THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI Abstract.We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds * *in the derived category of a ring R if and only if R is von Neumann regular. Th* *is extends results of the second author [Loc05]. We also characterize rings* * for which the original form (the faithful version) of the generating hypothe* *sis holds in the derived category of R. These must be close to von Neumann regular* * in a precise sense, and, given any of a number of finiteness hypotheses, mu* *st be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular, and therefore does not satisfy the strong fo* *rm of the generating hypothesis. Introduction The generating hypothesis was introduced by Peter Freyd [Fre66] in algebraic topology, where it is the assertion that any map f :X -!Y of finite spectra tha* *t is 0 on stable homotopy groups is in fact null homotopic. The generating hypothesi* *s is widely considered to be one of the most important and difficult problems in sta* *ble homotopy theory. It has many implications for the structure of the stable homot* *opy ring ss*S0 of the sphere, implying for example that it is totally non-coherent * *[Fre66] and that the p-completion ss*S0pis a self-injective ring [Hov ]. Somewhat surpr* *isingly, Freyd proved that the generating hypothesis in fact implies that the map [X, Y ] -!Hom ss*S0(ss*X, ss*Y ) from maps of finite spectra to maps of their stable homotopy modules is not only injective but also surjective. That is, the generating hypothesis implies that* * the stable homotopy functor is fully faithful on finite spectra. One approach to understanding the generating hypothesis is to look at analogo* *us questions in other categories. Following the second author [Loc05], we say that* * a ring R satisfies the generating hypothesis if whenever f :X -!Y is a map of perfect complexes in the derived category D(R) of R and H*f = 0, then f = 0. Recall that a perfect complex is a bounded chain complex of finitely generated projective (right) modules, and that f = 0 in D(R) exactly when f is chain ho- motopic to 0 (for maps of perfect complexes). Perfect complexes are the algebra* *ic analogue of finite spectra, as they are the small objects in D(R). Thus R satis* *fies the generating hypothesis exactly when the homology functor is faithful on perf* *ect complexes. Let us also say that R satisfies the strong generating hypothesis if the homology functor is fully faithful on perfect complexes. The second author noticed [Loc05, Section 4] that the homology functor is fai* *th- ful on all of D(R) if and only if all right R-modules are projective; that is, * *if and ____________ Date: October 5, 2006. 1 2 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI only if R is semisimple. Since perfect complexes are the small objects of D(R) and finitely presented modules are the small R-modules, it is natural to conjec- ture (as the second author did in [Loc05]) that the homology functor is faithfu* *l on perfect complexes (that is, R satisfies the generating hypothesis) if and only * *if all finitely presented right R-modules are projective; that is, if and only if R is* * von Neumann regular. The second author verified that all von Neumann regular rings do satisfy the generating hypothesis, and proved that if R satisfies the genera* *ting hypothesis and is either commutative or right coherent, then R is von Neumann regular [Loc05]. In this paper, we first prove that R satisfies the strong generating hypothes* *is if and only if R is von Neumann regular. We then consider the generating hypothesi* *s, in effect asking whether the generating hypothesis implies the strong generating hypothesis. We prove that R satisfies the generating hypothesis if and only if * *all short exact sequences of finitely presented modules split, and all submodules o* *f flat modules are flat. This makes R close to von Neumann regular, and in fact if R is local or satisfies one of several finiteness hypotheses it forces R to be von N* *eumann regular. However, we construct an example of a ring that satisfies the generati* *ng hypothesis but is not von Neumann regular. Over this ring, then, the homology functor is faithful on perfect complexes but not full. The authors would like to thank Grigory Garkusha for many helpful discussions. All R-modules M will be right R-modules in this paper, so that, for example, D(R) is the unbounded derived category of right R-modules. The differential d in a chain complex P will lower dimension, so that dn :Pn -!Pn-1. We will denote kerdn by ZnP and imdn by Bn-1P . If M is an R-module, then Dn(M) denotes the complex which is M in degree n and n - 1 and 0 elsewhere, with dn being the identity. SnM denotes the complex that is M in degree n and 0 elsewhere. 