THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION

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1 THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION THOMAS MARLEY AND MARCUS WEBB Abstract. Let R be a commutatve Noetheran local rng of prme characterstc p and f : R R the Frobenus rng homomorphsm. For e 1 let R (e) denote the rng R vewed as an R-module va f e. Results of Peskne, Szpro, and Herzog state that for fntely generated modules M, M has fnte projectve dmenson f and only f Tor R (R (e), M) = 0 for all > 0 and all (equvalently, nfntely many) e 1. We prove ths statement holds for arbtrary modules usng the theory of flat covers and mnmal flat resolutons. 1. Introducton Let R be a commutatve Noetheran rng of prme characterstc p and f : R R the Frobenus map. For e 1 let R (e) be the rng R consdered as an R-module va f e ;.e., for r R, s R (e), r s := r pe s. A classc result of Kunz [8] states that R s regular f and only f R (e) s flat as an R-module for all (equvalently, some) e 1. Subsequently, Peskne and Szpro [9, Théorèm 1.7] proved that f P s a fnte projectve resoluton of a fntely generated R-module M then for all e 1, R (e) R P s a projectve resoluton of R (e) R M; that s, fntely generated modules of fnte projectve dmenson are acyclc objects wth respect to the Frobenus functors R (e) R. A year later Herzog [6, Satz 3.1] showed the converse holds n the case the Krull dmenson of R s fnte; namely, f M s a fntely generated R-module and P s a projectve resoluton of M such that R (e) R P s acyclc for nfntely many ntegers e, then M has fnte projectve dmenson. An nterestng queston s whether these results hold for arbtrary R-modules, not just fntely generated ones. In Corollary 4.5(a) and Theorem 5.2, we gve an affrmatve answer to ths queston: Theorem 1.1. Let R be a Noetheran rng of prme characterstc and M an R-module. Then the followng hold: (a) If fd R M < then for all, e > 0, Tor R (R (e), M) = 0 and fd R (e) R (e) R M = fd R M. (b) If R has fnte Krull dmenson and Tor R (R (e), M) = 0 for all > 0 and nfntely many ntegers e, then pd R M <. Here fd R M and pd R M denote the flat and projectve dmensons of M, respectvely. Note that part (a) mples that f pd R M < then pd R (e) R (e) R M pd R M for all e (although we do not know of any examples where the nequalty s strct). We also prove an analogue of Theorem 1.1 for njectve dmenson n the case the Frobenus map s fnte (cf. Corollares 4.5(b) and 5.3). Date: December 29, Mathematcs Subject Classfcaton. Prmary 13D07; Secondary 13A35. Key words and phrases. flat dmenson, Frobenus map, Bass number, flat cover. 1

2 2 THOMAS MARLEY AND MARCUS WEBB A specal case of part (a) of Theorem 1.1 s well-known and s what nspred ths nvestgaton: Suppose x = x 1,..., x n s a regular sequence on R and C = C(x; R) s the Čech complex on R wth respect to x. Then C s a fnte flat resoluton of the local cohomology module H(x) n (R) and R(e) R C = C(x; R (e) ) = C(x pe ; R) s acyclc snce x pe 1,..., xpe n s an R-sequence. The proofs of Peskne-Szpro and Herzog both reduce to the case R s local and utlze mnmal projectve (free) resolutons. The mnmalty condton for such projectve resolutons can be expressed by sayng that all the dfferentals are zero when tensored wth the resdue feld. It can be easly shown that projectve resolutons of ths type do not necessarly exst for arbtrary modules. However, flat resolutons wth ths knd of mnmalty condton do exst for a large class of modules (.e., cotorson modules), whch we show s suffcent to prove Theorem 1.1. The theory of flat covers, cotorson modules, and mnmal flat resolutons, as developed by Enochs and Xu n [4] and [5], are essental ngredents n all our arguments. In Secton 2, we summarze some basc propertes of these notons as well as prove some auxlary results whch are used n later sectons. Mnmal flat resolutons have many propertes analagous to those of mnmal njectve resolutons. In partcular, the flat modules appearng n a mnmal flat resoluton of a module M are unquely determned (up to somorphsm) by nvarants whch we call the Enochs-Xu numbers of M. The Enochs-Xu numbers of a module are n some sense the dual of the Bass numbers of a module. In Secton 3, we prove a vanshng result for Enochs-Xu numbers whch wll be crucal n our proof of part (a) of Theorem 1.1. Sectons 4 and 5 are devoted to the proofs of parts (a) and (b), respectvely, of Theorem 1.1. For both parts t s suffcent to consder the case when R s a local rng, n whch case every module of fnte flat dmenson has fnte projectve dmenson by the Jensen-Raynaud- Gruson theorem ([7, Proposton 6] and [10, Seconde parte, Théorème 3.2.6]). Our strategy s to apply the methods of Peskne-Szpro and Herzog to mnmal flat resolutons. There are several dffcultes whch arse: the fntstc flat dmenson may exceed the depth of the rng (cf. [2, Corollary 5.3]); mnmal flat resolutons do not n general localze; and the modules appearng n mnmal flat resolutons are not generally fntely generated. We are able to overcome these dffcultes by, n part, provng that the depths of the modules n degrees exceedng depth R n fnte mnmal flat resolutons are nfnte, and showng that, n the case the Frobenus map s fnte, the acyclcty of Frobenus for mnmal flat resolutons of cotorson modules commutes wth the colocalzaton functor Hom R (R p, ). And whle the flat modules n queston are not typcally fntely generated, we are able to reduce to the case where they are completons of free R-modules. Acknowledgments: The authors would lke to thank Edgar Enochs for hs help n our understandng of flat covers. The authors also express ther grattude to everyone n the UNL algebra semnar, n partcular Lucho Avramov, Peder Thompson, Mark Walker, and Wenlang Zhang, for ther useful feedback on ths project. 2. Flat covers and cotorson modules In ths secton we collect the results we wll regardng flat covers, cotorson modules, and mnmal flat resolutons. We refer the reader to [13] for background on ths materal.

