Triangulations of projective modules

TRIANGULA TIONS OF PR OJECTI VE MODULES MARK HOVEY AND KEIR LO CKR IDGE Abstra ct. W e show that the category of pro jective mo dules o ver a graded comm utative ring admits a triangulation with respect to modu le susp ension if and only if the ring is a finite prod uct of graded fields and exterior algebras on one generator ov er a graded field (with a unit in th e appropriate degree). W e also classify the ungraded commutativ e rings for which the category of pro jectiv e modu les admits a triangulation with resp ect to the identit y susp en sion. Applications to tw o analogues of the generating hyp othesis in algebraic top ology are given, and we translate our results into the setting of mo dules ove r a symmetric ring sp ectru m or S -algebra. Contents 1. In tro d uction 1 2. T riangulations of pr o jectiv e mo d ules 3 3. The generating h yp othesis 8 4. Ring sp ectra 13 References 14 1. Introduction In the stable catego ry of sp ectra, F reyd’s generating h yp othesis ([5, § 9]) is the conjecture that an y map of fin ite sp ectra inducing the trivial map of homotop y groups m ust itself b e trivial. The global v ersion of this conjecture – that this is true for all maps of s p ectra – is easily seen to b e false. The p resen t w ork w as motiv ated by th e f ollo w in g question: what m ust b e true ab out a triangulated category in order for it to supp ort a global version of the generating hypothesis? W e will sh o w that this qu estion is related to the follo wing strictly algebraic one: for whic h rings do es the asso ciated category of pro jectiv e m o dules admit a triangulation? This question can b e add ressed without r eference to an y form of the generating hyp othesis; this is done in section 2 for commutat ive rings. In s ection 3, w e mak e explicit the relationship b etw een this algebraic question and t w o forms of the generating hyp othesis; there is sligh tly more at issue, h o w eve r, than whether pr o jectiv e mo dules ad m it triangulations. In the final section, we simply tran s late our w ork into the Date : Nov ember 21, 2018. 2000 Mathematics Subje ct C lassific ation. Primary: 18E30; Second ary: 55P43, 16L60. Key wor ds and phr ases. triangulated category , quasi-F rob enius ring, IF-ring, generating hypothesis, ring sp ectra. 1 2 MARK HO VEY AND KEIR LOC KRIDGE setting of ring sp ectra (symmetric ring s p ectra or S -algebras), where w e d efine semisimple and v on Neumann regular ring sp ectra and discuss their classification. First, we establish notation. Let R b e a graded r ing. By a mo d ule we mean a graded righ t R -mo d u le. Let P b e the category of pro jectiv e R -mo du les, and let P f denote the catego ry of fi nitely generated pro jectiv e R -mo d ules. F or an y R -mo du le M , wr ite M [ n ] for the shifted m o dule with M [ n ] i = M i − n . F or an y elemen t m ∈ M , denote b y | m | the degree of m . W rite Hom k ( M , N ) for degree k mo d ule maps from M t o N ; observ e that Hom k ( M , N ) = Hom 0 ( M [ k ] , N ). If M is an R - R -bimo dule ( or, if R is graded commutativ e), then, for any element x ∈ R of degree i , let x · M denote th e right m o dule map from M [ i ] to M induced by left multiplica tion by x . W e assu me that the reader is somewhat familiar with triangulated categories. In br ief, a triangulated catego ry is an add itiv e category T toge ther with an automorphism Σ of T called susp ension and a collectio n of diagrams called exact triangles of the form A / / B / / C / / Σ A satisfying s everal axioms (see [8, A.1.1] or [13] or [15 ]). Usin g the susp ension fu nctor, one can view the m orp hism sets in T as graded group s: define [ X, Y ] k = Hom T (Σ k X, Y ) and [ X, Y ] ∗ = M k ∈ Z [ X, Y ] k . It s eems natural, therefore, to consider one of tw o p ossibilities for Σ on P or P f . I f Σ( − ) = ( − )[1], then [ M , N ] ∗ is the g raded group of graded modu le maps from M to N . Alternative ly , w e could consider Σ = 1 , the identi ty f unctor. When R is concen trated in degree zero, one then ob tains a triangulation of the category of ungraded pro jectiv e (or finitely generated pro jectiv e) R -mo du les by identifying this category with th e th ick sub category of P (or P f ) generated by the m o dules concen trated in d egree zero. Usually , we will assum e that the susp ension fun ctor is of the form Σ = ( − )[ n ] for some n . F or con ve nience, w e will use the term ∆ n - ring to refer to any rin g for whic h P admits a triangulation with su sp ens ion Σ = ( − )[ n ] and ∆ n f - ring f or any ring for whic h P f admits a triangulation with susp ension Σ = ( − )[ n ]. Our goal is to c haracterize the graded comm u- tativ e rin gs R f or whic h the categories P and P f admit triangulations. In particular, w e pro ve Theorem 1.1. L et P b e the c ate gory of pr oje ctive mo dules over a gr ade d c ommutative ring R . P admits a triangulation with susp ension Σ = ( − )[1] if and only if R ∼ = R 1 × · · · × R n , wher e e ach factor ring R i is either a gr ade d field k or an exterior algebr a k [ x ] / ( x 2 ) over a gr ade d field k c ontaining a unit of de gr e e 3 | x | + 1 . In the u ngraded case, we h a v e Theorem 1.2. L et P b e the c ate gory of pr oje ctive mo dules over a c ommutative ring R . P admits a triangulation with susp ension Σ = 1 if and only if R ∼ = R 1 × · · · × R n , wher e e ach factor ring R i is either a field k , an exterior algebr a k [ x ] / ( x 2 ) over a field k of char acteristic 2, or T / (4) , wher e T is a c omplete 2 -ring. TRIANGULA TIONS OF PR OJECTIVE MODULES 3 A complete 2-ring is a complete lo cal d iscrete v aluation ring of c haracteristic zero whose maximal ideal is generated by 2 (see [11, p . 