1. The strong generating hypothesis We begin by recalling the second author's characterization of semisimple ring* *s. Lemma 1.1. Suppose P is a perfect complexLof R-modules with both BnP and HnP projective for all n. Then P ~= nSn(HnP ) in D(R). In this case, the natural map [P, Q] -!Hom R (H*P, H*Q) is an isomorphism for all complexes Q. Proof.We have Pn ~=ZnP Bn-1P ~=BnP HnP Bn-1P. L >FromLthis it follows that P ~= nDn(Bn-1P ) Sn(HnP ), which is isomorphic to nSn(HnP ) in D(R). A chain map from SnHnP to a complex Q is the same thing as a map f :HnP -! ZnQ, and such a map is chain homotopic to 0 exactly when there is a map D :HnP -! Qn+1 such that dD = f. Since HnP is projective, f is chain ho- motopic to 0 if and only if f lands in BnQ. Using projectivity of HnP again, we conclude that [SnHnP, Q] ~=Hom R(HnP, HnQ). Proposition 1.2. A ring R is semisimple if and only if the homology functor is faithful on D(R). Furthermore, in this case, the homology functor is in fact fu* *lly faithful on D(R). Proof.Suppose the homology functor is faithful in D(R). Take two R-modules M and N, and take a projective resolution P* of M. Then an element of Exts(M, N) THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 3 is represented by a map from P* to N, thought as a complex concentrated in degr* *ee s. This map is necessarily 0 in homology when s > 0. Thus Exts(M, N) = 0 for all s > 0 and all M, N, so every R-module is projective and R is semisimple. On the other hand, if R is semisimple, then Lemma 1.1 implies that homology is fully faithful. The analogue for the generating hypothesis is the following theorem. Theorem 1.3. A ring R satisfies the strong generating hypothesis if and only if* * R is von Neumann regular. In this case, the natural map [P, Q] -!Hom R (H*P, H*Q) is an isomorphism for all perfect complexes P and arbitrary complexes Q. Recall that R is von Neumann regular if and only if, for every x 2 R, there is a y 2 R with x = xyx. The standard reference for von Neumann regular rings is [Goo91 ]; the book [Lam99 ] takes an approach based on module categories, so contains some different and useful results about von Neumann regular rings. A standard characterization is that R is von Neumann regular if and only if all R- modules are flat, which is true if and only if all finitely presented R-modules* * are projective. Proof.Suppose R satisfies the strong generating hypothesis. Then, ann`annr(Rx) = Rx for all x 2 R, by [Loc05, Proposition 2.7]. Now take x 2 R, and consider the pe* *rfect complex P with Pi = R if i = 0, 1 and Pi = 0 otherwise, with the differential P1 -! P0 being left multiplication by x. This complex has H0(P ) = R=xR and H1(P ) = annr(x). By the strong generating hypothesis, there exists a chain map OE: P -! P such that H1(OE) = 0 and H0(OE) = 1, the identity of R=xR. Translati* *ng, this means there exist elements a, b 2 R such that xa = bx with a 2 ann`annr(x) (so that H1(OE) = 0) and b = 1 + xc for some c 2 R (so that H0(OE) = 1). But th* *en a = dx for some d 2 R, so we have xdx = xa = bx = (1 + xc)x = x + xcx. This means that x = x(d - c)x. Since x was arbitrary, R is von Neumann regular. Conversely, suppose R is von Neumann regular, and P is a perfect complex. In a von Neumann regular ring, finitely generated submodules of projectives are projective [Lam99 , p.44], so BnP is finitely generated projective for all n. * *Then ZnP , as the kernel of the (necessarily split) surjection Pn -!Bn-1P , is also * *finitely generated projective for all n. Hence HnP is finitely presented, and so is proj* *ective for all n. Now Lemma 1.1 implies that homology is fully faithful on maps out of perfect complexes. Recall from [Loc05] that if R is either commutative or right coherent and R satisifes the generating hypothesis, then R is von Neumann regular. Hence we get the following corollary. Corollary 1.4. If R is either commutative or coherent, then R satisfies the gen- erating hypothesis if and only if R satisfies the strong generating hypothesis. 