The Acyclicity of the Frobenius Functor for Modules of Finite Flat Dimension

0.19 MB·English

by  Thomas Marley

THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES
OF FINITE FLAT DIMENSION
5
1 THOMAS MARLEY ANDMARCUS WEBB
0
2
Abstract. Let R be a commutative Noetherian local ring of prime characteristic p and

n f :R−→RtheFrobeniusringhomomorphism. Fore≥1letR(e) denotetheringRviewed
a as an R-module via fe. Results of Peskine, Szpiro, and Herzog state that for finitely
J
generatedmodulesM,M hasfiniteprojectivedimensionifandonlyifTorRi (R(e),M)=0
5
for all i>0 and all (equivalently, infinitely many) e ≥1. We prove this statement holds

for arbitrary modules usingthe theory of flat covers and minimal flat resolutions.
]
C
A
.
h
t 1. Introduction
a
m
Let R be a commutative Noetherian ring of prime characteristic p and f : R−→R the
[ Frobenius map. For e≥ 1 let R(e) be the ring R considered as an R-module via fe; i.e., for

r ∈ R,s ∈ R(e),r·s := rpes. AclassicresultofKunz[8]states thatRisregularifandonlyif
2
v RisreducedandR(e) isflatasanR-moduleforall(equivalently, some)e ≥ 1. Subsequently,
6
Peskine and Szpiro [9, Th´eor`em 1.7] proved that if P is a finite projective resolution of a
3
3 finitely generated R-module M then for all e ≥ 1, R(e) ⊗R P is a projective resolution of
0 R(e) ⊗ M; that is, finitely generated modules of finite projective dimension are acyclic
R
0 objects with respect to the Frobenius functors R(e)⊗ −. A year later Herzog [6, Satz 3.1]
. R
1 showed the converse holds in the case the Krull dimension of R is finite; namely, if M is a
0 finitely generated R-module and P is a projective resolution of M such that R(e) ⊗ P is
5 R
1 acyclic forinfinitelymanyintegerse, thenM hasfiniteprojectivedimension. Aninteresting
: question is whether these results hold for arbitrary R-modules, not just finitely generated
v
i ones. In Corollary 4.5(a) and Theorem 5.2, we give an affirmative answer to this question:
X
r Theorem 1.1. Let R be a Noetherian ring of prime characteristic and M an R-module.
a
Then the following hold:
(a) If fd M < ∞ then for all i,e > 0, TorR(R(e),M) = 0 and fd R(e)⊗ M = fd M.
R i R(e) R R
(b) If R has finite Krull dimension and TorR(R(e),M) = 0 for all i > 0 and infinitely many
i
integers e, then pd M < ∞.
R
Here fd M and pd M denote the flat and projective dimensions of M, respectively. Note
R R
thatpart(a) implies that if pd M < ∞ then pd R(e)⊗ M ≤ pd M forall e(although
R R(e) R R
wedonotknowofanyexampleswheretheinequalityisstrict). Wealsoproveananalogueof
Theorem 1.1 for injective dimension in the case the Frobenius map is finite (cf. Corollaries
4.5(b) and 5.3).
Date: December 29, 2014.
2010 Mathematics Subject Classification. Primary 13D07; Secondary 13A35 .
Key words and phrases. flat dimension, Frobenius map, Bass number,flat cover.
1
2 THOMASMARLEYANDMARCUSWEBB
A special case of part (a) of Theorem 1.1 is well-known and is what inspired this inves-
tigation: Suppose x = x ,...,x is a regular sequence on R and C = C(x;R) is the Cˇech
1 n
complex on R with respect to x. Then C is a finite flat resolution of the local cohomology
module Hn (R) and R(e)⊗ C ∼= C(x;R(e)) ∼= C(xpe;R) is acyclic since xpe,...,xpe is an
(x) R 1 n
R-sequence.