223]). In [12], it is sh o wn that the category of finitely generated pro jective T / (4)-mod ules admits a unique triangulation, and we use their metho ds to construct triangulations for the rings ap p earing in the ab o v e tw o theorems. Let T b e a tr iangulated category , and let S ∈ T b e a distinguished ob ject. W rite π ∗ ( − ) for the fun ctor [ S, − ] ∗ . W e sa y that T satisfies the glob al g ener ating hyp othesis if π ∗ is a faithful functor from T to the catego ry of graded righ t mo d u les ov er π ∗ S . In the f ollo w in g application of Theorem 1.1, we assum e that T is a mon ogenic stable h omotop y category as defined in [8]. Certain conclusions can b e dra wn with wea ke r hypotheses; this sh ou ld b e clear in the p ro of. Corollary 1.3. L et T b e a mono genic stable homoto py c ate gory with unit obje ct S . T satisfies the glob al gene r ating hyp othesis if and only if (1) R ∼ = R 1 × · · · × R n , wher e R i is either a gr ade d field k or an exterior algebr a k [ x ] / ( x 2 ) over a gr ade d field c ontaining a u nit in de gr e e 3 | x | + 1 ( R is a ∆ 1 -ring), and (2) for every factor ring of R of the form k [ x ] / ( x 2 ) , x · π ∗ C 6 = 0 , wher e C is the c ofib er of x · S . The thick sub c ate gory gener ate d by S ( thic k h S i ) is the smallest f ull su b category of T that contai ns S and is closed un der sus p ension, retraction, and exact triangles. W e sa y that T satisfies the str ong gener ating hyp othesis if, for an y map f : X − → Y with X ∈ thick h S i and Y ∈ T , π ∗ f = 0 implies f = 0. Regarding the strong generating h yp othesis, we hav e the follo w ing corollary . If p is a prim e ideal, write R p for the lo calization of R at p . Corollary 1.4. A ssume the hyp otheses of Cor ol lary 1.3 . If T sat isfies the str ong gener ating hyp othesis, then, for any prime ide al p , R p is either a gr ade d field k or an exterior algebr a k [ x ] / ( x 2 ) with a unit in de gr e e 3 | x | + 1 . If R i s lo c al or N o etherian, then T satisfies the str ong gener ating hyp othesis if and only if it satisfies the g lob al gener ating hyp othesis. W e would lik e to thank Sun il Ch eb olu, who informed us of his work with Benson, Chr is- tensen, and Min´ a ˇ c on th e global generating hyp othesis in the stable mo dule category . It is his p resent ation of the material in [1] that led us to consider the questions r aised in the present p ap er. 2. Triangula tions of projective mod ules It is in this section th at we pr o v e T heorems 1.1 and 1.2. Before w e b egin, w e mak e a simple observ ation that w e w ill use frequently . Supp ose P (or P f ) is triangulated. If we apply the functor [ R , − ] 0 to an exact triangle A f / / B g / / C h / / Σ A, then we m ust obtain a long exact sequence of R -mo dules. Hence, any exact triangle is exact as a sequence of R -mo dules. (‘Exact’ at Σ A means ker( − Σ f ) = im h .) 4 MARK HOVEY A ND KEIR LOCKRIDGE First we s ho w that if P admits a triangulation, then R m ust b e a quasi-F rob eniu s rin g, and if P f admits a triangulation, then R m ust b e an IF-ring. A ring T is quasi-F r ob enius if it is r igh t No etherian and righ t self-injectiv e. This condition is righ t-left symm etric, and the qu asi-F r ob enius rings are exactly the rings f or w hic h th e collections of pro j ective and injectiv e mo d ules coincide. In fact, a ring is quasi-F r ob enius if and only if eve ry p ro jectiv e mo dule is injectiv e, if and only if eve ry in jectiv e mo du le is pr o jectiv e. A r ing T is inje ctiv e- flat (an IF-ring) if ev ery in jectiv e mo dule is flat. Note that IF-rin gs are coheren t ([4, 6.9]), so all ∆ n -rings and ∆ n f -rings are coherent . More information on quasi-F r ob enius and IF-rin gs ma y b e foun d in [9, § 15] and [4, § 6]. Prop osition 2.1. If P admits a triangulation, then R is quasi- F r ob e ni u s. Pr o of. Let M b e an R -mo du le. There is a map f : A − → B of pro jectiv e mo dules whose cok ernel is M . This map m ust lie in an exact triangle in P , A f / / B / / C / / Σ A. Exactness at B implies that M is isomorphic to a s ubmo d ule of the pr o jectiv e mo du le C . If M is injectiv e, then it must b e a summand of C and therefore pro j ectiv e. Hence , every injectiv e R -mo dule is p ro jectiv e, and R is qu asi-F rob eniu s.  Prop osition 2.2. If P f admits a triangulation, then R is an IF-ring. Pr o of. In ligh t of the p ro of of Pr op osition 2.1, we see that ev ery fi n itely presen ted mo dule em b ed s in a finitely generated pro j ectiv e m o dule. By [4, 6.8], R is an I F-ring.  