4 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI The second author also investigated the generating hypothesis from the view- point of global stable homotopy theory. Using the results of [Loc05], we get t* *he following corollary. Corollary 1.5. A ring R satisfies the strong generating hypothesis if and only * *if, in D(R), the thick subcategory generated by R is the collection of retracts of * *finite coproducts of suspensions of R. Recall that a full subcategory of a triangulated category is called thick if * *it is closed under shifts, retracts, and cofibers; the thick subcategory generated* * by R consists of the perfect complexes. This corollary follows from [Loc05, Propo- sition 5.1], and indicates how different stable homotopy theory must be from the derived category of a ring if the generating hypothesis in stable homotopy is t* *o be true, since there are many finite spectra that are not retracts of finite copro* *ducts of suspensions of the sphere. 2.Rings that satisfy the generating hypothesis Having dealt with the strong generating hypothesis, we now turn our attention to the generating hypothesis. The object of this section to prove the following theorem. Theorem 2.1. A ring R satisfies the generating hypothesis if and only if R has * *weak global dimension at most 1 and all finitely presented R-modules are FP-injectiv* *e. Weak global dimension at most 1 is of course equivalent to the statement that submodules of flat modules are flat. Recall that a module M is said to be FP- injective if Ext1(F, M) = 0 for all finitely presented modules F ; thus all fin* *itely presented modules are FP-injective if and only if all short exact sequences of finitely presented modules split. FP-injective modules seem to have been intro- duced in [Ste70]; a good guide to the literature can be found in [Fai99, Chapte* *r 6]. An FP-injective module is sometimes called absolutely pure, because M is FP- injective if and only if every short exact sequence 0 -!M -!N -!P -! 0 is pure (that is, remains exact upon tensoring with any left R-module). See [La* *m99 , Theorem 4.89(5)] for a proof of this equivalence. To compare the rings of Theorem 2.1 with von Neumann regular rings, the following lemma is helpful. Lemma 2.2. A ring R is von Neumann regular if and only if every R-module is FP-injective. This lemma is well-known, but does not appear in [Goo91 ] or [Lam99 ], so we include the proof for the convenience of the reader. Proof.Suppose R is von Neumann regular, and M is an R-module. Choose a short exact sequence E 0 -!M -!I -!N -!0 where I is injective. Since N is necesarily flat, this sequence is pure [Lam99* * , Theorem 4.85]. Hence, if F is finitely presented, Hom (F, E) is still exact [La* *m99 , Theorem 4.89(5)], and so Ext1(F, M) = 0 and M is FP-injective. THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 5 Conversely, if every module is FP-injective, another application of [Lam99 , * *The- orem 4.89(5)] shows that every short exact sequence is pure. Then [Lam99 , Theo- rem 4.85] shows that every module is flat, as required. We now begin the proof of Theorem 2.1. Our first task is to characterize the homology groups of perfect complexes. Proposition 2.3. Suppose R is a ring. An R-module M is a homology module of a perfect complex of R-modules if and only if there exists a finitely presented* * module F such that M embeds in F and the quotient F=M embeds in a projective module. Furthermore, in this case, there is a perfect complex P such that Pn = 0 unless n = 0, 1, 2 and M = H1P . Proof.Suppose M = HnP , where each Piis a finitely generated projective module. Then we have a short exact sequence 0 -!M -!Pn=BnP -dn!Bn-1P -! 0, Pn=BnP is finitely presented and Bn-1P embeds in the projective module Pn-1. Conversely, suppose M embeds in the finitely presented module F and the quo- tient F=M embeds in the projective module P0, which we can assume is finitely generated since F is so. Choose a presentation P2 d2-!P1 p-!F -! 