The proofs of Peskine-Szpiro and Herzog both reduce to the case R is local and utilize
minimal projective (free) resolutions. The minimality condition for such projective resolu-
tions can be expressed by saying that all the differentials are zero when tensored with the
residue field. It can be easily shown that projective resolutions of this type do not neces-
sarily exist for arbitrary modules. However, flat resolutions with this kind of minimality
condition do exist for a large class of modules (i.e., cotorsion modules), which we show is
sufficient to prove Theorem 1.1. The theory of flat covers, cotorsion modules, and minimal
flatresolutions, as developed by Enochs andXu in [4] and[5], are essential ingredients in all
ourarguments. InSection 2, wesummarizesome basicpropertiesof thesenotions as well as
prove some auxiliary results which are used in later sections. Minimal flat resolutions have
many properties analagous to those of minimal injective resolutions. In particular, the flat
modules appearing in a minimal flat resolution of a module M are uniquely determined (up
to isomorphism)by invariants which we call the Enochs-Xu numbersof M. TheEnochs-Xu
numbersofamoduleareinsomesensethedualoftheBassnumbersofamodule. InSection
3, we prove a vanishing result for Enochs-Xu numbers which will be crucial in our proof of
part (a) of Theorem 1.1.
Sections 4 and 5 are devoted to the proofs of parts (a) and (b), respectively, of Theorem
1.1. For both parts it is sufficient to consider the case when R is a local ring, in which case
everymoduleoffiniteflatdimensionhasfiniteprojectivedimensionbytheJensen-Raynaud-
Grusontheorem ([7, Proposition 6] and [10, Secondepartie, Th´eor`eme 3.2.6]). Ourstrategy
is to applythe methodsof Peskine-Szpiro andHerzog tominimalflat resolutions. Thereare
several difficulties which arise: the finitistic flat dimension may exceed the depth of the ring
(cf. [2, Corollary 5.3]); minimal flat resolutions do not in general localize; and the modules
appearing in minimal flat resolutions are not generally finitely generated. We are able to
overcome these difficulties by, in part, proving that the depths of the modules in degrees
exceeding depthR in finite minimal flat resolutions are infinite, and showing that, in the
case the Frobenius map is finite, the acyclicity of Frobenius for minimal flat resolutions of
cotorsion modules commutes with the colocalization functor Hom (R ,−). And while the
R p
flat modules in question are not typically finitely generated, we are able to reduce to the
case where they are completions of free R-modules.
Acknowledgments: The authors would like to thank Edgar Enochs for his help in our
understanding of flat covers. The authors also express their gratitude to everyone in the
UNL algebra seminar, in particular Lucho Avramov, Peder Thompson, Mark Walker, and
Wenliang Zhang, for their useful feedback on this project.
2. Flat covers and cotorsion modules
In this section we collect the results we will regarding flat covers, cotorsion modules, and
minimal flat resolutions. We refer the reader to [13] for background on this material.
THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION 3
Definition 2.1. Let M be an R-module. An R-homomorphism ϕ : F−→M is called a flat
cover of M if the following hold:
(a) F is flat;
(b) for every map ψ : G−→M with G flat, there exists a homomorphism g : G−→F such
that ψ = ϕg; and
(c) if h :F−→F satisfies ϕ = ϕh then h is an isomorphism.
By abuse of language, we sometimes refer to the module F as the ‘flat cover’ of M, rather
than the homomorphism ϕ. In [3], it is proved that flat covers exists for all modules over
all rings. However, in this work we will only need the existence of flat covers for modules
over commutative Noetherian rings of finite Krull dimension, which was proved in [12]. It
is easily seen that flat covers are surjective (since every module is a homomorphic image of
a flat module), that a flat cover of a module is unique up to isomorphism, and that the flat
cover of a flat module is an isomorphism.
Definition 2.2. An R-module M is called cotorsion if Ext1(F,M) = 0 for every flat
R
R-module F.