Remark 2.3. When R is qu asi-F r ob enius, one can form S tMo d ( R ), the stable mo dule c ate gory of R . The ob jects of StMod ( R ) are R -mo dules and the morphisms are R -mo du le maps m o dulo an equiv alence relation: t wo maps are equiv alen t if their difference factors through a pro jectiv e mo dule. StMo d( R ) is a tr iangulated catego ry . If M is an R mo d ule, then the H el ler shift of M , wr itten Ω M , is the k ernel of a pro jectiv e co ver P ( M ) / / M . This descends to a w ell-defined, inv ertible endomorph ism of the stable mo dule category , and Ω − 1 is the susp ension functor for the triangulation of StMo d( R ). In Prop osition 2.1, the exact triangle gives rise to three short exact sequences: 0 / / M / / C / / Σ k er f / / 0 0 / / k er f / / A / / im f / / 0 0 / / im f / / B / / M / / 0 . T ogether they imp ly that, if P admits a triangulation, then Ω 3 Σ M ∼ = M . Th is observ ation w as made b y Heller in [6]. TRIANGULA TIONS OF PR OJECTIVE MODULES 5 The follo wing prop osition sho ws that a tr iangulation imp oses a sev ere restriction on the lo cal rings that ma y o ccur. Prop osition 2.4. Supp ose P (or P f ) admits a triangulation. If R is a gr ade d c ommutative lo c al ring with maximal ide al m , then m is princip al and c ontains no nontrivial pr op e r ide als. Pr o of. First, w e observe that, eve n if R is neither graded commutat ive nor lo cal, it must satisfy the double annihilator condition ann l ann r Rx = R x . Let x ∈ R , and let i = | x | . W e ha v e an exact triangle in P R [ i ] x / / R ψ / / P φ / / Σ R [ i ] . Observe that im φ = ann r Rx , and if y ∈ ann l ann r Rx , then ( y · R ) ◦ φ = 0. Exactness at R [ i ] therefore implies that ann l ann r Rx = R x . F or the r emainder of the pr o of, we assume R is graded comm utativ e and lo cal. Since P (or P f ) is triangulated, R is coheren t; this implies that ann( x ) is fin itely generated for all x ∈ R . Consider x ∈ m . W e n ow s ho w that x 2 = 0. S ince ( x · P ) ◦ ψ = ψ ◦ ( x · R ) = 0, w e obtain a factorizati on x · P = f ◦ φ , wh er e f : im φ − → P . Sin ce im φ = ann ( x ), ( x · P ) ◦ f = 0. Hence, x 2 · P = 0. Since R is lo cal, P is free ( P is non trivial since x is n ot a u nit). Th er efore x 2 = 0. Fix a nonzero element x ∈ m . W e next sh o w th at ann ( x ) = ( x ). Note that P is an extension of fin itely generated mo d ules and is therefore finitely generated. Let e 1 , . . . , e n generate P , let φ i = φ ( e i ), and let ψ (1) = Σ e i t i . Since x is nonzero, im φ = ann( x ) ⊆ m ; hence, φ ( e i φ i ) = φ 2 i = 0, and exactness at P pr o vides an elemen t z i ∈ R such that ψ ( z i ) = e i φ i . This implies that t j z i = 0 when i 6 = j , and t i z i = φ i . Since im φ = ann( x ), the elemen ts φ i generate ann( x ). Let t = t 1 + · · · + t n and consider q = Σ φ i a i ∈ ann( x ). W e ha v e t (Σ z i a i ) = q , so ann( x ) ⊆ ( t ). Since ψ ◦ ( x · R ) = 0, we obtain t i x = 0 for all i . Hence, ( t ) = ann( x ). Sin ce x 6 = 0, t is not a unit, so ( t ) ⊆ ann( t ) = ann ann( x ) = ( x ). This prov es that ann ( x ) = ( t ) = ( x ). F u rther, it is worth remarking that P m ust b e f ree of rank 1. Supp ose n ≥ 2. It is clear that e 1 x and e 2 x are in the k ernel of φ . Hence, for s ome a, b ∈ R , e 1 x = ψ ( a ) and e 2 x = ψ ( b ). Th is m eans th at e 1 xb − e 2 xa = 0, forcing a, b ∈ ann( x ) = ( x ). But sin ce t i x = 0 for all i , it must b e the case that e 1 x = ψ ( a ) = Σ e i t i a = 0, a contradicti on. Finally , w e show th at m m ust b e the unique, p rop er, non trivial, principal ideal of R . Supp ose a ∈ m and b ∈ R hav e the pr op erty that ab 6 = 0. Then, a ∈ ann( ab ) = ( ab ). Hence, there is an elemen t k ∈ R s u c h that a (1 − bk ) = 0. Since R is lo cal, this forces a to b e zero or b to b e a un it. No w consider t wo n on zero elemen ts x, y ∈ m . W e hav e x ∈ ann( y ) = ( y ) and y ∈ ann( x ) = ( x ), so ( x ) = ( y ). This forces m to b e the u nique, prop er, n on trivial, principal ideal of R .  Prop osition 2.5. L et R b e a gr ade d c ommutative lo c al ring with r esidue field k . If P (or P f ) admits a triangulation with susp ension Σ = ( − )[ n ] , then R is either (1) the gr ade d field k , 6 MARK HOVEY A ND KEIR LOCKRIDGE (2) an exterior algebr a k [ x ] / ( x 2 ) with a unit in de gr e e 3 | x | + n , wher e c har k = 2 if n = 0 , or (3) n is even and R ∼ = T / (4) , wher e T is the unique (up to isomorph ism) c omplete 2 -ring with r esidue field k (of char acteristic 2) c ontaining a u nit in de gr e e n . Pr o of. In the pr o of of Prop osition 2.4, w e show ed th at the cofib er of x · R is free of rank 1 for any nontrivial elemen t x in the maximal ideal m . Hence, an y such x fits into an exact triangle of th e form (1) R [ i ] x / / R vx / / R [ j ] w x / / R [ i + n ] , where i = | x | and v and w are u nits. One can c hec k that | ( v w ) − 1 | = 3 i + n . Supp ose R cont ains a field. As such, R is a r ing of equal c haracteristic; sin ce it is complete, it con tains a field isomorph ic to k = R / m ([11, 28.3]) . S o either m = 