0 of F , where P1 and P2 are finitely generated projectives. Define the map d1: P* *1 -! P0 to be the composite P1 p-!F -! F=M -!P0. This defines a three-term perfect chain complex P . Pulling back the presentati* *on of F through the inclusion M -!F shows that H1P ~=M. Corollary 2.4. Suppose P is a perfect complex. Then each cycle module ZnP is a homology module of some perfect complex. Proof.Note that ZnP is a submodule of the finitely presented module Pn and the quotient Pn=ZnP embeds in the projective module Pn-1. We now take a significant step towards Theorem 2.1 by showing how FP-injective modules get involved. Theorem 2.5. Let R be a ring, and let Q be an arbitrary object of D(R). Then the generating hypothesis with target Q is true in D(R) if and only if HnQ is F* *P- injective for all n. In particular, R satisfies the generating hypothesis if an* *d only if all homology modules of perfect complexes are FP-injective. The generating hypothesis with target Q is the statement that any map f :P -! Q in D(R) where P is a perfect complex and H*f = 0 has f = 0. So R satisfies the generating hypothesis if and only if R satisfies the generating h* *ypothesis with target Q for all perfect complexes Q. Note in particular that this theorem and Lemma 2.2 imply that R satisfies the generating hypothesis with target Q for all (not necessarily perfect) Q, if and* * only if R is von Neumann regular. 6 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI Proof.Suppose first that the generating hypothesis with target Q holds, and con- sider a finitely presented module F and an integer n. Choose a finite presentat* *ion Pn dn-!Pn-1 -!F -! 0 of F , so that, by letting Pi= 0 for i 6= n, n - 1, we get a perfect complex P** * with Hn-1P* = F . To prove that Ext1(F, HnQ*) = 0, it suffices to show that any map f :Pn=ZnP -! HnQ* ___ ___ extends to a map g :Pn-1 -! HnQ* with gdn = f, where dn is the map induced by dn. Since Pn is projective, there is a map OEn :Pn -!Qn such that the composite Pn OEn--!Qn q-!Qn=BnQ is the composite Pn p-!Pn=ZnP -f!HnQ* i-!Qn=BnQ. Now let OEn-1: Pn-1 -!Qn-1 be the zero map. Then OE: P* -!Q* is a chain map. Indeed, write dn :Qn -!Qn-1 as dn = rq. Then dnOEn = rqOEn = rifp = 0 since ri = 0. Furthermore, OE induces the zero map on homology, because if x 2 ZnP , then qOEnx = 0, so OEnx is a boundary. If the generating hypothesis is true, then OE must be chain homotopic to 0. T* *his gives us maps Dn-1: Pn-1 -! Qn and Dn :Pn -! Qn+1 such that dnDn-1 = 0 and Dn-1dn_+ dn+1Dn = OEn. Since_dn :Qn -!Qn-1 factors through Qn=BnQ as dn = dnq, we conclude that dnqDn-1 = 0, so there exists a map g :Pn-1 -!HnQ* such that ig = qDn-1. Of course, we claim that g is the desired extension. To s* *ee this, apply q to the relation Dn-1dn + dn+1Dn = OEn to get qDn-1dn = ifp origdn = ifp. ___ Writing dn = dnp and using_the_fact that i is a monomorphism and p is an epimor- phism, we conclude that gdn = f, as required. Now suppose that every homology group of Q is FP-injective, and OE: P* -!Q* is a map of perfect complexes that induces 0 on homology. We will construct a chain homotopy Dn :Pn -!Qn+1 such that dn+1Dn + Dn-1dn = OEn by induction on n. Our induction hypothesis will be that we have constructed Di for i n - 1 and that OEn - Dn-1dn, which is a map from Pn to Qn, in fact lands in the boundaries BnQ. Getting started is easy since P is bounded below. For the induction step, our hypothesis gives us the commutative square below, ____OE Pn+1=Zn+1P ----! n+1Qn+1=Bn+1Q ? ? dn+1?y ?ydn+1 Pn --------!OE BnQ n-Dn-1dn THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 7 _____ where OEn+1exists because OE_is_zero on homology, so must take cycles to bounda* *ries. We will construct a lifting Dn :Pn -! Qn+1=Bn+1Q in this square. First of all, there is obviously a map En :Pn -!Qn+1=Bn+1Q such that dn+1En = OEn - Dn-1dn, simply because Pn is projective. Then _____ _____ dn+1(OEn+1- Endn+1) = dn+1OEn+1- OEndn+1 + Dn-1dndn+1 = 0. _____ Hence OEn+1-Endn+1 is a map from Pn+1=Zn+1P to Hn+1Q*. Since Hn+1Q*_is FP- injective, there is a map Fn :Pn -!Hn+1Q* such that Fndn+1 = OEn+1- Endn+1. Hence ___ Dn = En + Fn :Pn -!Qn+1=Bn+1Q defines a lift in our commutative square._ We now choose Dn :Pn -! Qn+1 lifting Dn , which we can do because Pn is projective._ Then_one_can easily check that dn+1Dn = OEn - Dn-1dn, and also, because Dndn+1 = OEn+1, that OEn+1 - Dndn+1 lands in Bn+1Q. This completes the induction step and the proof. We can now prove Theorem 2.1. Proof of Theorem 2.1.Suppose the generating hypothesis holds in D(R). In view of Theorem 2.5, we need only show that R has weak dimension at most 1. Since Tor*(-, M) commutes with direct limits, it suffices to show that the weak dimen* *sion of any finitely presented module is at most 1. Since any finitely presented mod* *ule is a homology group of a perfect complex, it is enough to show that the cycles ZnP and the boundaries BnP are flat for all perfect complexes P and integers n. But ZnP is itself a homology group of a perfect complex by Corollary 2.4, and so Theorem 2.5 implies that ZnP is FP-injective. This means that the short exact sequence 0 -!ZnP -! Pn -!Bn-1P -! 