3 THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION 3 Defnton 2.1. Let M be an R-module. An R-homomorphsm ϕ : F M s called a flat cover of M f the followng hold: (a) F s flat; (b) for every map ψ : G M wth G flat, there exsts a homomorphsm g : G F such that ψ = ϕg; and (c) f h : F F satsfes ϕ = ϕh then h s an somorphsm. By abuse of language, we sometmes refer to the module F as the flat cover of M, rather than the homomorphsm ϕ. In [3], t s proved that flat covers exsts for all modules over all rngs. However, n ths work we wll only need the exstence of flat covers for modules over commutatve Noetheran rngs of fnte Krull dmenson, whch was proved n [12]. It s easly seen that flat covers are surjectve (snce every module s a homomorphc mage of a flat module), that a flat cover of a module s unque up to somorphsm, and that the flat cover of a flat module s an somorphsm. Defnton 2.2. An R-module M s called cotorson f Ext 1 R (F, M) = 0 for every flat R-module F. It s easly seen that f M s cotorson then Ext R (F, M) = 0 for all 1 and all flat R- modules F. By [4, Lemma 2.2 and Corollary], the kernel of a flat cover s cotorson and a flat cover of a cotorson module s cotorson. For an R-module M we let C R (M) denote the cotorson envelope of M, whch exsts by [13, Theorem 3.4.6]. The cotorson envelope of a flat module s flat by [13, Theorem 3.4.2]. If F s flat and cotorson then F = p Spec R T (p) where each T (p) s the completon wth respect to the pr p -adc topology of a free R p -module G(p); furthermore, the ranks of the free R p -modules G(p) are unquely determned by F ([4, Theorem]). For each p Spec R, let π(p, F ) denote the rank (possbly nfnte) of G(p). Gven an R-module M, a mnmal flat resoluton of M s a complex F : F F 1 F 0 0 such that H (F) = 0 for > 0, H 0 (F) = M, and for each, the natural map F coker +1 s a flat cover. It s clear that every R-module has a mnmal flat resoluton, and that any two mnmal flat resolutons of M are (chan) somorphc. Snce the flat cover of a flat module s an somorphsm, t follows that f fd R M = n < then the length of any mnmal flat resoluton of M s n. Note that, by the remarks n the precedng paragraph, f F s a mnmal flat resoluton of M then F s cotorson for all 1. For each 1 and p Spec R we set π (p, M) := π(p, F ). For = 0 and p Spec R, we set π 0 (p, M) := π(p, C R (F 0 )). Note that f M s cotorson, so s F 0 and thus π 0 (p, M) = π(p, F 0 ) for all p Spec R. We call the the nvarants π (p, M) the Enochs -Xu numbers of M. Note that fd R M n f and only f π (p, M) = 0 for all p Spec R and > n. We now state a theorem of Enochs and Xu: Theorem 2.3. ([5, Theorem 2.1 and 2.2]) Let R be a Noetheran rng. Then for any R-module M we have: (a) π (p, M) = π (p, C R (M)) for all 0 and p Spec R. (b) If M s cotorson then π (p, M) = dm k(p) Tor Rp (k(p), Hom R (R p, M)) for all 0 and p Spec R. In partcular, note that part (a) of ths theorem mples that fd R M = fd R C R (M).