0 and R is a graded field, or m = ( x ) for some x 6 = 0 (by Prop osition 2.4) and R is isomorphic to the exterior algebra k [ x ] / ( x 2 ). In the latter case, we just obs er ved that k must con tain a unit of degree 3 i + n . If p is a unit for all p rimes p , then R con tains the fi eld Q . If p is not a unit for some prime, th en either R has c haracteristic p and con tains a field, or p is nonzero and is in the maximal ideal. If p is in the m aximal ideal m , then b y Prop osition 2.4, we must hav e m = ( p ). If n is o dd, this is not p ossible: since p has d egree zero, we saw ab o ve that there m ust b e a un it of degree 3 · 0 + n , w hic h is o dd . But if s is an o dd unit, then 2 s 2 = 0 by graded comm utativit y; this f orces 2 = 0, con tradicting m = ( p ). It r emains to consider the case where n is even and m = ( p ) for some prime p . Let T b e the uniqu e (up to isomorphism) complete p -ring with residue field k (see [11, § 29]). By the discussion on p. 225 of [11], sin ce R is a complete lo cal ring of u nequal c haracteristic, it con tains a co efficien t r ing A ∼ = T / ( p 2 ). F urther, it is observe d in the pr o of of [11, 29.4] that ev ery elemen t can b e exp anded as a p o w er series in the generators of the maximal ideal with co efficien ts in A . S ince the maximal id eal is generated by p ∈ A , we obtain R ∼ = A . T o see why p = 2, consider the rotation of triangle (1), R vp / / R [ j ] w p / / R [ n ] − p / / R [ n ] . The map κ in the diagram R p / / R vp / / v   R [ j ] w p / / κ      R [ n ] R vp / / R [ j ] w p / / R [ n ] − p / / R [ n ] m ust exist. Hence, κv p = w pv = − pκv , so 2 p = 0. This forces p = 2. When n = 0 and R = k [ x ] / ( x 2 ), an argument similar to the one just giv en sho ws that c har k = 2.  TRIANGULA TIONS OF PR OJECTIVE MODULES 7 W e next sho w that, for n = 0 and n = 1, the rings in the conclusion of Prop osition 2.5 are ∆ n -rings. Prop osition 2.6 ([12]) . If T i s a c omplete 2 -ring, then T / (4) is a ∆ 0 -ring. Pr o of. In [12], a triangulation of P f is constructed for an y comm utativ e lo cal ring of charac- teristic 4 with maximal ideal (2), but the pr o of in fact sho ws th at P admits a triangulation. These r ings are exactly th e rings of the form T / (4) (where T i s a complete 2-ring). Cer- tainly , any ring of the form T / (4) is of the t yp e d iscussed in [12], and in Pr op osition 2.5 it is sh o wn that an y commutat ive lo cal ring of characte ristic 4 with maximal ideal (2) is of the form T / (4).  Prop osition 2.7. Every gr ade d field k is a ∆ n -ring for al l n . Every exterior algebr a k [ x ] / ( x 2 ) with a unit in de gr e e 3 | x | + n is a ∆ n -ring, pr ovide d n and | x | ar e not b oth even when char k 6 = 2 . Pr o of. W e will use a graded v ersion of the construction pr esen ted in [12 ]. Fix n ≥ 0. W e will work in the category of d ifferen tial graded mo d ules o v er a differen tial graded algebra A , where the degree of all deriv ations is − n . S o if x, y ∈ A , d ( xy ) = d ( x ) y + ( − 1) n | x | xd ( y ) . This category admits a triangulation w ith sus p ension functor ( − )[ n ]. F or example, if u ∈ A has degree i , then there is an exact triangle A [ i ] u / / A / / A ⊕ A [ i + n ] / / A [ i + n ] , where the d ifferen tial on A [ j ] is ( − 1) j d and the d ifferential D on A ⊕ A [ i + n ] is D ( a, b ) = ( da + ub, ( − 1) i + n db ) . Let A = k with zero differen tial. T rivially , the homology of an y differential graded A - mo dule is pro jective , so h omology indu ces an equiv alence of categorie s from D ( A ) (the deriv ed category of A ) to P = Mo d ( k ) by Prop osition 3.1 (4). F or the exterior algebra k [ x ] / ( x 2 ) w ith a un it v in degree 3 | x | + n , w e use the differen tial graded algebra constructed in [12]. Let i = | x | , and let a and u b e s y mb ols w ith degrees | a | = 2 i + n and | u | = i . Let A = k h a, u i /I , where I is the t wo sid ed ideal generated by the homogeneous elements a 2 and au + ua + v (here we see w h y the existence of the unit v is necessary). Define the d ifferen tial on A b y da = u 2 and du = 0. One can chec k that th is differen tial is well-defined; if i and n ha ve opp osite parit y , then v h as o dd d egree, and so c har k = 2 by graded commuta tivit y and signs do not matter. If i and n are b oth o d d, th en the signs w ork out indep enden t of th e c haracteristic of k . The d ifferen tial is not well-defined if i and n are b oth even and char k 6 = 2; fortunately , if n = 0, char k = 2 is forced (see Prop osition 2.5). As in [12], it is straigh tforward to c hec k that H ∗ A ∼ = k [ x ] / ( x 2 ), where x is the h omology class of u , and for any differential graded A -mo dule M , H ∗ M is pro jectiv e. Again we see that homology induces an equ iv alence of catego ries from D ( A ) to P .  