THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF R-MODULES KEIR H. LOCKRIDGE Abstract.In this paper, we prove a version of Freyd's generating hypothe* *sis for trian- gulated categories: if D is a cocomplete triangulated category and S 2 D* * is an object whose endomorphism ring is graded commutative and concentrated in degree* * zero, then S generates (in the sense of Freyd) the thick subcategory determined by * *S if and only if the endomorphism ring of S is von Neumann regular. As a corollary, we* * obtain that the generating hypothesis is true in the derived category of a commutati* *ve ring R if and only if R is von Neumann regular. We also investigate alternative formul* *ations of the generating hypothesis in the derived category. Finally, we give a charac* *terization of the Noetherian stable homotopy categories in which the generating hypothesis* * is true. Contents 1. Introduction 1 2. Generalities and Criteria 4 3. Von Neumann Regular Rings 7 4. Variations 11 5. Noetherian Stable Homotopy 14 References 15 1.Introduction In his 1966 paper Stable Homotopy, Freyd introduces the following conjectu* *re: Conjecture 1.1 (The Generating Hypothesis). If f : X -! Y is a morphism of fini* *te spectra and ss*f = 0, then f is trivial. The conjecture remains open, but progress has been made using the methodology s* *ur- rounding the Nilpotence Theorem of Devinatz, Hopkins, and Smith ([4], [8]). Let* * L1(-) ___________ Date: October 16, 2005. 2000 Mathematics Subject Classification. Primary: 55p42; Secondary: 18e30. Key words and phrases. generating hypothesis, derived category, von Neumann r* *egular. 1 2 KEIR H. LOCKRIDGE denote E(1)-localization. In [3], Devinatz proves the E(1)-approximate generat* *ing hy- pothesis when the target spectrum is the sphere: if ss*f = 0, then L1f ' 0. An * *axiomatic approach to stable homotopy theory has led to the study of other triangulated c* *ategories from a homotopy theoretic point of view ([10]), and a natural extension of this* * study is to try to formulate and prove the generating hypothesis in these structurally simi* *lar settings. Examples of general stable homotopy categories include localizations of the usu* *al stable category, the derived category of a ring, the stable module category (arising i* *n represen- tation theory), the homotopy category of complexes of injective comodules over * *a Hopf algebra (e.g., the Steenrod algebra), among others. It is not completely clear how the generating hypothesis should be stated * *in general stable categories; one can leave the statement as it is and get something that * *makes perfect sense, but there are several ways of characterizing the finite spectra from an * *axiomatic point of view that do not always coincide in general. Difficulties arise, for e* *xample, when the sphere is not a weak generator; i.e., ss*X = 0 does not necessarily imply t* *hat X is trivial. If such a finite X exists, then GH would clearly be false (an example * *will be given later: the derived category of quasi-coherent sheaves over P1). One might ins* *ist that [X, f] = 0 for each weak generator X. We now give two relevant definitions and formulate the version of GH with * *which we will work. Let D be a triangulated category, and let S 2 D be a distinguishe* *d object. The thick subcategory generated by S, thick, is the smallest class of object* *s in D that contains S and is closed under suspension, retraction, and cofiber sequences. I* *n the usual stable category, this is exactly the finite spectra; in general,`it may not des* *cribe the class of small objects. An object Y is small if, whenever the coproduct ffXffexists, t* *he natural map L ` [Y, Xff]___//[Y, Xff] ff ff is an isomorphism. Note that 'small' and 'finite' have the same meaning in the * *usual stable category. Definition 1.2. Let D be a triangulated category and let S 2 D be a distinguish* *ed object. Write ss*(-) for the functor Hom *D(S, -). The generating hypothesis is the sta* *tement: If f : X -! Y is a morphism of objects in thickand ss*f = 0, then f ' 0. In a stable homotopy category, S will be the sphere object. In the derived* * category of a ring, for example, the sphere is the chain complex with R in degree zero a* *nd zero elsewhere. The following theorem is our main result. Theorem A. Let D be a triangulated category with arbitrary coproducts, and let * *S be an object in D such that R = ss*S is commutative and concentrated in degree zero. * * GH is true in D if and only if R is von Neumann Regular. A ring R is von Neumann regular if, for every element x 2 R, there exists an el* *ement y 2 R such that xyx = x. This class of rings, in the context of noncommutative ring t* *heory, was THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF R-MODULES 3 originally introduced by von Neumann to study operator algebras in