4 4 THOMAS MARLEY AND MARCUS WEBB We now establsh some addtonal results on cotorson modules whch are needed n sectons 3 and 4. We thank Edgar Enochs for showng us a proof of part (b) of the followng lemma, whch s mplct n [5]: Lemma 2.4. Let R be a Noetheran rng and S a flat R-algebra. (a) If C s a cotorson R-module and T a flat S-module then Hom R (T, C) s a cotorson S-module. (b) If F s a flat and cotorson R-module and p Spec R then Hom R (R p, F ) s a flat and cotorson R p -module. Proof. To prove (a), let A be a flat S-module. As S s flat over R, any flat S-module s also flat as an R-module. Thus, Ext R (P S A, C) = 0 for all > 0 and all projectve S-modules P. Thus, we have a Grothendeck spectral sequence Ext p R (TorS q (T, A), C) Ext p+q S (A, Hom R(T, C)) (cf. [11, Theorem 11.54]). The spectral sequence collapses, gvng Ext S (A, Hom R(T, C)) = Ext R (T S A, C) = 0 for > 0, as T S A s a flat S-module and therefore flat as an R-module as well. Hence, Hom R (T, C) s a cotorson S-module. For (b), as F s flat and cotorson we have F = q Spec R T (q) where each T (q) s the completon of a free R q -module. We ll show that Hom R (R p, F ) = q p T (q), whch s flat and cotorson by [4, Theorem]. For each q Spec R let ρ q : F T (q) be the projecton map. Let ϕ Hom R (R p, F ). Then, as R p s dvsble by each element n R \ p, the same s true for the mage (ρ q ϕ)(r p ) for all q. Suppose q p. Let r q \ p. Then for all n 1 we have (ρ q ϕ)(r p ) r n (ρ q ϕ)(r p ) q n T (q). As T (q) s separated n the qr q -adc topology, we conclude that (ρ q ϕ)(r p ) = 0. Thus, Hom R (R p, F ) = Hom R (R p, G), where G = q p T (q). But snce each T (q) for q p s an R p-module, G s an R p -module. Thus, Hom R (R p, F ) = G. Lemma 2.5. Let R be a Noetheran rng of fnte Krull dmenson and M an R-module. Suppose M has a (left) cotorson resoluton C;.e., there exsts an exact sequence where each C s cotorson. Then ϕ C C 1 C 0 M 0 (a) M s cotorson. (b) For any flat R-module T, Hom R (T, C) s a cotorson resoluton of Hom R (T, M). Proof. For each nteger 0 let K = coker ϕ +1. Let T be a flat R-module. Snce Ext R (T, C j) = 0 for all 1 and j 0, we have that Ext 1 R (T, M) = Ext +1 R (T, K ) for all 0. Let n = dm R. Then Ext n+1 R (T, K n) = 0 as pd R T n by the Jensen- Raynaud-Gruson theorem ([7, Proposton 6] and [10, Seconde parte, Théorème 3.2.6]). Thus, Ext 1 R (T, M) = 0 and M s cotorson. Ths proves (a). For part (b), frst note that Hom R (T, C ) s cotorson for all by Lemma 2.4(a). Let K = coker ϕ +1 as above. Snce each K has a cotorson resoluton, K s cotorson by part (a). Applyng Hom R (T, ) to the exact sequences 0 K +1 C K 0, we obtan that 0 Hom R (T, K +1 ) Hom R (T, C ) Hom R (T, K ) 0 s exact for all. Splcng these short exact sequences, we see that Hom R (T, C) s a cotorson resoluton of Hom R (T, M).

5 THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION 5 We remark that for any R-modules L, M, and N there s a natural R-module homomorphsm L R Hom R (M, N) ψ Hom R (M, L R N) such that for l L, f Hom R (M, N) and m M, ψ(l f)(m) = l f(m). Lemma 2.6. Let R be a Noetheran rng of fnte Krull dmenson. Let M, T, and F be R-modules such that M s fntely generated, T s flat, and F s flat and cotorson. Then the natural map ψ : M R Hom R (T, F ) Hom R (T, M R F ) s an somorphsm. Proof. We frst note that the lemma s clear n the case M s a fntely generated free R- module. Let G be a free resoluton of M consstng of fntely generated free R-modules. As F s flat and cotorson, G R F s a cotorson resoluton of M R F. By Lemma 2.5(b), we obtan that Hom R (T, G R F ) s a cotorson resoluton of Hom R (T, M R F ). Now consder the commutatve dagram: G 1 R Hom R (T, F ) G 0 R Hom R (T, F ) M R Hom R (T, F ) 0 = = Hom R (T, G 1 R F ) Hom R (T, G 0 R F ) Hom R (T, M R F ) 0 where the rows are exact and the vertcal arrows are the natural maps. Snce the frst two vertcal maps are somorphsms, so s the thrd by the Fve Lemma. We wll need the followng colocalzaton result for Tor: Proposton 2.7. Let R be a Noetheran rng of fnte Krull dmenson. Let L be a fntely generated R-module and M a cotorson R-module. Suppose Tor R (L, M) = 0 for all > 0. Then Tor Rp (L p, Hom R (R p, M)) = 0 for all p Spec R and > 0. Proof. Let p Spec R. It suffces to prove that Tor R (L, Hom R (R p, M)) = 0 for all > 0. Let F be a mnmal flat resoluton of M. As M s cotorson, so s F for all. As Tor R (L, M) = 0 for all > 0 we have that L R F s acyclc. We remark that L R F s cotorson for all. To see ths, let G be a free resoluton of L by fntely generated free R-modules. Then G R F s a cotorson resoluton of L R F. Hence, L R F s cotorson by Lemma 2.5(a). By Lemma 2.5(b), Hom R (R p, L R F) s acyclc. By Lemma 2.6, L R Hom R (R p, F) = Hom R (R p, L R F) as complexes. Thus, L R Hom R (R p, F) s acyclc. However, Hom R (R p, F) s a flat resoluton of Hom R (R p, M) by Lemmas 2.4(a) and 2.5(b). Thus, Tor R (L, Hom R (R p, M)) = H (L R Hom R (R p, F)) = 0 for > A vanshng result for Bass numbers and Enochs-Xu numbers Recall that for an R-module M and p Spec R, the Bass number µ (p, R) s defned to be dm k(p) Ext R p (k(p), M p ), where k(p) s the feld R p /pr p. If I s a mnmal njectve resoluton of M, then µ (p, M) s the cardnalty of the number of copes of E R (R/p) n any decomposton of I nto ndecomposable modules. It s well-known that f M s fntely generated and d R M < then µ (p, M) = 0 for all p Spec R and > depth R p. The followng proposton shows ths result stll holds even f M s not fntely generated. Proposton 3.1. Let R be a Noetheran rng and M an R-module such that d R M <. Then µ (p, M) = 0 for all p Spec R and > depth R p.