8 MARK HOVEY A ND KEIR LOCKRIDGE Observe that, for any integ er n , every ∆ n -ring is a ∆ n f -ring. F or if P admits a triangula- tion, then thic k h R i admits a triangulation. Since R m ust b e No etherian, thick h R i = P f . Prop ositions 2.6 and 2.7 n o w imply Prop osition 2.8. L et n ∈ { 0 , 1 } . E very ring R in the c onclusion of Pr op osition 2.5 is a ∆ n -ring. Henc e, the classes of gr ade d c ommutative lo c al ∆ n -rings and gr ade d c ommutative lo c al ∆ n f -rings c oincide. According to [9, 15.27], a commuta tive rin g is quasi-F rob enius if and only if it is a finite pro d uct of lo cal Artinian rings w ith simple so cle. The follo wing prop osition allo w s us to restrict our atten tion to the lo cal case. Com bined with the ab ov e characte rization of comm utativ e lo cal ∆ n -rings for n ∈ { 0 , 1 } , this completes th e p ro ofs of T heorems 1.1 and 1.2. Prop osition 2.9. Supp ose R ∼ = A × B . The c ate gory of pr oje ctive R - mo dules admits a triangulation if and only if the c ate gories of pr oje ctive A -mo dules and pr oje c tive B -mo dules e ach admit a triangulation. The same is true for finitely g e ner ate d pr oje ctive mo dules. Pr o of. This is a consequence of th e f act th at Mo d ( R ) ∼ = Mo d ( A ) × Mod ( B ).  As a fi nal n ote, we b roaden the scop e of the second statemen t of P r op osition 2.8. Prop osition 2.10. L et n ∈ { 0 , 1 } . The classes of gr ade d c ommutative No e therian ∆ n -rings and gr ade d c ommutative No etherian ∆ n f -rings c oincide. Pr o of. It su ffices to chec k that every comm utativ e No etherian ∆ n f -ring R is a ∆ n -ring. Since R must b e an IF-ring (Prop osition 2.2), it m us t b e a commutativ e No etherian self-injectiv e ring ([4, 6.9]). Th is makes R quasi-F r ob enius by defin ition, an d therefore a pro du ct of lo cal rings, eac h of whic h m ust b e a ∆ n -ring b y Prop osition 2.8 (sin ce eac h lo cal ring is a ∆ n f -ring). Hence, R is a ∆ n -ring.  3. The genera t ing hypothe sis W e n o w giv e an application of the ab o v e classification to the global generating hypothesis. First, we mak e a general observ ation. A triangulated category T is c o c omplete if arbitrary copro ducts exist in T . W e call a π ∗ S -mo du le M r e alizable if there is an ob ject X ∈ T such that π ∗ X = M . If T is co complete, then idemp oten ts split in T ([2]), so every p ro jectiv e π ∗ S -mo du le is realizable as π ∗ Y , wh ere Y ∈ h S i 0 , the collection of retracts of copro ducts of susp ensions of S . Hence, π ∗ induces an equiv alence of categ ories Φ : h S i 0 − → P (recall that P is th e category of pr o jectiv e π ∗ S -mo du les). W rite lo c h S i for the lo c alizing sub c ate gory ge ner ate d b y S (the smallest fu ll su b category of T con taining S that is thic k and closed u nder arbitrary copro ducts). If T = lo c h S i , then we say that S gener ates T . W e ha v e the follo w in g equiv alen t forms of the global generating hyp othesis; for this prop osition, w e do not require π ∗ S to b e graded commutat ive. TRIANGULA TIONS OF PR OJECTIVE MODULES 9 Prop osition 3.1. L et T b e a c o c omplete triangulate d c ate gory. The fol lowing ar e e qu iva- lent: (1) T satisfies the glob al gener ating hyp othesis. (2) T = h S i 0 , the c ol le ction of r etr acts of arbitr ary c opr o ducts of susp ensions of S . (3) The functor π ∗ ( − ) i s ful l and faithful. If S is a c omp act gener ator of T , then these c onditions ar e e q uivalent to (4) Eve ry r e alizable mo dule is pr oje ctive. If any one of the ab ove c onditions is satisfie d, then the c ate gory P of pr oje ctive right π ∗ S - mo dules admits a triangulation with shift functor Σ = ( − )[1] . Pr o of. (1) = ⇒ (2). Assume T sa tisfies the global generating hyp othesis. Let Y b e an ob ject of T . Using a generating set for the π ∗ S -mo du le π ∗ Y , construct a map f : X − → Y suc h that X ∈ h S i 0 and π ∗ f is su rjectiv e. Th is map fits in to an exact triangle X f / / Y g / / Z / / Σ X . Since π ∗ f is su rjectiv e, π ∗ g = 0. By the global generating hyp othesis, g = 0, forcing Y to b e a r etract of X . Hence, T = h S i 0 . (2) = ⇒ (3). This implication is trivial, since Φ is an equiv alence of categories. (3) = ⇒ (1). This implication is tru e by definition of the global generating hyp othesis. (2) = ⇒ (4). This implication is trivial. (4) = ⇒ (2 ). Assu me S is compact. F or any X ∈ lo c h S i , π ∗ X = 0 if and only if X is trivial. Consequen tly , for any map f in lo c h S i , π ∗ f is an isomorphism if and only if f is an equiv alence. No w fix X ∈ lo c h S i . Since π ∗ X is pr o jectiv e, there is a map f : Y − → X suc h that Y ∈ h S i 0 ⊆ lo c h S i and π ∗ f is an isomorphism. Hence, X is equiv alen t to Y . Since T = lo c h S i , the imp lication is established. If an y of these conditions hold, then Φ ma y b e used to end ow P with a triangulation. In this triangulation, Σ = ( − )[1] since π ∗ Σ X = π ∗− 1 X = π ∗ X [1].  