It is easily seen that if M is cotorsion then Exti (F,M) = 0 for all i ≥ 1 and all flat R-
R
modules F. By [4, Lemma 2.2 and Corollary], the kernel of a flat cover is cotorsion and a
flat cover of a cotorsion module is cotorsion. For an R-module M we let C (M) denote the
R
cotorsion envelope of M, which exists by [13, Theorem 3.4.6]. The cotorsion envelope of a
flatmoduleisflatby[13, Theorem3.4.2]. IfF is flatandcotorsion then F ∼= T(p)
Qp∈SpecR
whereeachT(p)isthecompletion withrespecttothepR -adictopologyofafreeR -module
p p
G(p); furthermore, the ranks of the free R -modules G(p) are uniquely determined by F
p
([4, Theorem]). For each p ∈ SpecR, let π(p,F) denote the rank (possibly infinite) of G(p).
Given an R-module M, a minimal flat resolution of M is a complex
F : ···−→F −∂→i F −→···−→F −→0
i i−1 0
such that H (F) = 0for i >0, H (F) ∼= M, and foreach i, the naturalmap F −→coker∂
i 0 i i+1
isaflatcover. ItisclearthateveryR-modulehasaminimalflatresolution,andthatanytwo
minimal flat resolutions of M are (chain) isomorphic. Since the flat cover of a flat module
is an isomorphism, it follows that if fd M = n < ∞ then the length of any minimal flat
R
resolutionofM isn. Notethat,bytheremarksintheprecedingparagraph,ifFisaminimal
flat resolution of M then F is cotorsion for all i ≥ 1. For each i ≥ 1 and p ∈ SpecR we
i
set π (p,M) := π(p,F ). For i = 0 and p ∈ SpecR, we set π (p,M) := π(p,C (F )). Note
i i 0 R 0
that if M is cotorsion, so is F and thus π (p,M) = π(p,F ) for all p ∈SpecR. We call the
0 0 0
the invariants π (p,M) the Enochs -Xu numbers of M. Note that fd M ≤ n if and only if
i R
π (p,M) = 0 for all p ∈ SpecR and i > n.
i
We now state a theorem of Enochs and Xu:
Theorem 2.3. ([5, Theorem 2.1 and 2.2]) Let R be a Noetherian ring. Then for any
R-module M we have:
(a) π (p,M) = π (p,C (M)) for all i ≥ 0 and p ∈ SpecR.
i i R
(b) If M is cotorsion then π (p,M) = dim TorRp(k(p),Hom (R ,M)) for all i ≥ 0 and
i k(p) i R p
p ∈ SpecR.
In particular, note that part (a) of this theorem implies that fd M = fd C (M).
R R R
4 THOMASMARLEYANDMARCUSWEBB
We now establish some additional results on cotorsion modules which are needed in
sections 3 and 4. We thank Edgar Enochs for showingus a proof of part (b)of thefollowing
lemma, which is implicit in [5]:
Lemma 2.4. Let R be a Noetherian ring and S a flat R-algebra.
(a) If C is a cotorsion R-module and T a flat S-module then Hom (T,C) is a cotorsion
R
S-module.
(b) If F is a flat and cotorsion R-module and p ∈ SpecR then Hom (R ,F) is a flat and
R p
cotorsion R -module.
p
Proof. To prove (a), let A be a flat S-module. As S is flat over R, any flat S-module is
also flat as an R-module. Thus, Exti (P ⊗ A,C) = 0 for all i > 0 and all projective
R S
S-modules P. Thus, we have a Grothendieck spectral sequence Extp(TorS(T,A),C) ⇒
R q
p+q
Ext (A,Hom (T,C)) (cf. [11, Theorem 11.54]). The spectral sequence collapses, giving
S R
Exti (A,Hom (T,C)) ∼= Exti (T ⊗ A,C) = 0 for i > 0, as T ⊗ A is a flat S-module and
S R R S S
therefore flat as an R-module as well. Hence, Hom (T,C) is a cotorsion S-module.
R
For (b), as F is flat and cotorsion we have F ∼= T(q) where each T(q) is the
Qq∈SpecR
∼
completion of a free R -module. We’ll show that Hom (R ,F) = T(q), which is flat
q R p Qq⊆p
and cotorsion by [4, Theorem]. For each q ∈ SpecR let ρ : F−→T(q) be the projection
q
map. Let ϕ ∈ Hom (R ,F). Then, as R is divisible by each element in R\p, the same
R p p
is true for the image (ρ ϕ)(R ) for all q. Suppose q 6⊆ p. Let r ∈ q \ p. Then for all
q p
n ≥ 1 we have (ρ ϕ)(R ) ⊆ rn(ρ ϕ)(R ) ⊆ qnT(q). As T(q) is separated in the qR -adic
q p q p q
∼
topology, we conclude that (ρ ϕ)(R ) = 0. Thus, Hom (R ,F) = Hom (R ,G), where
q p R p R p
G = T(q). But since each T(q) for q ⊆ p is an R -module, G is an R -module. Thus,
Qq⊆p p p
Hom (R ,F) ∼= G. (cid:3)
R p
Lemma 2.5. Let R be a Noetherian ring of finite Krull dimension and M an R-module.
Suppose M has a (left) cotorsion resolution C; i.e., there exists an exact sequence
···−→C −ϕ→i C −→···−→C −→M−→0
i i−1 0
where each C is cotorsion. Then
i
(a) M is cotorsion.
(b) For any flat R-module T, Hom (T,C) is a cotorsion resolution of Hom (T,M).
R R
Proof. For each integer i ≥ 0 let K = cokerϕ . Let T be a flat R-module. Since
i i+1
Exti (T,C ) = 0 for all i ≥ 1 and j ≥ 0, we have that Ext1(T,M) ∼= Exti+1(T,K )
R j R R i
for all i ≥ 0. Let n = dimR. Then Extn+1(T,K ) = 0 as pd T ≤ n by the Jensen-
R n R
Raynaud-Gruson theorem ([7, Proposition 6] and [10, Seconde partie, Th´eor`eme 3.2.6]).
Thus, Ext1(T,M) = 0 and M is cotorsion. This proves (a).
R
For part (b), first note that Hom (T,C ) is cotorsion for all i by Lemma 2.4(a). Let
R i
K = cokerϕ as above. Since each K has a cotorsion resolution, K is cotorsion by part
i i+1 i i
(a). Applying Hom (T,−) to the exact sequences 0−→K −→C −→K −→0, we obtain
R i+1 i i
that
0−→Hom (T,K )−→Hom (T,C )−→Hom (T,K )−→0
R i+1 R i R i
isexactforalli. Splicingtheseshortexactsequences, weseethatHom (T,C)isacotorsion
R
resolution of Hom (T,M). (cid:3)
R
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 is a natural R-module homomor-
phism
ψ
L⊗ Hom (M,N) −→ Hom (M,L⊗ N)
R R R R
such that for ℓ ∈ L, f ∈ Hom (M,N) and m ∈ M, ψ(ℓ⊗f)(m)= ℓ⊗f(m).
R
Lemma 2.6. Let R be a Noetherian ring of finite Krull dimension. Let M, T, and F be
R-modules such that M is finitely generated, T is flat, and F is flat and cotorsion. Then
the natural map ψ : M ⊗ Hom (T,F)−→Hom (T,M ⊗ F) is an isomorphism.
R R R R
Proof. We first note that the lemma is clear in the case M is a finitely generated free R-
module. Let Gbeafree resolution of M consisting of finitely generated free R-modules. As
F is flat and cotorsion, G⊗ F is a cotorsion resolution of M ⊗ F. By Lemma 2.5(b), we
R R
obtain that Hom (T,G⊗ F) is a cotorsion resolution of Hom (T,M⊗ F). Now consider
R R R R
the commutative diagram:
G ⊗ Hom (T,F) −−−−→ G ⊗ Hom (T,F) −−−−→ M ⊗ Hom (T,F) −−−−→ 0
1 R R 0 R R R R
∼= ∼= 
  
y y y
Hom (T,G ⊗ F) −−−−→ Hom (T,G ⊗ F) −−−−→ Hom (T,M ⊗ F) −−−−→ 0
R 1 R R 0 R R R
where the rows are exact and the vertical arrows are the natural maps. Since the first two
vertical maps are isomorphisms, so is the third by the Five Lemma. (cid:3)
We will need the following colocalization result for Tor:
Proposition 2.7. Let R be a Noetherian ring of finite Krull dimension. Let L be a finitely
generated R-module and M a cotorsion R-module. Suppose TorR(L,M) = 0 for all i > 0.