6 6 THOMAS MARLEY AND MARCUS WEBB Proof. By localzng at p, we can assume R s local wth maxmal deal m and that p = m. Let µ (M) := µ (m, M), s := depth R, and k := R/m. Certanly µ (M) = 0 for > d R M. Suppose r > s and µ (M) = 0 for all > r. It suffces to show that µ r (M) = 0. Let x 1,..., x s m be a regular sequence and I = (x 1,..., x s ). Snce depth R/I = 0 we have an exact sequence 0 k R/I R/J 0 where J = (I, y) for some y m. Snce pd R R/I = s, Ext R (R/I, M) = 0 for > s. From the long exact sequence on Ext, we have Ext R (k, M) = Ext +1 R (R/J, M) for > s. As r > s and µ (M) = 0 for all > r, we obtan Ext R (R/J, M) = 0 for > r + 1. Let n := d R M, I a mnmal njectve resoluton of M and L := Hom R (R/J, I). Then H (L) = Ext R (R/J, M) = 0 for > r + 1, gvng us an exact sequence 0 K L r+1 r+1 L r+2 L n 0, where K = ker r+1. Now, for each 0 we have L = Hom R (R/J, I ) = Hom R (R/J, E R (R/p) µ (p,m) ) = E R/J (R/p) µ (p,m). p V (J) Note that for p m, Tor R j (k, E R/J (R/p)) = 0 for all j snce E R/J (R/p) s an R p -module. Snce µ (M) = 0 for r + 1, we see that Tor R j (k, L ) = 0 for all j and all r + 1. Hence, from the long exact sequence above, we see that Tor R j (k, K) = 0 for all j. Now, we have an exact sequence K Ext r+1 R (R/J, M) 0. Tensorng ths sequence wth k and notng that Ext r+1 R that k R Ext r R (k, M) = 0. Hence, µ r(m) = 0. (R/J, M) = Ext r R (k, M), we conclude Corollary 3.2. Let R be a Noetheran rng and M an R-module such that fd R M <. Then Tor Rp (k(p), M p ) = 0 for all p Spec R and > depth R p. Proof. By localzng at p, we may assume R s local and p = m. Let k = R/m, E = E R (k), and let ( ) v = Hom R (, E) be the Matls dual functor. Snce the Matls dual s exact and T v s njectve for any flat R-module T, we see that d R M v <. By Proposton 3.1 and [11, Corollary 11.54], we have Tor R (k, M) v = Ext R (k, M v ) = 0 for > depth R. Hence, Tor R (k, M) = 0 for > depth R. We now prove the analogue of Proposton 3.1 for the Enochs-Xu numbers of a module of fnte flat dmenson: Proposton 3.3. Let R be a Noetheran rng and M an R-module such that fd R M <. Then π (p, M) = 0 for all p Spec R and > depth R p. Proof. By Theorem 2.3(a), we may assume M s a cotorson module. Let p Spec R. Note that Hom R (R p, M) s a cotorson R p -module by Lemma 2.4(a) and fd Rp Hom R (R p, M) < by Lemmas 2.4(b) and 2.5(b). Usng part (b) of Theorem 2.3, we have that π (p, M) =

7 THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION 7 π (pr p, Hom R (R p, M)) for all 0. Resettng notaton, we can now assume R s a Noetheran local rng and p = m, the maxmal deal of R. By Theorem 2.3(b) and Corollary 3.2, we have π (m, M) = dm R/m Tor R (R/m, M) = 0 for > depth R. 4. An acyclcty theorem for fnte flat resolutons We begn by makng a conventon regardng depth for (possbly) non-fntely generated modules. Let R be a Noetheran local rng wth maxmal deal m and M an arbtrary R-module. For 0, let Hm(M) denote the th local cohomology module wth support n m. We defne the depth of M by depth M := nf{ N 0 H m(m) 0}. Note that under ths defnton, depth M = f and only f H m(m) = 0 for all. We remark that f F s a flat R-module then H m(f ) = H m(r) R F ; hence, depth F depth R. We ll need the followng result concernng the depths of certan cotorson flat modules: Lemma 4.1. Let ϕ : R S be a homomorphsm of Noetheran local rngs. Let m and n be the maxmal deals of R and S, respectvely, and assume ϕ(m) n. Let F be a cotorson flat R-module such that π(m, F ) = 0. Then for every S-module N we have depth S N R F =. In partcular, depth F =. Proof. We frst remark that as F s a flat R-module, Hn(N R F ) = Hn(N) R F for all. Note that, as ms n, Supp R Hn(N) {m} for all. Thus, t suffces to prove that gven any R-module M wth Supp R M {m} then M R F = 0. As F s flat and cotorson, we have F = p Spec R T (p) where T (p) s the pr p-adc completon of a free R p -module of rank π(p, F ). As π(m, F ) = 0, we can wrte ths decomposton as F = p m T (p). Let M be an R-module of fnte length. As R s Noetheran, M s fntely presented and hence M R F = p m (M R T (p)) = 0, snce M R R p = 0 for all p m. Suppose now that M s an arbtrary R-module such that Supp R M {m}. Then M s the drect lmt of the drect system {M α } consstng of the fnte length R-submodules of M (wth morphsms the ncluson maps). Thus M R F = (lm M α ) R F = lm (M α R F ) = 0. Our next result s the acyclcty lemma of Peskne and Szpro [9], but stated for complexes whose modules are not necessarly fntely generated. The proof s largely mutats mutands, but we nclude t for the convenence of the reader. Proposton 4.2. ([9, Lemme d acyclcté 1.8]) Let R be a Noetheran local rng and consder a bounded complex T of R-modules f s T : 0 T s Ts 1 f0 T 0 0. Suppose the followng two condtons hold for each > 0: (1) depth T ; (2) depth H (T) = 0 or H (T) = 0. Then H (T) = 0 for all > 0.

8 8 THOMAS MARLEY AND MARCUS WEBB Proof. If s = 0 there s nothng to show, so we assume s > 0. Snce H s (T) s a submodule of T s and depth T s s 1, we see that depth H s (T) > 0. Thus, H s (T) = 0. Suppose for some j 1 we have that H (T) = 0 for j < s. It suffces to show that H j (T) = 0. Let C j = m f j+1. Then the truncated complex 0 T s f s Ts 1 fj+2 T j+1 C j 0 s exact. Usng the nequaltes depth T and nducton, one can show that depth C j j + 1. Let K j = ker f j and consder the exact sequence 0 C j K j H j (T) 0. Snce K j T j and depth T j j 1, we conclude that depth K j 1. Snce depth C j 2, we see that depth H j (T) 1. Thus, H j (T) = 0. We now prove the man result of ths secton, whch generalzes [9, Corollary 1.10] to modules of fnte flat dmenson: Theorem 4.3. Let ϕ : R S be a homomorphsm of Noetheran rngs such that for every q Spec S and p = ϕ 1 (q) one has that depth S q depth R p. Let M be an R-module of fnte flat dmenson. Then (a) Tor R (S, M) = 0 for all > 0. (b) fd S S R M fd R M. (c) If k(p) R S 0 for all p Spec R, then fd S S R M = fd R M. Proof. We prove part (a) by way of contradcton. Suppose Tor R (S, M) 0 for some 1. Let q Spec S be a prme mnmal wth respect to the property that Tor R (S, M) q 0 for some 1 and let p := ϕ 1 (q). By replacng R, S, and M wth R p, S q, and M p, we may assume (R, m) and (S, n) are local rngs, ϕ(m) n, Tor R (S, M) 0 for some 1, and Supp S Tor R (S, M) {n} for all 1. Let F be a mnmal flat resoluton of M and L := S R F. Snce H (L) = Tor R (S, M) and Supp S Tor R (S, M) {n} for all 1, we have that depth H (L) = 0 or H (L) = 0 for all 1. We clam that depth S L for all. Snce L s a flat S-module, depth S L depth S. Suppose > depth S. Then F s a cotorson flat R-module and π(m, F ) = π (m, M) = 