functional a* *nalysis. Theorem A has the following corollary. Corollary 1.3. Let R be a commutative ring. GH is true in the derived category* * of R- modules if and only if R is von Neumann regular. The main result actually applies to any triangulated category where ss*S i* *s graded commutative and concentrated in even degrees. If ss*S is graded commutative an* *d not concentrated in degree zero, we prove that if ss*S is nonzero in only finitely * *many degrees or if ss*S connective and concentrated in even degrees, then the generating hyp* *othesis is necessarily false. We also give an application of Theorem A to the derived cat* *egory of quasicoherent sheaves over certain schemes in x3. There is a more abstract way to view the content of Theorem A. An analysis* * of the proof shows that, for categories covered by the main result, the generating* * hypothesis is true if and only if the category is trivial in the following sense: thickmust be exactly the collection of retracts of finite wedges of suspensions of S. In ord* *inary stable homotopy, this condition would imply that every finite spectrum is a wedge of s* *uspensions of the sphere (the main theorem, of course, does not apply in this case). We co* *njecture that this characterization is valid for the derived category of any Grothendiec* *k abelian category. Note that S is not of this form - for any object X in the derived cat* *egory of an abelian category, the cofiber Y of 2 : X -! X has the property that 2 : Y - * *! Y is trivial. However, 2 : M(2) -! M(2) is nontrivial in S, where M(2) is the mod 2 * *Moore spectrum (I learned this from Neil Strickland). Unless otherwise indicated, for the remainder of the paper we consider a t* *riangulated category D and an object S 2 D which has the property that its endomorphism ring R = ss*S = [S, S]* is commutative and concentrated in degree zero. We further a* *ssume that D has arbitrary coproducts; this guarantees that idempotents split in D ([* *2, 3.2]). This paper is organized as follows. In x2, we develop two important neces* *sary conditions for GH to hold: the Nilpotence Criterion (2.5) and the Annihilator C* *riterion (2.7). In x3, we introduce (von Neumann) regular rings and show that the two c* *riteria from x2 are equivalent to regularity. We then show that regularity is also suff* *icient. In x4, we explore variants of GH by changing its domain of definition. In D(R), for ex* *ample, we study GH when stated for maps of objects in thick, where I is a finitely g* *enerated ideal of R. In x5, we show that GH holds in a Noetherian stable homotopy catego* *ry if and only if thickis exactly the collection of retracts of finite wedges of suspe* *nsions of S. I would like to thank my advisor, Ethan Devinatz, for many helpful convers* *ations regarding the content and preparation of this paper. Further, I wish to acknow* *ledge the contribution of the referee, who suggested the given proof of Proposition 3* *.6. The original proof did not apply to noncommutative regular rings, and in its presen* *t form, the proposition allows us to make Remark 3.9. The referee also suggested the inclu* *sion of Proposition 5.3 which is used to establish the noncommutative aspect of Corolla* *ry 5.5. 4 KEIR H. LOCKRIDGE 2. Generalities and Criteria By way of motivation, let us consider the derived category of a commutativ* *e ring D(R), where S is the chain complex with R concentrated in degree zero. D(R) is* * a monogenic stable homotopy category ([10, 9.3.1]). If R is a field, then every * *object in D(R) is equivalent to a wedge of suspensions of the sphere; therefore, GH is tr* *ivially true. The following proposition gives an apposite connection between GH and direct pr* *oducts. Proposition 2.1. Suppose R ~=R1 x R2. GH is true in D(R) if and only if it is * *true in D(R1) and D(R2). Proof.This is essentially a consequence of the fact that every R-module is the * *direct sum of an R1-module and an R2-module and every R-module map decomposes similarly. Remark 2.2. Hence, GH is true in D(R) whenever R is a finite product of fields* *. In particular, GH is true for Z=(n), provided n is square free. This condition is * *also necessary. In [11] we prove that if GH is true in a category where S is connective and ss0* *S is projective- free (meaning that every projective ss0S module is free), then ss*S is either a* * field or totally non-coherent. A graded ring is totally non-coherent if no proper, nonze* *ro, finitely generated ideal is finitely presented. Hence, GH is false for Z=(pn)when n > 1 * *since Z=(pn) is local and therefore projective-free but obviously neither a field nor totall* *y non-coherent. We also give a specific counterexample in Example 2.4. The next proposition is based upon Devinatz' approach to GH in [3], where * *he proves the E(1)-approximate version when the target is the sphere. The Spanier-Whiteh* *ead duality pairing DX^X _____//Sinduces a map (1) ss-i(DX) R ssi(X)___//_R. Let M be an R-module. After tensoring with M, we may examine the adjoint map (2) ss-i(DX) R M ____//_HomR(ssi(X), M). If M is both flat and injective, then this is a natural transformation of cohom* *ology theories. Since it is an isomorphism when X = S, it is an isomorphism for all X 2 thick. Now suppose there is an injection R ____//_Mand M=R is flat. Then ss-i(DX)_____//ss-i(DX) R M is also injective since the failure of this map to be injective is measured by Tor1R(ss-i(DX), M=R), which is trivial. Now consider a degree zero map f : X -! S where X 2 thick.