Remark 3.2. Supp ose T satisfies Bro wn Representa bility (i.e., cohomologi cal functors are represent able); for example, this holds in an y co complete, compactly generated triangulated catego ry . If I is an injectiv e π ∗ S -mo du le, then the functor Hom 0 ( π ∗ ( − ) , I ) is a cohomological functor on T , hence representa ble by an ob ject E ( I ) ∈ T . I n this situation, π ∗ E ( I ) = I , so ev ery injectiv e mo dule is r ealizable. I f ev ery realizable mo dule is pro jectiv e, then ev ery injectiv e mo du le is pro jectiv e. Similarly , if ev ery realizable mo du le is injectiv e, then ev ery pro jectiv e mo d ule is in j ectiv e. In either case, π ∗ S is quasi-F rob enius, and the inj ectiv e, pro jectiv e, and realizable mo dules coincide. Hence, if T satisfies Brown Representa bilit y , then it satisfies the strong generating hyp othesis if and only if th e realizable mo dules and the pro jectiv e mo dules coincide, if an d only if the realizable mo du les and the in jectiv e mo dules coincide. In the graded comm utativ e case, then, we see that if T satisfies the global generating h yp othesis, then π ∗ S m ust b e a pro d uct of fields and exterior algebras on one generator 10 MARK HOVEY A ND KEIR LOCKRIDGE b y Theorem 1.1. Ho wev er, the conv erse of this statemen t if false; th e triangulation on P ma y not b e indu ced by π ∗ . F or example, consider the stable mo dule category StMo d( k [ G ]) asso ciated to a fi nite p -group G and field k of c haracteristic p . In StMo d( k [ G ]), π ∗ S = ˆ H ( G ; k ), the T ate cohomology of G . F or n ≥ 1, ˆ H ( Z / (3 n ); F 3 ) ∼ = F 3 [ y , y − 1 ][ x ] / ( x 2 ), w h ere | x | = 1 an d | y | = 2. Ho w ev er, it is s ho wn in [1] that StMo d ( F 3 [ Z / (3)]) satisfies the global generating hypothesis, though StMo d( F 3 [ Z / (3 n )]) do es not for n ≥ 2. Th e relev an t condition is in cluded in Corollary 1.3, whic h w e now pro ve. Pr o of of Cor ol lary 1.3 . First, note that eve ry s table homotopy category has a s ymmetric monoidal structure (for which S is the unit) compatible with the triangulation. This forces R = π ∗ S to b e graded commutat ive ([8, A.2.1]). F urth er , we may u se the monoidal pro du ct to take an y elemen t x ∈ π ∗ S and obtain a map x · X for an y X ∈ T . F or example, if e ∈ π ∗ S is idemp oten t, then e · X is an id emp otent en d omorphism of X . S ince idemp oten ts split, there is a decomp osition X ≃ Y ∨ Z s uc h that π ∗ Y ∼ = eπ ∗ X and π ∗ Z = (1 − e ) π ∗ X (see [8, 1.4.8]) . Supp ose T satisfies the global generating hyp othesis. By Pr op osition 3.1, the category P of p ro jectiv e mo d ules o ve r π ∗ S admits a triangulation with susp ension f unctor ( − )[1]. By Theorem 1.1, condition (1) is s atisfied, and π ∗ S ∼ = R 1 × · · · × R n , where R i is either a graded field k or an exterior algebra k [ x ] / ( x 2 ). S ince idemp oten ts s p lit in T , S ∼ = A 1 ∨ · · · ∨ A n , wh ere π ∗ A i = R i . The cofib er C i of x · A i is a summand of C , the cofib er of x · S . Sin ce π ∗ C i m ust b e a nontrivia l fr ee R i -mo dule, x · C i 6 = 0. Th is pro ve s (2). Con ve rsely , if condition (1) holds, then R and S admit decomp ositions as ab ov e. W e now sho w that it suffi ces to assume R is lo cal. Let T i = lo c h A i i ; lo c h A i i is an ideal b y [8, 1.4.6]. Ev ery X ∈ T admits a decomp osition X ∼ = X 1 ∨ · · · ∨ X n , where X i ∈ T i (tak e X i = A i ∧ X ). W e claim that this decomp osition is orthogonal, in th at Hom T ( T i , T j ) = 0 whenever i 6 = j (i.e., ev ery map f rom an ob ject in T i to an ob ject in T j is trivial). First, obs er ve that if i 6 = j , then an y m ap f : A i − → A j is trivial, as follo ws. It s u ffices to sho w that the induced map f ∗ : R i − → R j is zero. Let e k ∈ π ∗ S b e the idemp otent corresp ond ing to R k . W e no w hav e f ∗ ( x ) = f ∗ ( e i x ) = e i f ∗ ( x ) = 0, for all x ∈ R i . So in deed, [ A i , A j ] ∗ = 0. No w, [ A i , − ] ∗ v anishes on lo c h A j i since A i is compact; therefore [ − , X ] ∗ v anishes on lo c h A i i for an y X ∈ lo c h A j i . In summary , there is an orthogonal decomp osition T = lo c h S i ∼ = T 1 ∨ · · · ∨ T n . W e now see that π ∗ is f aithful on T if and only if the fun ctors [ A i , − ] ∗ are faithful on T i for i = 1 , . . . , n . It th er efore su ffices to assum e that R is a lo cal rin g. W e m ust no w sho w that if R = π ∗ S is lo cal and satisfies conditions (1) and (2), then π ∗ is faithful. By Prop osition 3.1 (4), we n eed only chec k that π ∗ X is alwa ys p r o jectiv e (or f r ee, since R is lo cal). If R is a graded field, th en this is trivially tr ue. If R ∼ = k [ x ] / ( x 2 ), then condition (2) tells us that π ∗ C is f r ee, where C is th e cofib er of x · S . Sin ce ev ery map of free k [ x ] / ( x 2 )-mo dules is the copro duct of a trivial map, an isomorphism , an d m ultiplication by x , it is easy to c hec k that π ∗ X is free for all X ∈ thic k h S i . F or arbitrary X ∈ T , π ∗ X is the direct limit of a sys tem of m o dules of th e form π ∗ X α , where X α ∈ thick h S i ([8, 2.3.11]). TRIANGULA TIONS OF PR OJECTIVE MODULES 11 Since an y direct limit of fl at mo dules is fl at, π ∗ X is flat. Ov er k [ x ] / ( x 2 ), flat implies free. This completes th e pro of.  