0 by Proposton 3.3 snce > depth R. Then depth S L = by Lemma 4.1. Ths proves the clam. By Proposton 4.2, we obtan that H (L) = Tor R (S, M) = 0 for all 1, whch s a contradcton. Ths proves part (a). To see part (b), let F be a flat resoluton of M of length n = fd R M. Then S R F s an S-flat resoluton of S R M, snce Tor R (S, M) = 0 for all > 0 by part (a). Hence, fd S S R M n. To prove part (c) t suffces by part (b) to show that fd S S R M fd R M. Let n = fd R M and J an deal maxmal wth respect to Tor R n (R/J, M) 0. Then by [1, Proposton 2.2], J = p s prme and Tor R n (R/p, M) s a (nonzero) k(p)-module, where k(p) = R p /pr p. Let (F, ) be a flat resoluton of M of length n. Then we have an exact sequence 0 Tor R n (R/p, M) k(p) R F n 1 n k(p) R F n 1. Now, tensorng wth S over R (whch s the same as tensorng by S R k(p) over k(p), whch s flat over k(p)), we have 0 S R Tor R n (R/p, M) S R k(p) R F n 1 1 n S R k(p) R F n 1

9 THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION 9 s exact. Reassocatng, we obtan an exact sequence 0 S R Tor R n (R/p, M) (k(p) R S) S (S R F n ) (1 1) (1 n) (k(p) R S) S (S R F n 1 ). Snce S R F s a flat S-resoluton of S R M, ths sequence shows that Tor S n(k(p) R S, S R M) = S R Tor R n (R/p, M) = S R k(p) l, whch s nonzero, snce l 0 and S R k(p) 0. Hence, fd S S R M n. In the case the rng homomorphsm ϕ : R S s fnte, we have the followng dual result for modules of fnte njectve dmenson: Corollary 4.4. Let ϕ : R S be as n n Theorem 4.3 and assume S s a fntely generated R-module. Let M be an R-module of fnte njectve dmenson. Then (a) Ext R (S, M) = 0 for all > 0. (b) d S Hom R (S, M) d R M. (b) If k(p) R S 0 for all p Spec R, then d S Hom R (S, M) = d R M. Proof. For part (a), t suffces to prove the statement locally at every prme deal (as S s a f.g. R-module). So assume R s local and let ( ) v denote the Matls dual functor for R. By [11, Theorem 11.57], Tor R (A, M v ) = Ext R (A, M)v for all 0 and all fntely generated R-modules A. In partcular, we have that fd R M v = d R M <. Applyng part (a) of Theorem 4.3, we have Ext R (S, M)v = Tor R (S, M v ) = 0 for all > 0. Thus, Ext R (S, M) = 0 for all > 0. As n Theorem 4.3, part (b) follows readly from part (a). To prove (c), t suffces to show that d S Hom R (S, M) d R M. Let n = d R M. Then there exsts p Spec R such that Ext n R p (k(p), M p ) 0. An argument analagous to the one n the proof of part (c) of Theorem 4.3 shows that Ext n S p (k(p) R S, Hom R (S, M) p ) = Hom Rp (S R k(p), Ext n R p (k(p), M p )), whch s nonzero snce k(p) R S 0 and Ext n R p (k(p), M p ) 0. Thus, d S Hom R (S, M) n. We now apply our results to the Frobenus map. The followng corollary generalzes Théorèms 1.7 and 4.15 of [9], whch were proved for fntely generated modules. Corollary 4.5. Let R be a Noetheran rng of prme characterstc, M an R-module, and e 1 an nteger. (a) If fd R M < then Tor R (R (e), M) = 0 for all > 0 and fd R (e) R (e) R M = fd R M. (b) If the Frobenus map of R s fnte and d R M < then Ext R (R(e), M) = 0 for all > 0 and d R (e) Hom R (R (e), M) = d R M. Proof. It suffces to prove both (a) and (b) n the case e = 1. If f : R R s the Frobenus map, then for all q Spec R, f 1 (q) = q and k(q) R R (e) 0 for all e 1. The conclusons now follow from Theorem 4.3 and Corollary 4.4.