* * The map f corresponds to a map f0 : S -! DX. If f is nontrivial, then so is f0; f0 * *is also nontrivial in ss0(DX) R M. Since (2) is an isomorphism and the pairing (1) cor* *responds to composition, there is a map g : S -! X such that fg is nontrivial. Hence, ss* **f 6= 0. We have proved: THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF R-MODULES 5 Proposition 2.3. If there is an R-module M that is flat and injective and if th* *ere is an injective map R ____//_Msuch that M=R is flat, then GH for the target spectrum* * S is true in D(R). For example, if R is a von Neumann regular ring (see x3), then every R-module i* *s flat ([13, 4.2.9]); taking M to be the injective hull of R, we obtain that GH for th* *e target S is true in D(R). Also, if R is self injective, then GH for the target S is true in* * D(R). As we will see in the next example, however, there are self-injective rings R for whi* *ch the general form of GH is false in D(R). Example 2.4. Z=(n) is self injective for all n. Hence, GH for the target S is* * true in D(Z=(p2)). We have observed, however, that GH is false in D(Z=(p2)) in general.* * Here is a counterexample: consider the map of chain complexes 0OO 0OO | | | | | p | Z=(p2)____//_Z=(p2)OOOO |p| p|| | 0 | Z=(p2)____//_Z=(p2)OOOO | | | | | | 0 0 f X ________//X. Though ss*f = 0, f is nontrivial; a null-chain-homotopy of f would correspond t* *o an element s 2 Z=(p2)such that p = ps and ps = 0, implying p = 0. Our next proposition gives a general reason for the failure of GH in the a* *bove ex- ample. In this proposition, we return to the general situation, where D is a tr* *iangulated category with distinguished object S such that ss*S is commutative and concentr* *ated in degree zero. Proposition 2.5 (Nilpotence Criterion). If GH is true in D, then R contains no * *nonzero nilpotent elements. Proof.For now, we do not assume that R is commutative. Let g 2 R be nonzero nil* *potent; replacing g with some power if necessary, we may assume g2 = 0. Consider the fo* *llowing commutative diagram, where the rows are cofiber sequences. The map h is any map* * that 6 KEIR H. LOCKRIDGE makes the diagram commute. g OE ffi S ____//_S___//Y____//O S |0| |g| OhO |0| fflffl|fflffl|fflfflg|fflfflOEffi S ____//_S___//Y____// S Notice that h factors through both S and S; since ss*S is concentrated in one * *degree, we must have ss*h = 0. Hence, any h that fills in the above diagram of cofiber seq* *uences must be trivial by GH . Now, since g2 = 0, g = fOE. One may take h to be OEf, and co* *nsequently OEf ' 0. Hence, f = gk. Let x = kOE; then, g = fOE = gx and xg = kOEg = 0. If g* * lies in the center of R, then we have a contradiction. In particular, if R is commutati* *ve, then it contains no nontrivial nilpotent elements. Remark 2.6. For this remark, assume ss*S is graded commutative but not necessar* *ily concentrated in degree zero. If the map g in the proof has degree k, then one o* *btains that h factors through both -kS and k+1S. Hence, the Nilpotence Criterion holds wh* *enever ss*S is concentrated in even degrees. Notice also that if GH is true, then a no* *nzero element g of degree k cannot be nilpotent if ss*S is zero outside a range of 2k+1 conse* *cutive degrees. In particular, if ssiS is zero for all but finitely many i and is not concentra* *ted in degree zero, then GH is false. In [5], Freyd proves that if the generating hypothesis is true in the stab* *le category of spectra, then ann ann(x) = (x) for all x 2 ss*S. We now give a generalizati* *on of this result for D. We have so far assumed that ss*S is commutative and concentr* *ated in degree zero; the following proposition is true without these assumptions. We wr* *ite (x)L for the left ideal generated by x and (x)R for the right ideal generated by x. * * We also define the right annihilator of a left ideal IL and the left annihilator of a r* *ight ideal IR by Rann IL = {x 2 R | ix = 0 for alli 2 IL}, and Lann IR = {x 2 R | xi = 0 for all* *i 2 IR}. Rann IL is a right ideal and Lann IR is a left ideal. Proposition 2.7 (Annihilator Criterion). If GH is true in D, then for any f 2* * R, LannRann (f)L = (f)L. Proof.Let k be the degree of f. The containment (f)L LannRann (f)L is always * *true. Take any g 2 LannRann (f)L. Consider the cofiber sequence f OE ffi kS ____//_S___//_Y___//_ k+1S. Since imss*ffi = kerss*f = Rann(f)L, we have that ss*(gffi) = 0. By GH , gffi '* * 0. Hence, g = _f and so (f)L = LannRann (f)L as desired. Remark 2.8. One could just as easily prove that Rann Lann(f)R = (f)R. THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF R-MODULES 7 Example 2.9. Using the Annihilator Criterion, we give a family of examples of t* *otally non-coherent local (and hence projective-free) rings for which GH in D(R) is st* *ill false. Let k be a field and consider R = k[x1, x2, . .].