Remark 3.3. Let T b e the category of pro jectiv e mo dules ov er Z / (4) endo wed with the triangulation asso ciated to Σ = 1 , and let S = Z / (4). It is, of course, the case that π ∗ acts faithfully in th is situation. Ho wev er, the ring π ∗ S = L i ∈ Z Z / (4) is not graded comm utativ e, so this do es not con tradict Corollary 1.3. Finally , we study the strong generating hypothesis. W rite h S i 0 f for the collection of retracts of fi nite copro d ucts of s usp ensions of S . As ab o v e, π ∗ induces an equiv alence of catego ries Φ f : h S i 0 f − → P f . W e hav e the follo wing equiv alen t forms of the strong generating h yp othesis. Prop osition 3.4. L e t T b e a c o c omplete triangulate d c ate gory, and assume S is c omp act. The f ol lowing ar e e quivalent: (1) T satisfies the str ong gener ating hyp othesis. (2) thick h S i = h S i f 0 , the c ol le ction of r etr acts of finite c opr o ducts of susp ensions of S . (3) F or al l X ∈ thic k h S i , π ∗ X is pr oje ctive. If any one of the ab ove c onditions is satisfie d, then the c ate g ory P f of finitely g ener ate d pr oje ctive right π ∗ S -mo dules admits a triangulation with su sp ension functor Σ = ( − )[1] . Pr o of. (1) ⇐ ⇒ (2). Supp ose T satisfies the s trong generating h yp othesis. As in the p r o of of Pr op osition 3.1, we s ee that any X ∈ thic k h S i is a retract of a copro du ct of susp ensions of S . Since S is compact, so is X ; it must th er efore b e a retract of a finite copro duct. Hence, (1) implies (2). T h e conv erse is tr ivial. (2) ⇐ ⇒ (3). The pro of of this equiv alence is similar to the pro of of its analogue in Prop osition 3.1. Note that we do not need to assu me T = lo c h S i sin ce thic k h S i ⊆ lo c h S i . If any of these equiv alent conditions hold, th en Φ f ma y b e used to endo w P f with a triangulation with susp ension Σ = ( − )[1].  Remark 3.5. This remark is a companion to Remark 3.2. Assume T is a monogenic s table homotop y categ ory and a Bro wn category (see [8, 4.1.4]). F or all X ∈ T , π ∗ X is the direct limit o v er a system of mo dules of th e f orm π ∗ X α , where X α ∈ thick h S i . Consequently , if T satisfies th e strong generating hyp othesis, then every realizable mo dule is flat ([9, 4.4]), and eve ry flat mo dule is realizable (since ( − ) ⊗ π ∗ S F is representable for an y flat m o dule F ). On the other hand, representa bility of cohomologica l functors imp lies that arb itrary pro du cts of π ∗ S are r ealizable. If ev ery realizable mo dule is fl at, this forces π ∗ S to b e coheren t ([9, 4.47]). No w, for all X ∈ thick h S i , π ∗ X is a finitely presented flat mo dule and therefore p ro jectiv e ([9, 4.30]) . In summary , if T is a monogenic stable homotopy category and Brown category , th en it satisfies the strong generating hyp othesis if and only if the fl at mo dules and realizable mo dules coincide. F or example, in the d eriv ed category of a ring R , 12 MARK HOVEY A ND KEIR LOCKRIDGE ev ery R -mo dule is realizable, and the strong generating hyp othesis is true if and only if all mo dules are flat, if and only if R is von Neumann regular ([7]). W e next s ho w that, if either form of the generating h yp othesis holds, then it also holds lo cally . F or any π ∗ S -mo du le M , let M p denote the lo calizat ion of M at the p rime id eal p . Supp ose T is a co complete triangulated category with compact generator S . The pro ofs of [8, 2.3.17] and [8, 3.3.7] sho w that there exists a lo calization fu nctor L p on T suc h that π ∗ L p X ∼ = ( π ∗ X ) p . W e mean ‘lo calization functor’ in th e s ense of d efinition [8, 3.1.1], omitting the condition inv olving smash pr o ducts, sin ce T ma y not h a v e a pro duct stru ctur e. In particular, the n atural map Y ι Y / / L p Y induces an isomorphism (2) [ L p Y , L p X ] ∗ ∼ = / / [ Y , L p X ] ∗ . Call an ob ject p -lo c al if it lies in th e image of L p , and let T p denote the fu ll su b category of p -local ob j ects. S ince S is compact, L p comm utes with copro ducts; h ence, T p = lo c h L p S i , and L p S is compact in T p . Prop osition 3.6. Su pp ose T is a c o c omplete triangulate d c ate gory with c omp act gener ator S . If T satisfies the str ong or glob al g ener ating hyp othesis, then so do es T p for al l prime ide als p in π ∗ S Pr o of. Consid er a map f : L p X − → L p Y . Because of the n atural isomorp hism (2), f = 0 if and only if ˜ f = f ◦ ι X = 0. F or an y α ∈ π ∗ X , w e hav e a comm utativ e diagram S α / / ι S   X ι X   ˜ f # # G G G G G G G G G L p S L p α / / L p X f / / L p Y . Hence, if [ L p S, f ] ∗ = 0, then [ S, ˜ f ] ∗ = 0. W e immediately see that if T satisfies the global generating hyp othesis, then so do es T p . The same argument app lies if T satisfies the strong generating hypothesis, b ecause X is compact in T if and only if L p X is compact in T p .  