0 is pure. Now choose a left R-module M and apply - R M to this short exact sequence. By purity, it remains exact, and so the Tor long exact sequence shows that TorR1(Bn-1P, M) = 0. Since M was arbitrary, Bn-1P is flat. But then ZnP , as a kernel of a surjection of flat modules, is also flat. Conversely, assume R has global weak dimension at most 1 and all finitely pre- sented R-modules are FP-injective. We need to show that an arbitrary homology group M of a perfect complex is FP-injective, by Theorem 2.5. By Proposition 2.* *3, there is a finitely presented module F and an exact sequence 0 -!M -!F -! F=M -!0, where F=M embeds in a projective module. Since R has global weak dimension at most 1, F=M is flat. But then the above exact sequence is pure [Lam99 , Theo- rem 4.85]. Applying Hom R(N, -) to this sequence we get a long exact sequence 0 -!Hom R (N, M) -!Hom R (N, F ) -!Hom R (N, F=M) -! Ext1R(N, M) -!Ext1R(N, F ) -!. . . If N is finitely presented, though, the map Hom R(N, F ) -! Hom R(N, F=M) is surjective, since our original sequence is pure [Lam99 , Theorem 4.89(5)]. By h* *y- pothesis, Ext1R(N, F ) = 0, so we conclude that Ext1R(N, M) = 0. Thus M is FP-injective. 8 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI 3.Examples and counterexamples In this section, we give conditions under which rings that satisfy the genera* *ting hypothesis must be von Neumann regular, and also give an example of a ring that satisfies the generating hypothesis yet is not von Neumann regular, and thus do* *es not satisfy the strong generating hypothesis. Theorem 3.1. A ring R is von Neumann regular if and only if the generating hypothesis holds in D(R) and finitely generated flat submodules of projective r* *ight R-modules are projective. Proof.Assume that the generating hypothesis holds in D(R) and finitely generated flat submodules of projectives are projective. We will show that all finitely p* *resented modules, and hence all modules, are flat. Given a finitely presented module M, choose a perfect complex P with M ~= HnP for some n. We then have a short exact sequence 0 -!BnP -! ZnP -! M -!0. Now BnP is finitely generated and flat (since it is a submodule of Pn) by Theo- rem 2.1. By hypothesis, then, BnP is finitely generated projective. Hence BnP is FP-injective by Theorem 2.1 again, and so the above exact sequence splits. Thus M is a summand of ZnP , which is flat as well, since it is also a submodule of * *Pn. So M is flat. Conversely, if R is von Neumann regular, then any finitely generated submodule of a projective module is projective [Lam99 , Example 2.32(d)]. This immediately gives the following corollary, implicit in [Loc05]. Corollary 3.2. A ring R is von Neumann regular if and only if R satisfies the generating hypothesis and is right coherent. Proof.If R is right coherent, then a finitely generated submodule of a projecti* *ve module is finitely presented. If it is also flat, then it is projective. There are a great many rings where finitely generated flat modules are known to be projective [PR04 ]. The following theorem contains some cases of this, wh* *ich are somewhat less satisfactory since not all von Neumann regular rings satisfy * *the hypotheses. Theorem 3.3. Suppose the generating hypothesis holds in D(R) and one of the following hypotheses holds. (1) R is local (unique maximal right ideal). (2) R is semiperfect (every finitely generated module has a projective cover* *). (3) R is reduced (no nonzero nilpotents)and has finite uniform dimension (R contains no infinite direct sum of nonzero right ideals). (4) R has zero Jacobson radical and finite uniform dimension. (5) R is right nonsingular (the only element whose right annihilator is esse* *ntial in R is 0)and has finite uniform dimension. (6) R is simple (no nontrivial two-sided ideals)and has finite uniform dimen- sion. Then R is von Neumann regular. THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 9 Note that these conditions may not all be independent of each other. For ex- ample, the authors suspect that if R is both right FP-injective (as it must be * *if it satisfies the generating hypothesis) and has finite uniform dimension, then R m* *ay have to be semiperfect. Proof.For a local, semiperfect, or right nonsingular ring with finite uniform d* *imen- sion, every finitely generated flat module