=(xixj, i 6= j, xni, i 1) or k[x* *1, x2, . .].=(xixj, i 6= j). R is totally non-coherent (exercise). However, (x1) 6 (x1+ x2) though ann(* *x1+ x2) ann(x1). The above two necessary conditions, as we shall see, are also sufficient. 3.Von Neumann Regular Rings There is a general term for a ring that satisfies both the Annihilator Cri* *terion and the Nilpotence Criterion. A ring (not necessarily commutative) R is (von Neuma* *nn) regular if, for every x 2 R, there exists y 2 R such that xyx = x. For commuta* *tive rings, this is the same as the requirement that (x2) = (x) for all x 2 R, which* * is in turn equivalent to the condition that every principal ideal be generated by an idemp* *otent: if there exists an element s 2 R such that sx2 = x, then (x) = (sx) and sx is idem* *potent. [6] is a general reference for rings of this type. One can say something concre* *te about the class of commutative regular rings. The following two propositions must be well* * known; we include proofs for the reader's convenience. Proposition 3.1. If R is commutative von Neumann regular, then it is a subring * *of a direct product of fields. Proof.First, we show that every prime ideal of R is maximal. Let P R be a pr* *ime ideal. For any x 2 R, there is some y 2 R such that x2y = x by regularity. He* *nce, x(xy - 1) = 0. Since R=P is an integral domain, either x is zero mod P or xy * *1 mod P . Therefore R=P is a field and P is a maximal ideal. Next, we observe that the ni* *lradical N (R) of R is trivial: N (R) cannot contain nonzero idempotent elements. If e 2* * N (R) is idempotent, then since e is contained in every prime ideal, 1 - e is contain* *ed in no prime ideal; therefore, 1 - e is a unit. But since e(1 - e) = 0, e must be zer* *o. Now consider the map OE : R -! mR=m defined by ssmOE(x) = x mod m. This is an injec* *tive ring homomorphism. Any direct product of fields is regular, but the converse of the above Pro* *position is easily seen to be false; for example, Z Q is not a regular ring (and, of cour* *se, GH is false in D(Z)). Proposition 3.2. A commutative ring R is regular if and only if it satisfies bo* *th the Annihilator Criterion and the Nilpotence Criterion. Proof.Suppose R is commutative regular. In general, if e is idempotent, then an* *n(e) = (1 - e). Since each principal ideal (x) is generated by an idempotent, we obtain annann(x) = (x) 8 KEIR H. LOCKRIDGE for all x 2 R. Further, since (x) = (x2), it is clear that there are no nonzero* * nilpotent elements of R. Conversely, suppose the two criteria hold. Every element of ann(x) \ (x) * *has the property that its square is zero; hence, by the Nilpotence criterion, this inte* *rsection is trivial. If x2y = 0, then xy 2 ann(x), forcing xy = 0. Thus ann(x2) ann(x), a* *nd by the Annihilator Criterion, (x) (x2). Hence R is regular. We have proved: Theorem 3.3. If GH is true in D, then R is von Neumann regular. Remark 3.4. In Remark 2.6, we noted that if GH is true, then the Nilpotence Cri* *terion holds when ss*S is concentrated in even degrees. Since the Annihilator Criterio* *n also holds in this case, we can conclude, as above, that (x2) = (x) for all x 2 ss*S. Ther* *efore, GH is false in D if S is connective and ss*S is concentrated in even degrees but not * *concentrated in degree zero. In order to prove the converse of Theorem 3.3, we first give circumstances* * under which thickis exactly the collection of retracts of finite wedges of suspens* *ions of S. In this situation, GH is trivially true. For the next proposition, it is not neces* *sary to assume that ss*S is commutative or concentrated in degree zero. Proposition 3.5. If every finitely generated submodule of a free R-module is a * *summand, then every element of thickis a retract of a finite wedge of suspensions of * *S. Proof.Let C be the collection of objects in D which satisfy the conclusion of t* *he Propo- sition. C is contained in thick and contains S; hence, it suffices to show * *that C is thick. It is clearly closed under suspension and retraction. To show closure un* *der cofiber sequences, we first consider maps between wedges of spheres. Let Sn be an n-fol* *d wedge of suspensions of S and