W e end this section with the p ro of of Corollary 1.4. Pr o of of Cor ol lary 1.4 . Assume T satisfies the s trong generating hyp othesis. Since T is a monogenic stable homotopy category , the un it ob ject S is a compact generator. F urth er , R = π ∗ S is comm utativ e. By Prop ositions 3.6 and 3.4, R p is a ∆ 1 f -ring. S ince it is a graded comm utativ e lo cal rin g, it must b e ∆ 1 -ring by Prop osition 2.8 . By Prop osition 2.5, R is either a grad ed field k or an exterior algebra k [ x ] / ( x 2 ) o ve r a field with a u nit in degree 3 | x | + 1. W e turn no w to th e last statemen t of the corollary . Certainly , the global generating hypothesis alw a ys implies the strong generating hyp oth- esis. Assum e th e s tr ong generating h yp othesis is true. By Prop osition 3.4, R is a ∆ 1 f -ring. If R is lo cal or No etherian, then it is also a ∆ 1 -ring by Prop osition 2.8 or 2.10. W e no w wish to inv ok e C orollary 1.3. T o do so, w e must ve rify condition (2). Arguing as in the TRIANGULA TIONS OF PR OJECTIVE MODULES 13 pro of of C orollary 1.3, it su ffices to assume that R ∼ = k [ x ] / ( x 2 ). F or this ring, condition (2) is imp lied by the fact that π ∗ C must b e fr ee by th e strong generating hyp othesis.  4. Ring sp ectra In this brief section, w e translate our results in to the setting of r ing sp ectra, wh ere by ‘ring sp ectrum’ w e mean either a sym metric ring sp ectrum or an S -algebra. Let E b e a ring sp ectrum, and let D ( E ) denote the deriv ed category of E -mo du les. As in § 3, write h E i 0 for the collecti on of retracts of copro d ucts of su sp en sions of E ; call these ob j ects pr oje ctive E -mo dule sp e ctr a . Define E to b e semisimple if ev ery E -mo dule sp ectrum is p ro jectiv e (i.e., if D ( E ) = h E i 0 ), and call E von Neumann r e gular if ev ery compact E -mo dule sp ectrum is pro jectiv e (i.e., if thick h E i = h E i 0 f ). The relationship b etw een s u c h r ing sp ectra and the generating h yp othesis is giv en by Prop osition 4.1. L et E b e a ring sp e ctrum. E is semisimple if and only if D ( E ) satisfies the glob al gene r ating hyp othesis. E is v on Neumann r e gu lar if and only if D ( E ) satisfies the str ong gener ating hyp othesis. Pr o of. The category D ( E ) is a co complete tr iangulated category with compact generator E . Th e c haracterization of semisimplicit y follo ws from Prop osition 3.1; the characte rization of v on Neumann regularit y follo ws f rom Prop osition 3.4.  The follo wing prop osition sho ws that these definitions are consisten t with the ones in ordinary ring theory . Prop osition 4.2. L et H R denote the Eilenb er g- M ac L ane sp e ctrum asso ciate d to the ring R . H R is semisimple if and only if R is semisimple, and H R is von Neumann r e gu lar if and only if R is v on Neumann r e gular. Pr o of. It is well-kno wn (see, for example, [3 , IV.2.4]) that the d er ived categ ory of H R - mo dule sp ectra is equiv alen t to the derive d category of R -mo dules, D ( R ). In [10, § 4] it is sho wn that D ( R ) satisfies the global generating hyp othesis if and on ly if R is semisimple, and in [7, 1.3] it is shown that D ( R ) satisfies the strong generating hypothesis if and only if R is v on Neumann regular.  W e w ould like to h a v e a full classification of the semisimp le and v on Neumann regular ring sp ectra; w e now summarize the application of the foregoing work to this problem. Prop osition 4.3. L e t E b e a ring sp e ctrum. If π ∗ E is c ommutative, then (1) If E is semisimple, then π ∗ E ∼ = R 1 × · · · × R n , wher e R i is e i ther a gr ade d field k or an exterior algebr a k [ x ] / ( x 2 ) over a gr ade d field c ontaining a u ni t in de g r e e 3 | x | + 1 (i.e., π ∗ E is a gr ade d c ommutative ∆ 1 -ring). (2) If E is von Neu mann r e gular, then ( π ∗ E ) p is a gr ade d c ommutative lo c al ∆ 1 -ring for eve ry prime ide al p of π ∗ E . If E is c ommutative, then 14 MARK HOVEY A ND KEIR LOCKRIDGE (3) E is semisimple if and only if π ∗ E is a gr ade d c ommutative ∆ 1 -ring and for ev ery factor ring of π ∗ E of the f orm k [ x ] / ( x 2 ) , x · π ∗ C 6 = 0 , wher e C is the c ofib er of x · E . (4) If π ∗ E is lo c al or No etherian, then E is semisimple if and only if E is von Ne umann r e gu lar. According to [14], ev ery simplicial, cofibran tly generated, prop er, s table mo del category with a compact generator P is Quillen equiv alen t to the mo d ule category of a certain endomorphism ring sp ectrum En d( P ). This is true in particular for the d eriv ed category of a differential graded algebra. In ligh t of the pro of of Prop osition 2.7, we see that ev ery graded comm utativ e ∆ 1 -ring arises as π ∗ E for some (not necessarily commutati ve) ring sp ectrum E . Because of Prop osition 4.3 (3), how ever, π ∗ E b eing a graded commutativ e ∆ 1 -ring is not sufficien t to conclude that E is semisimp le. Referen ces [1] David J. Benson, Sunil K. Cheb olu, J. Daniel Christensen, and J´ an Min´ a ˇ c. The generating hyp othesis for th e stable modu le category of a p -group. arXiv :math/061140 3 v 3. [2] Marcel B¨ okstedt and Amnon N eeman. Homotopy limits in triangulated categories. Com p ositio Math. , 86(2):209– 234, 1993. [3] A. D. Elmendorf, I. Kriz, M. A. 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An intr o duction to homolo gic al algebr a , volume 38 of Cambridge Studies i n A dvanc e d Mathematics . Cam bridge Un iversi ty Press, Cambridge, 1994. TRIANGULA TIONS OF PR OJECTIVE MODULES 15 Dep ar tment of Ma thema tics, W esley an Unive rsity, Mi ddletown, CT 06459 E-mail addr ess : hovey@member.ams .org Dep ar tment of Ma thema tics, W esley an Unive rsity, Mi ddletown, CT 06459 E-mail addr ess : keir@alumni.rice .edu