is projective; the local case is due* * to Endo and can be found in [Lam99 , Theorem 4.38]. The semiperfect case is due to Bass and is [Lam99 , Exercise 4.21]. The right nonsingular case is due to Sandomier- ski [San68, Corollary 1,p. 228]. Every reduced ring is right nonsingular by [La* *m99 , Lemma 7.8]; since the singular elements form a two-sided ideal, every simple ri* *ng is also right nonsingular [Lam99 , Section 7A]. If R is FP-injective, or in fact o* *nly has Ext1(R=aR, R) = 0 for all a 2 R, then having zero Jacobson radical is equivalent to being right nonsingular, by [NY95 , Theorem 2.1]. Not every von Neumann regular ring has finite uniform dimension. They all, however, are right nonsingular [Lam99 , Corollary 7.7]. This leads to the follo* *wing theorem. Theorem 3.4. A ring R is von Neumann regular if and only if it satisfies the generating hypothesis, is right nonsingular, and its maximal right ring of quot* *ients Q is a flat left R-module. The maximal right ring of quotients of R is the endomorphism ring of the inje* *ctive hull of R as a right R-module, and is much studied in ring theory. See [Lam99 , Section 13] for an introduction. When R is right nonsingular, Q is just equal t* *o the injective hull of R. Proof.Sandomierski [San68, Theorem 2.9] proves that if R is right nonsingular a* *nd the maximal right ring of quotients Q is flat as a left R-module, then finitely* * gen- erated flat submodules of free R-modules (and hence also of projective R-module* *s) are projective. Theorem 3.1 completes the proof. Theorem 3.5. There exists a ring S that satisfies the generating hypothesis but* * is not von Neumann regular. Of course, such a ring will not satisfy the strong generating hypothesis. Bef* *ore proving this theorem, we need the following lemmas. Lemma 3.6. Every principal right ideal of a ring R is flat if and only if whene* *ver ab = 0 in R there is an x 2 R such that ax = 0 and xb = b. Proof.Consider the short exact sequence 0 -!annr a -!R ax--!aR -!0. By [Lam99 , Theorem 4.23], aR is flat if and only if for every b 2 annra, there* * is a map ` :R -!annr(a) with `(b) = b. Translating, this means that aR is flat if and only if whenever ab = 0, there is an x such that ax = 0 and xb = b. Lemma 3.7. A ring R has global weak dimension 1 if and only if for every integer m and every pair of m x m matrices A, B over R with AB = 0, there is an m x m matrix X over R such that AX = 0 and XB = B. 10 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI Proof.In view of Lemma 3.6, the matrix condition of this lemma is equivalent to every principal right ideal of Mm (R) being flat, for all m 1. We will use t* *he Morita equivalence between R and Mm (R) to prove that this is equivalent to R having global weak dimension 1. Indeed, if R has global weak dimension 1, so does Mm (R) [Lam99 , p. 481], and so every ideal of Mm (R) is flat. Conversely, suppose every principal right ideal of Mm (R) is flat for all m * * 1. Suppose I is an m-generated right ideal of R. Then I corresponds under the Mori* *ta equivalence to a principal right ideal of Mm (R) [Lam99 , Remark 17.23(C)]. This principal ideal is flat, and so I is flat as well, since Morita equivalences pr* *eserve flatness [Lam99 , p. 481]. Hence all finitely generated ideals of R are flat. Since Torcommutes with dir* *ect limits, all ideals of R are flat. But then R has weak dimension 1 [Lam99 , Lemma 4.66]. Proof of Theorem 3.5.We will use the method of [PRZ95 ], who introduce and study indiscrete rings. For us, the salient property of indiscrete rings is th* *at all finitely presented modules over an indiscrete ring are FP-injective [PRZ95 , Th* *eo- rem 2.4]. Thus, we must find an indiscrete ring that also has weak dimension on* *e. The construction given in [PRZ95 , p. 359] begins with a finite-dimensional alg* *ebra R of finite representation type over an infinite field F . Because we want to * *end up with something of weak dimension one, we will take R to have right (and left) global dimension 1. For example, we can take R to be the ring of 2 x 2 upper triangular matrices over F , which is a classical example of a ring of right (a* *nd left) global dimension 1 that is not von Neumann regular [Lam99 , Example 2.36]. The method of [PRZ95 ] is then to construct a map o :R -!MnR and then let S = Ro be the direct limit Mn2o S = Ro = colim(R o-!MnR Mno---!Mn2R ----! . .). Then Prest, Rothmaler, and Ziegler show that S is always