let Sm be an m-fold such wedge. Consider the cofiber se* *quence __ffi//__f__// __OE_// -1C Sn Sm C. By hypothesis, imf* is a summand of ss*Sm , so we get a decomposition ss*Sm ~=i* *m f* cokerf*. Since idempotents split in D (we assume D has arbitrary coproducts), w* *e obtain an associated splitting Sm ' K_L such that ss*K ~=imf* and ss*L ~=cokerf*. Sinc* *e imf* is projective, kerf* is a summand of ss*Sn. Hence we have a decomposition ss*Sn* * ~=kerf* coimf* and an associated splitting Sn ' M_N with ss*M ~=coimf* and ss*N ~=kerf*. Now, f : M_N -! K_L has matrix form ` ' ` ' ssK f'M ssK f'N ssK f'M 0 = . ssLf'M ssLf'N 0 0 All entries but the upper left are zero since they are zero on homotopy groups * *and have domains that are retracts of wedges of spheres. We now need to show that ssK f'* *M is an THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF R-MODULES 9 equivalence; if so, then C ' N_L, making C a retract of a wedge of suspensions* * of the sphere. Certainly, ssK f'M induces an isomorphism of homotopy groups. This suf* *fices, as our next claim demonstrates. Claim. Let q : A -! B be a morphism in thick. If q induces an isomorphism * *of homotopy groups, then it is an equivalence in D. Proof.First, suppose M 2 thick has the property ss*M = [S, M]* = 0. Then the functor [-, M]* must vanish on all of thick. In particular, [M, M]* = 0, for* *cing the identity map of M to be trivial. Hence M is trivial. Since the cofiber of q sat* *isfies this property, it must be trivial, forcing q to be an equivalence. Now we consider the general case. Consider the following commutative diag* *ram, where the rows are cofiber sequences, Wk is a wedge of suspensions of S, and th* *e vertical compositions are the identity. f X ___________//Y__________//_C i1|| |i2| |_| |fflffli2fr1 fflffl| fflffl| W1 __________//W2__________//D r1|| |r2| |OE| |fflfflf fflffl| fflffl| X ___________//Y__________//_C. Let ` = OE_. By the five-lemma, the induced map `* : [C, C]* -! [C, C]* is an i* *somor- phism; hence, C is a retract of D, which is a retract of a wedge of suspensions* * of S by the above special case. This completes the proof. Next we show that regularity guarantees that the hypothesis of Proposition* * 3.5 is satisfied. The following proposition follows from [6, 2.7]. We include a simple* * proof for the reader's convenience. Proposition 3.6. If T is a von Neumann regular ring, then every finitely genera* *ted submodule of a projective T -module is a summand. Proof.It suffices to consider a finitely generated submodule K of T n. We have * *a short exact sequence 0_____//K____//_T_n__//Q____//0. This is a finite presentation of Q. Since T is von Neumann regular, all modules* * are flat ([13, 4.2.9]). The module Q is therefore projective by [13, 3.2.7], so the sequ* *ence splits and K is a summand of T n. Combining this proposition with Theorem 3.3, we have proved: 10 KEIR H. LOCKRIDGE Theorem 3.7. Let D be a triangulated category with arbitrary coproducts, and le* *t S 2 D have the property that R = ss*S is commutative and concentrated in degree zero.* * GH is true in D if and only if R is von Neumann regular. Corollary 3.8. For commutative rings, GH is true in D(R) if and only if R is vo* *n Neu- mann regular. Remark 3.9. Observe that since Proposition 3.6 applies to noncommuative rings, * *we may drop the commutativity assumption for one direction of this theorem: GH is tru* *e in the derived category of (right) modules over a von Neumann regular ring. Remark 3.10. Notice that GH is true in D if and only if every object of thickis a retract of a wedge of suspensions of S. This is a very trivializing set of cir* *cumstances. In D(R), for example, this says that every finite object must be a wedge of sus* *pensions of finitely generated projective modules. For any connective monogenic stable h* *omotopy category where ss0(S) is projective-free, this trivializing condition is equiva* *lent to the requirement that all finite objects be finite wedges of suspensions of the sphe* *re. Remark 3.11. Continuing the discussion of Remark 3.4, notice that Theorem 3.7 i* *s true when ss*S is concentrated in even degrees; all that must be changed is the defi* *nition of regularity, in the obvious way: a graded ring R* is regular if for every elemen* *t x 2 Rk there exists an element y 2 R-k such that xyx = x. One would also replace `fiel* *d' with `graded field' in Proposition 3.1. Example 3.12. As a final example, we consider the derived category D(Qcoh (X)) * *of quasi-coherent sheaves of OX -modules over a scheme X, where O = OX is the stru* *cture sheaf of X. Since the categories R - Mod and Qcoh(X) are equivalent as abelian* * cate- gories for the affine scheme X = SpecR, we have already addressed GH in the aff* *ine case. According to [9, 2.6], whenever X is a finite-dimensional Noetherian