk. Thus the decomposition is unique. For the converse, define D be the full subcategory of all finite coproducts of objects from Dp. We just have to show that D is a thick subcategory and then the rest follows. Note that [X, Y ] = 0 whenever X is a p-torsion spectrum and Y is* * a q-torsion spectrum [Mar83 , 8.2, Lemma 21]. Therefore thickness of D follows fr* *om lemma 2.5. The following corollary is clear from these decompositions. Corollary`7.4. Let A be be a thick subcategory of F. Then either A = F, or A = Ci,p, where i ranges over some subset of positive integers and p over some subset of primes. 20 SUNIL K. CHEBOLU 8. Questions 8.1. Krull-Schmidt decompositions in Db(projR). We have shown that the thick subcategories of perfect complexes over a noetherian ring admit Krull-Sch* *midt decompositions. Note that proposition 4.10 was crucial for this decomposition. * *Now the question that remains to be answered is whether this proposition holds for * *ar- bitrary commutative rings. In other words, if R is any commutative ring, is it * *true that every thick support of Spec(R) admits a unique Krull-Schmidt decomposition? An affirmative answer to this question would imply, by theorem 4.8, that thick * *sub- categories of perfect complexes over commutative rings (not necessarily noether* *ian) admit Krull-Schmidt decompositions. 8.2. The tensor product property. Recall that our decompositions for the thick subcategories of small objects in K(Proj B) and StMod(B) assumed that the finite dimensional co-commutative Hopf algebra B satisfies the tensor product property. Now the question that arises is which finite dimensional Hopf algebras satisfy * *the tensor product property? This seems to be a question of independent interest; s* *ee [HP01 , Corollary 3.7]. Friedlander and Pevtsova [FP04 ] have shown recently th* *at the finite dimensional co-commutative ungraded Hopf algebras satisfy the tensor product property. Using their ideas, or otherwise, we would like to know if th* *is property also holds for finite dimensional graded co-commutative Hopf algebras. Also we do not know if the tensor product product property in [FP04 ] is equiva* *lent to the one given in [HP01 ]. References [AM69] M. F. Atiyah and I. G. Macdonald. Introduction to commutative algebra. A* *ddison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. [BCR96]D. J. Benson, Jon F. Carlson, and J. Rickard. Complexity and varieties f* *or infinitely generated modules. II. Math. Proc. Cambridge Philos. Soc., 120(4):597-61* *5, 1996. [FP04] Eric M. Friedlander and Julia Pevtsova. ss-supports for modules for fini* *te group schemes over a field. Preprint, 2004. [FS97] Eric M. Friedlander and Andrei Suslin. Cohomology of finite group scheme* *s over a field. Invent. Math., 127(2):209-270, 1997. [HP00] Mark Hovey and John H. Palmieri. Galois theory of thick subcategories in* * modular representation theory. J. Algebra, 230(2):713-729, 2000. [HP01] Mark Hovey and John H. Palmieri. Stably thick subcategories of modules o* *ver Hopf algebras. Math. Proc. Cambridge Philos. Soc., 130(3):441-474, 2001. [HPS97]Mark Hovey, John H. Palmieri, and Neil P. Strickland. Axiomatic stable h* *omotopy theory. Mem. Amer. Math. Soc., 128(610):x+114, 1997. [HS98] Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy * *theory. II. Ann. of Math. (2), 148(1):1-49, 1998. [Kra99]Henning Krause. Decomposing thick subcategories of the stable module cat* *egory. Math. Ann., 313(1):95-108, 1999. [Lan91]Steven E. Landsburg. K-theory and patching for categories of complexes. * *Duke Math. J., 62(2):359-384, 1991. [Mar83]H. R. Margolis. Spectra and the Steenrod algebra, volume 29 of North-Hol* *land Mathe- matical Library. North-Holland Publishing Co., Amsterdam, 1983. [Mit85]Stephen A. Mitchell. Finite complexes with A(n)-free cohomology. Topolog* *y, 24(2):227- 246, 1985. [Nee92]Amnon Neeman. The chromatic tower for D(R). Topology, 31(3):519-532, 199* *2. [Tho97]R. W. Thomason. The classification of triangulated subcategories. Compos* *itio Math., 105(1):1-27, 1997. [Wil81]Clarence Wilkerson. The cohomology algebras of finite-dimensional Hopf a* *lgebras. Trans. Amer. Math. Soc., 264(1):137-150, 1981. KRULL-SCHMIDT DECOMPOSITIONS FOR THICK SUBCATEGORIES 21 Department of Mathematics, University of Western Ontario, London, ON. N6A 5B7 E-mail address: schebolu@uwo.ca