indiscrete. Now, in our case, our ring R has global dimension 1, and therefore all of the MkR also have global dimension 1 since they are Morita equivalent to R. Now, if we take a pai* *r of m x m matrices A, B over S with AB = 0, then we can choose k large enough such that A, B are actually matrices over MnkR, and AB = 0 as such matrices. Then Lemma 3.7 shows that there is a matrix X over MnkR, and hence over S, with AX = A and XB = 0. Thus Lemma 3.7 implies that S has weak dimension 1, and S cannot be von Neumann regular because R is not (see [PRZ95 , p. 359]). The indiscrete rings of [PRZ95 ], of which our counterexample S is one, have been generalized by Garkusha and Generalov [GG99 ] to the class of almost regul* *ar rings, in which all (left or right) finitely presented modules are FP-injective* *. The indiscrete rings are the simple almost regular rings. We also note that the ring S of Theorem 3.5 is in fact weakly semihereditary in the sense of Cohn [Coh85 , p. 13]. This means that if A and B are (not neces* *sarily square) matrices such that AB = 0, then there is an idempotent matrix E such th* *at AE = A and EB = 0. Since hereditary implies weakly semihereditary, each Mn(R) in the above proof is weakly semihereditary, and so the same argument shows that S is as well. One can then use (the left module version of) Lemma 3.7 to see th* *at weakly semihereditary implies global weak dimension 1. THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 11 We now turn to some questions we have not been able to answer. First of all, the stable homotopy category in topology is more like D(R) for a graded ring R (or, better yet, a differential graded algebra R), though, it must be stressed,* * these categories are still much simpler than the stable homotopy category. We have not considered the generating hypothesis for these R. We could ask whether there is a ring R that satisfies the generating hypothes* *is for right R-modules but not left R-modules. Such a ring could not be von Neumann regular, of course. Also, recall that there is a strongly convergent spectral sequence whose E2 t* *erm is Ext**R(H*P, H*Q) converging to D(R)(P, Q)*. It seems intuitively evident that for the strong generating hypothesis to hold, this spectral sequence must colla* *pse to the 0-line for perfect complexes P and Q. This is in fact true, since in thi* *s case R is von Neumann regular, hence coherent, so the homology groups H*P are finitely presented modules and therefore projective. However, the situation for R satisfying the generating hypothesis but not the strong generating hypothesis is less clear. To satisfy the generating hypothesi* *s, it must be that every element of Exts,*(H*P, H*Q) with s > 0 does not survive the spectral sequence. But in order not to satisfy the strong generating hypothesi* *s, there must be an element of Hom *(H*P, H*Q) for some perfect P and Q that supports a differential. It would be intriguing to understand how this happens. Finally, one could define R to satisfy the n-fold generating hypothesis if whenever f1, . .,.fn are composable maps of perfect complexes such that H*(fi) * *= 0 for all i, then fn O . .O.f1 = 0 in D(R). If we ask for this condition to hold* * for all n-tuples of composable maps with H*fi = 0, not just maps between perfect complexes, then the second author has shown in his thesis, using work of Chris- tensen [Chr98], that R has projective dimension n. One could then ask for an analogous characterization of rings R, probably in terms of weak dimension, that satisfy the n-fold generating hypothesis, or some strong version of the n-* *fold generating hypothesis. References [Chr98]J. Daniel Christensen, Ideals in triangulated categories: phantoms, ghos* *ts and skeleta, Adv. Math. 136 (1998), no. 2, 284-339. MR MR1626856 (99g:18007) [Coh85]P. M. Cohn, Free rings and their relations, second ed., London Mathemati* *cal Society Monographs, vol. 19, Academic Press Inc. [Harcourt Brace Jovanovich Publ* *ishers], Lon- don, 1985. 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(2) 2 (1970), 323-329. MR MR0258888 (41 #3533) Department of Mathematics, Wesleyan University, Middletown, CT 06459 E-mail address: hovey@member.ams.org Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195 E-mail address: lockridg@math.washington.edu Department of Mathematics, University of Manchester, Booth Street East, Manch- ester M13 PL, United Kingdom E-mail address: gpuninski@maths.man.ac.uk