scheme wit* *h enough locally frees, D(Qcoh (X)) is a unital algebraic stable homotopy category with * *weak gen- erators the locally free sheaves of finite rank. The sphere S is the chain com* *plex with O concentrated in degree zero. By [7, III.6.3], ss*S = ExtQcoh(X)(O, O) = H*(X;* * O). If X = PnR, n dimensional projective space over a commutative Noetherian ring R, t* *hen ss*S = R concentrated in degree zero ([7, III.2.7, III.5.1]). Hence, by Coroll* *ary 5.5, we conclude that GH is true if and only if R is a finite product of fields. More g* *enerally, GH is true in D(Qcoh (X)) for any smooth Fano variety X over a field of characteri* *stic zero; such X have no higher cohomology by the Kodaira Vanishing Theorem ([7, III.7.15* *]). We are now able to give an example where GH is true for thickbut not tr* *ue if stated for morphisms between any two small objects. Over P1R, for example, ever* *y locally free sheaf of finite rank is a direct sum of invertible sheaves ([7, V.2.6]), a* *nd the invertible sheaves over P1Rare exactly the twists of the structure sheaf, O(k) for k 2 Z (* *[7, II.6.17]). Hence, the collection of small objects in D(Qcoh (P1R)) is exactly the thick su* *bcategory generated by the O(k). Now, [O, O(-1)]* = ExtQcoh(P1R)(O, O(-1)) = H*(P1R, O(-1* *)) = 0 (again, use [7, III.5.1, III.6.3]). But the identity map on O(-1) is nontrivial. THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF R-MODULES 11 4. Variations In this section, we change the domain of definition of GH . To distinguish amo* *ng the variants, we will write GH C for the statement: If f : X -! Y is a morphism of * *objects in C and ss*f = 0, then f ' 0. Note that GH = GH thick. We continue to assu* *me that R = ss*S is commutative and concentrated in degree zero. Consider, for example, C = D = D(R). Suppose GH D(R) is true. Then GH is a* *lso true, so R is regular. Let A and B be R-modules with projective resolutions Po * *and Qo, respectively. Now, ss*Po = A and ss*Qo = B (both concentrated in degree zero),* * and [Po, Qo]* = Ext*R(A, B). If k 0, then any map f 2 [Po, Qo]k must induce the z* *ero map of homotopy groups. It must therefore be trivial by GH D(R). Hence, ExtkR(A, B)* * = 0 for all k 0 and all R-modules A and B. This forces every R-module to be projectiv* *e, making R Noetherian. By Proposition 5.5, R must be a finite product of fields. Convers* *ely, if R is a finite product of fields, then GH D(R)is true (this follows from the fact tha* *t Proposition 2.1 is also true for GH D(R)(the same proof works) and GH D(R)is true when R is* * a field). For the next variant, we require that D be a closed symmetric monoidal cat* *egory (see [10, xA.2] for all relevant definitions); let S be the unit for the smash * *product. We will make use of the product structure and the existence of function objects (a* *nd hence a notion of duality). Corresponding to every finitely generated ideal I = (x1, * *. .,.xn) of R, there is a spectrum S=I = S=x1^. .^.S=xn, where S=xi is the cofiber of xi as* * a self map of S. The definition of S=I depends upon the choice of generators for I, t* *hough thickis independent of this choice ([10, 6.0.9]). Motivated by the fact t* *hat thick subcategories of this form are involved in the classification of all thick subc* *ategories of thick in D(R), we let C = thickand consider GH C. First, we prove a ser* *ies of Propositions establishing the relevant homotopy theoretic properties of S=I. Let Ik = (x1, . .,.xk). S=Ik possesses a 'unit' map jk : S -! S=Ik defined* * to be the smash product of the obvious maps S ____//_S=xi. Write j for jn. Proposition 4.1. ss0jk is the quotient map ss0S = R____//R=Ik = ss0S=Ik, and ss* *lS=Ik = 0 for l < 0. Proof.An examination of the cofiber sequence x1 j1 S ____//_S___//S=x1___//_ S, coupled with the fact that ss*S is concentrated in degree zero, yields the desi* *red result for k = 1. Now proceed by induction, using the diagram xk+1 S=Ik_____//S=Ik___//_S=Ik+1__//_OS=Ik,O:: uu jk|| uuuu |uuuujk+1 S where the triangle commutes and the top row is a cofiber sequence. 12 KEIR H. LOCKRIDGE Corollary 4.2. z 2 I if and only if jz = 0. Dually, there are maps ffik : S=Ik -! kS defined to be the smash product * *of the obvious maps S=xi____//_.S Write ffi for ffin. The proof of the following Prop* *osition is dual to the proof of 4.1. Proposition 4.3. sskffik is the inclusion sskS=Ik = annIk___//_R = ss0S, and ss* *lS=Ik = 0 for l > k. Corollary 4.4. z 2 annI if and only if z = ffiz0 for some z02 sskS=I. Proposition 4.5. If GH thickis true, then annann I = I. Proof.Consider z 2 ann annI. Since im ssnffi = ann I, ss*jzffi = 0. Hence, jz* *ffi ' 0 by GH thick. j fits into a cofiber sequence ` // j // ae// F _____S_____S=I _____ F. Therefore zffi = `q for some q : S=I -! F . At this point, we need to make use* * of the duality functor D(-) = F (-, S), where the objects F (X, Y ) are the function o* *bjects in the closed symmetric monoidal structure on D. S is equivalent to its dual, and * *self maps of S are self dual up to unit. Further, using the definitions, one can check t* *hat j and ffi are dual. Thus zffi = `q implies that jz = "qD`, "q: DF -! S=I. Consider * *the map ae"q: DF -! F 2 ss0F ^ F . We will prove that ae"q= 0 by showing that ss0F ^ F* * = 0. For each k, we have cofiber sequences Fk_____//S___//_S=Ik___//_ Fk which are related via the diagram of cofiber sequences jk S __________//_S=Ik_____// Fk || || | | || | | || jk+1 fflffl| fflffl| S _________//_S=Ik+1___// Fk+1 | | | | | | | fflffl| fflffl| fflffl| .