10 10 THOMAS MARLEY AND MARCUS WEBB 5. Proof of the converse In ths secton we prove part (b) of Theorem 1.1. Before dong so, we need to establsh some results on completons of free modules. Let (R, m) be a Noetheran local rng and M an R-module. We let M denote the m-adc completon of M;.e., M := lm M/m n M. If M s separated (.e., n m n M = 0), then the natural map M M s njectve. We note that for any n 1, M/m n M = M/m n M = M/ m n M. In partcular, f M s separated, then m n M M = m n M. Fnally, as m n s fntely generated, we have that m n M = m n M. Lemma 5.1. Let (R, m) be a Noetheran local rng, F a free R-module, and N a fntely generated R-module. Let t := t(r) be the least nteger such that m t H 0 m(r) = 0. (Such a t exsts by the Artn-Rees lemma.) Then the followng hold: (a) Hm( 0 F ) = Hm(F 0 ). (b) For all n t, m n F H 0 m ( F ) = 0. (c) F R N = F R N. Proof. Let X be a bass for F. Then F = α X R α, where R α = R for all α. We wrte ths as F = α R for short. Note that m n F = α m n for all n 1 and Hm(F 0 ) = α Hm(R). 0 In partcular, F s separated. We also observe that the topology on Hm(F 0 ) nduced from the m-adc topology on F concdes wth the m-adc topology on Hm(F 0 ). (One can check ths on each component.) Thus, F/H 0 m (F ) = F /Ĥ0 m(f ). Clearly, Ĥm(F 0 ) = Hm(F 0 ) and Hm(F 0 ) Hm( 0 F ). Therefore, to prove (a) t suffces to show that Hm( 0 F/H 0 m (F )) = 0. If R = Hm(R), 0 ths s clear. Otherwse, we can replace R by R/Hm(R) 0 and assume Hm(R) 0 = 0. In ths case, Hm(F 0 ) = 0 and t suffces to prove that Hm( 0 F ) = 0. Let x m be a non-zero-dvsor on R. Then 0 F µx F s exact, where µ x s multplcaton by x. We µx clam that 0 F F s exact. To see ths, t suffces to show that the topology nduced by µ 1 x (m n F ) on F concdes wth the m-adc topology on F. As x s a non-zero-dvsor on R, there exsts (by the Artn-Rees lemma) an nteger t such that (m n : R x) m n t for all n t. Then µ 1 x (m n F ) = α (m n : R x) m n t F for all n t. Hence the topologes concde, and x s a non-zero-dvsor on F. Thus, Hm( 0 F ) = 0 and part (a) s proved. Usng part (a) and that m n F F = m n F, we have for all n t m n F H 0 m ( F ) = m n F H 0 m(f ) = α (m n H 0 m(r)) = 0. Thus, (b) s proved. We now prove part (c). Let R r ϕ R s N 0 be a presentaton for N. Let K = m ϕ. By the Artn-Rees lemma, there exsts an nteger t 1 such that m n R s K m n t K for all n t. As F s free, we have 0 F R K F R R s F R N 0 s exact. Then

11 THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION 11 for n t, m n (F R R s ) (F R K) = ( α m n R s ) α K = α (m n R s K) α m n t K m n t (F R K). Thus, the nduced and m-adc topologes on F R K concde. Therefore, 0 F R K F R R s F R N 0 s exact. Composng wth the surjecton F R R r F R K and notng that  R R n s naturally somorphc to Ân for any R-module A and any postve nteger n, we obtan the commutatve dagram: 1 ϕ Hence, F R R r = F R R r 1 ϕ F R R s = F R N = F R N by the Fve Lemma. F R N 0 1 ϕ F R R s F R N 0. Before provng part (b) of Theorem 1.1, we ntroduce some notaton used n the proof. For e 1 we let m e denote the maxmal deal of R (e). For x R, we let x e denote the element x consdered as an element n R (e). Thus, xr (e) = x pe e R (e). Gven an R-module N we let N (e) denote the R (e) -module R (e) R N. If ψ : L N s an R-module homomorphsm, ψ (e) denotes the map 1 R ψ : L (e) N (e). Theorem 5.2. Let R be a Noetheran rng of prme characterstc p and M an R-module. Assume R has fnte Krull dmenson. Suppose Tor R (R (e), M) = 0 for all > 0 and nfntely many ntegers e. Then pd R M <. Proof. We frst note that t suffces to prove fd R M < by the Jensen-Raynaud-Gruson theorem. Let d := dm R. As fd R M < f and only f Tor Rq d+1 (M q, N q ) = 0 for every R-module N and q Spec R, we can assume R s local. By a standard argument, there exsts a fathfully flat local R-algebra (T, n) such that T (e) s f.g. as a T -module for all e (e.g, [8, Secton 3]). Note that fd R M = fd T (T R M) and Tor T (T (e), T R M) = T (e) R (e) Tor R (R (e), M) for all and e. Hence, by replacng R by T, we may assume R (e) s a fntely generated R-module for all e. Let ϕ : F M be a flat cover and K = ker ϕ. By [4, Lemma 2.2], K s cotorson. Note that fd R K < f and only f fd R M < and Tor R (R (e), K) = Tor R +1(R (e), M) for all 1 and all e. Hence, we may assume that M s cotorson. We assume that fd R M = and derve a contradcton. Choose q Spec R mnmal wth respect to the property that fd Rq Hom R (R q, M) =. By Proposton 2.7, f Tor R (R (e), M) = 0 for some e and all > 0, then Tor Rq (R q (e), Hom R (R q, M)) = 0 for all > 0. Note also that Hom R (R q, M) s a cotorson R q -module by Lemma 2.4(a) and that R (e) q s fntely generated as an R q -module for all e 1. Thus, by replacng R wth R q and