__________//_ S=Ik_____ S=Ik. After smashing this diagram with F , one can prove, by induction on k, that ssl Fk^F = ssl S=Ik = 0 forl 0. In particular, ss0F ^ F = 0. Now, since ae"q= 0, we have "q= jh and jz = jhD`. * *Since hD` and `Dh differ by a unit, we have hD` = `"h. Hence jz = j`"h= 0, and z 2 I * *by Corollary 4.2. This completes the proof. Corollary 4.6. If GH thickis true, then annann J = J whenever S=J 2 thick<* *S=I>. THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF R-MODULES 13 Proof.If GH thickis true and S=J 2 thick, then GH thickis true. Our next goal is to prove that this Corollary applies to I2. The following* * proposition follows from a straightforward manipulation of cofiber sequences. Proposition 4.7. S=I2 2 thick. We now prove the main result for this subsection. Theorem 4.8. Let I be a finitely generated ideal of R such that I \ annI = 0. * *Then, GH thickis true if and only if R = I x R=I as a direct product of rings an* *d R=I is regular. Proof.We prove the `only if' direction first. Assume GH thickis true. Th* *en, by Corollary 4.6 and the previous Proposition, ann annI = I and ann annI2 = I2. N* *ow take x 2 annI2 and fix an element a 2 I. For all b 2 I, xab = 0 since ab 2 I2.* * Thus xa 2 I \ annI = 0; i.e., x 2 annI. This shows that annI2 annI, which in turn * *implies that I I2. Therefore I = I2. By Nakayama's lemma (as stated, for example, in * *[12, x2.8, Corollary 1]), there exists an element r in R such that rI = 0 and 1 - r 2 I. F* *or i 2 I, i = i(1 - r); hence, I is a principal idempotent ideal and therefore a ring dir* *ect factor of R. The ring direct product decomposition R ~=I x T corresponds to a splitting S* * ' A_B with ss*A = I and ss*B = T . Further, S=I ' B_ B. Therefore thick= thick. Since there are no maps from A to B, ss*(-) and [B, -]* are identical on thick<* *B>. T is therefore von Neumann regular by Theorem 3.7. Conversely, if R=I is regular and a ring direct factor of R, then we obtai* *n, as above, a decomposition S ' A_B with thick= thick and [B, B]* = ss*B = R=I. By Theorem 3.7, GH thickis true. The next proposition helps us identify situations where the condition I \ * *annI = 0 holds. Proposition 4.9. If GH thickis true and x^S=I ' 0 for all x 2 I, then I \a* *nn I = 0. Proof.Let z 2 I \ annI. Consider the diagram of cofiber sequences ` // j // ae// F _____S _____S=I _____ F | | O 0|| |z| hOO |0| fflffl|fflffl| fflffl fflffl| F __`__//S_j_//_S=I_ae//_ F, where h is any map that makes the diagram commute. Note that h must factor thro* *ugh both S and F (jz ' 0 since z 2 I). Since ss0 F = 0, ss*h = 0. By GH thick* *, h ' 0. Since z 2 ann I, z = ffiz0. By duality, z = kj for some k 2 [S=I, S]0. Since * *we may take h = jk in the above diagram, we have jk ' 0. Hence z is of the form `lj fo* *r some 14 KEIR H. LOCKRIDGE l 2 [S=I, F ]0. We now prove `*[S=I, F ]0 = 0, forcing z to be trivial. By na* *turality of duality, we have the following commutative diagram, where the vertical maps are* * duality isomorphisms and the bottom sequence is exact: `* [S=I, F ]0____________//[S=I, S]0 || || || || || || [S, DS=I^F ]0_(DS=I^`)//_[S, DS=I^S]0_______//[S, DS=I^S=I]0. * (DS=I^j)* Since x^S=I ' 0 for all x 2 I (and therefore the same is true of the maps x^DS=* *I), the map DS=I^j splits. Hence `* = 0, as desired. This Proposition applies to the derived category of a ring; it is straight* *forward to check that x^S=x ' 0 for all x 2 R. We therefore have Corollary 4.10. Let D = D(R) , and let I be a finitely generated ideal of R = s* *s*S. GH thickis true if and only if I is a ring summand of R with R=I von Neuma* *nn regular. 5. Noetherian Stable Homotopy In this section, we consider the case where ss*S is a Noetherian ring; we * *do not require that ss*S be commutative or concentrated in degree zero. To begin, we * *use the following generalization of a result of Freyd's ([11]). All ss*S-modules are ri* *ght modules. Proposition 5.1. Let D be a triangulated category with distinguished object S. * *Suppose GH is true in D. For every X 2 thick, if ss*X is finitely generated as a gra* *ded ss*S- module, then X is a retract of a finite wedge of suspensions of S. We now observe that when ss*S is Noetherian, GH can only be true under trivial * *circum- stances. Proposition 5.2. Let D be a triangulated category and let S be an object in D s* *uch that ss*S is Noetherian. Then, GH is true in D if and only if thickis exactly the* * collection of retracts of finite wedges of suspensions of S. Proof.The `if' direction is trivial. For the `only if' direction, we show that * *the collection C of retracts of finite wedges of suspensions of S is thick. Since it contains * *S, this gives the desired conclusion. C is trivially closed under suspension and retraction. * *If X and Y are in C, then ss*X and ss*Y are finitely generated right modules over a Noethe* *rian ring; hence, the homotopy of any cofiber C of a map from X to Y is also a finitely ge* *nerated right ss*S-module. By Proposition 5.1, this implies that C is a retract of a fi* *nite wedge of suspensions of S. THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF R-MODULES 15 Further, we record the following related observation concerning the derived cat* *egory of a ring. Proposition 5.3. Let R be a ring, and let D(R) be the derived category of right* * R-modules. If thickis the collection of retracts of finite wedges of suspensions of S, * *then R is von Neumann regular. Proof.Let x be an element of R, and let S=x be the cofiber of S _x__//_S. By hy* *pothesis, ss*S=x is projective; hence, ss0S=x = R=(x)R is projective. Consequently, there* * is a pro- jection R ____//_(x)Rmapping 1 to xk for some k 2 R, and xkx = x. R is therefo* *re von Neumann regular. We conjecture that the collection of graded Noetherian rings for which GH * *holds is exactly the collection of semisimple rings. A ring R is semisimple if, as a rig* *ht module over itself, it decomposes as a finite direct sum of simple modules. It is a fa* *ct ([1, 9.4]) that every finitely generated submodule of a free module over a semisimple ring* * is a summand. Hence, by Proposition 3.5, GH is true if ss*S is semisimple, establish* *ing part of our conjecture. We will prove the remaining half of the conjecture for the deri* *ved category of a ring using the following well-known result ([6, 2.16]). Proposition 5.4. If a ring R is right Noetherian regular, then it is semisimple. Proof.A ring is semisimple if and only if every ideal is a direct summand ([1, * *9.6]). Since R is Noetherian, every ideal I of R is finitely generated. Since R is regular, * *every finitely generated ideal of R is a summand ([13, 4.2.8]). Hence, R is semisimple. Corollary 5.5. Let R be a Noetherian ring. Then, GH is true in D(R) if and only* * if R is semisimple. If R is also commutative, then GH is true in D(R) if and only if R * *is a finite product of fields. Proof.We observed that the 'if' direction is true earlier in this section. For * *the 'only if' direction, simply combine Propositions 5.2, 5.3, and 5.4. The final observatio* *n follows from the Artin-Wedderburn theorem, which classifies semisimple rings. References [1]Frank W. Anderson and Kent R. Fuller. Rings and categories of modules, volu* *me 13 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1992. [2]Marcel B"okstedt and Amnon Neeman. Homotopy limits in triangulated categori* *es. Compositio Math., 86(2):209-234, 1993. [3]Ethan S. Devinatz. K-theory and the generating hypothesis. Amer. J. Math., * *112(5):787-804, 1990. 16 KEIR H. LOCKRIDGE [4]Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith. Nilpotence and* * stable homotopy theory. I. Ann. of Math. (2), 128(2):207-241, 1988. [5]Peter Freyd. Stable homotopy. In Proc. Conf. Categorical Algebra (La Jolla,* * Calif., 1965), pages 121-172. Springer, New York, 1966. [6]K. R. Goodearl. von Neumann regular rings. Robert E. Krieger Publishing Co.* * Inc., Malabar, FL, second edition, 1991. [7]Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Grad* *uate Texts in Mathe- matics, No. 52. [8]Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy the* *ory. II. Ann. of Math. (2), 148(1):1-49, 1998. [9]Mark Hovey. Model category structures on chain complexes of sheaves. Trans.* * Amer. Math. Soc., 353(6):2441-2457 (electronic), 2001. [10]Mark Hovey, John H. Palmieri, and Neil P. Strickland. Axiomatic stable homo* *topy theory. Mem. Amer. Math. Soc., 128(610):x+114, 1997. [11]Keir Lockridge. The generating hypothesis in general stable homotopy catego* *ries. Ph.D. thesis. Uni- versity of Washington, 2006. [12]Miles Reid. Undergraduate commutative algebra, volume 29 of London Mathemat* *ical Society Student Texts. Cambridge University Press, Cambridge, 1996. [13]Charles A. Weibel. An introduction to homological algebra, volume 38 of Cam* *bridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 1994. Department of Mathematics, University of Washington, Seattle, WA 98195 E-mail address: lockridg@math.washington.edu