12 12 THOMAS MARLEY AND MARCUS WEBB M wth Hom R (R q, M), we may assume fd R M = and fd Rq Hom R (R q, M) < for all q m. Let F be a mnmal flat resoluton of M and let ϕ : F F 1 denote the dfferentals of F. Snce M s cotorson, so s F for all. By the proof of [5, Theorem 2.2], ϕ R 1 R/m = 0 for all ;.e., ϕ (F ) mf 1 for all. Then ϕ (e) (F (e) ) m [pe ] e F (e) 1 for all and e. Let s := depth R and x m a regular sequence on R of length s. Let t := t(r/(x)) as defned n Lemma 5.1, and let e be an nteger such that m [pe] m t and Tor R (R (e), M) = 0 for all > 0. Let S denote the R (e) -algebra R (e) /(x e ) and n = m e S, the maxmal deal of S. Then Hn(S) 0 0 and n t Hn(S) 0 = 0 by defnton of t. Snce pd R (e) S = s, we have that Tor R(e) (S, M (e) ) = 0 for all > s. As F (e) s an R (e) -flat resoluton of M (e), we obtan that H (S R (e) F (e) ) = 0 for all > s. For each 0 let C = m ϕ +1. Then for all 1 we have an exact sequence of R (e) -modules 0 C (e) F (e) C (e) 1 0. From the remarks above, we have that C (e) = ϕ (e) +1 Snce H (S R F (e) ) = 0 for all > s, we have that 0 S R (e) C (e) (F (e) +1 ) ] m[pe e S R (e) F (e) S R (e) C (e) F (e) 1 0 m t ef (e) for all. s exact for all > s 1. Note that S R (e) C (e) n t (S R (e) F (e) ) for all > s 1. By our assumptons, we have that fd Rq Hom R (R q, M) d 1 for all q m, where d = dm R. By Theorem 2.3(b), ths mples that π(q, F ) = π (q, M) = π (qr q, Hom R (R q, M)) = 0 for all d and all q m. Consequently, for d, F s the completon of a free R-module G of rank r := π (m, M). By Lemma 5.1(c), for d we have that F (e) = R (e) R Ĝ = Ĝ (e) and S R (e) F (e) = S R (e) G (e), whch s the n-adc completon of a free S-module of rank r. Note that snce fd R M =, r 0 for all d. For all d 1, let B = S R (e) C (e) and V = S R (e) F (e). From above, for d we have exact sequences of S-modules 0 B V B 1 0 where V s the completon of a (nonzero) free S-module and B n t V. In partcular, by Lemma 5.1(a), Hn(V 0 ) 0 for all d. By Lemma 5.1(b), Hn(B 0 ) n t V Hn(V 0 ) = 0 for all d. Ths mples that Hn(V 0 ) = 0 for d + 1, a contradcton. In the case the Frobenus map s fnte, we obtan the converse to part (b) of Corollary 4.5. Ths generalzes [6, Satz 5.2], whch was proved n the case the module M s fntely generated. Corollary 5.3. Let R be a Noetheran rng of prme characterstc. Assume that R has fnte Krull dmenson and that the Frobenus map s fnte. Let M be an R-module and suppose that Ext R (R(e), M) = 0 for all > 0 and for nfntely many ntegers e. Then d R M <. Proof. Snce R (e) s a fntely generated R-module for all e and dm R <, we may assume R s local. Let ( ) v be the Matls dual functor. Then, as n the proof of Corollary 4.4(a), d R M = fd R M v. By [11, Theorem 11.57], Tor R (R (e), M v ) = Ext R (R(e), M) v = 0 for all > 0 and nfntely many e. By Theorem 5.2, d R M = fd R M v <.

13 THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION 13 References 1. A. Auslander and D. Buchsbaum, Homologcal Dmenson n Noetheran rngs II, Trans. Amer. Math. Soc. 88 (1958), H. Bass, Injectve dmenson n Noetheran rngs, Trans. Amer. Math. Soc. 102 (1962), L. Bcan, R. El Bashr, and E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, E. Enochs, Flat covers and flat cotorson modules, Proc. Amer. Math. Soc. 92 (1984), E. Enochs and J. Xu, On Invarants dual to the Bass numbers, Proc. Amer. Math Soc. 125 (1997), J. Herzog, Rnge der Charakterstk p und Frobenusfunktoren, Math. Z. 140 (1974), C. Jensen, On the vanshng of lm (), J. Algebra 15 (1970), E. Kunz, Characterzatons of regular local rngs for characterstc p, Amer. J. Math. 91 (1969), C. Peskne and L. Szpro, Dmenson projectve fne et cohomologe locale. Applcatons à la démonstraton de conjectures de M. Auslander, H. Bass et A. Grothendeck, Inst. Hautes Études Sc. Publ. Math. No. 42 (1973), M. Raynaud and L. Gruson, Crtères de plattude et de projectvé. Techques de platfcaton d un module, Invent. Math. 13 (1971), J. Rotman, An Introducton to Homologcal Algebra, Pure and Appled Mathematcs no. 85, Academc Press, Inc., New York-London, J. Xu, The exstence of flat covers over Noetheran rngs of fnte Krull dmenson. Proc. Amer. Math. Soc. 123 (1995), J. Xu, Flat covers of modules, Lecture Notes n Mathematcs, vol. 1634, Sprnger-Verlag, Berln, Department of Mathematcs, Unversty of Nebraska-Lncoln, Lncoln, NE E-mal address: tmarley1@unl.edu Department of Mathematcs, Unversty of Nebraska-Lncoln, Lncoln, NE E-mal address: mwebb4@math.unl.edu