i
Then TorRp(L ,Hom (R ,M)) = 0 for all p ∈ SpecR and i> 0.
i p R p
Proof. Let p ∈ SpecR. It suffices to prove that TorR(L,Hom (R ,M)) = 0 for all i >
i R p
0. Let F be a minimal flat resolution of M. As M is cotorsion, so is F for all i. As
i
TorR(L,M) = 0 for all i > 0 we have that L ⊗ F is acyclic. We remark that L ⊗ F
i R R i
is cotorsion for all i. To see this, let G be a free resolution of L by finitely generated
free R-modules. Then G⊗ F is a cotorsion resolution of L ⊗ F . Hence, L ⊗ F is
R i R i R i
cotorsion by Lemma 2.5(a). By Lemma 2.5(b), Hom (R ,L⊗ F) is acyclic. By Lemma
R p R
2.6, L⊗ Hom (R ,F) ∼= Hom (R ,L⊗ F) as complexes. Thus, L⊗ Hom (R ,F) is
R R p R p R R R p
acyclic. However, Hom (R ,F) is a flat resolution of Hom (R ,M) by Lemmas 2.4(a) and
R p R p
2.5(b). Thus, TorR(L,Hom (R ,M)) ∼= H (L⊗ Hom (R ,F)) = 0 for i> 0. (cid:3)
i R p i R R p
3. A vanishing result for Bass numbers and Enochs-Xu numbers
Recall that for an R-module M and p ∈ SpecR, the Bass number µ (p,R) is defined
i
to be dim Exti (k(p),M ), where k(p) is the field R /pR . If I is a minimal injective
k(p) Rp p p p
resolution of M, then µ (p,M) is the cardinality of the number of copies of E (R/p) in
i R
any decomposition of Ii into indecomposable modules. It is well-known that if M is finitely
generated and id M < ∞ then µ (p,M) = 0 for all p ∈ SpecR and i > depthR . The
R i p
following proposition shows this result still holds even if M is not finitely generated.
Proposition 3.1. Let R be a Noetherian ring and M an R-module such that id M < ∞.
R
Then µ (p,M) = 0 for all p ∈ SpecR and i> depthR .
i p
6 THOMASMARLEYANDMARCUSWEBB
Proof. By localizing at p, we can assume R is local with maximal ideal m and that p = m.
Let µ (M) := µ (m,M), s := depthR, and k := R/m. Certainly µ (M) = 0 for i > id M.
i i i R
Suppose r > s and µ (M) = 0 for all i > r. It suffices to show that µ (M) = 0. Let
i r
x ,...,x ∈ m be a regular sequence and I = (x ,...,x ). Since depthR/I = 0 we have an
1 s 1 s
exact sequence
0−→k−→R/I−→R/J−→0
where J = (I,y) for some y ∈ m. Since pd R/I = s, Exti (R/I,M) = 0 for i > s.
R R
From the long exact sequence on Ext, we have Exti (k,M) ∼= Exti+1(R/J,M) for i > s.
R R
As r > s and µ (M) = 0 for all i > r, we obtain Exti (R/J,M) = 0 for i > r + 1.
i R
Let n := id M, I a minimal injective resolution of M and L := Hom (R/J,I). Then
R R
Hi(L) = Exti (R/J,M) = 0 for i> r+1, giving us an exact sequence
R
0 −−−−→ K −−−−→ Lr+1 −−∂−r+−→1 Lr+2 −−−−→ ··· −−−−→ Ln−→0,
where K = ker∂r+1. Now, for each i≥ 0 we have
Li = Hom (R/J,Ii)
R
∼= Hom (R/J,⊕E (R/p)µi(p,M))
R R
∼= M ER/J(R/p)µi(p,M).
p∈V(J)
Note that for p 6= m, TorR(k,E (R/p)) = 0 for all j since E (R/p) is an R -module.
j R/J R/J p
Since µ (M) = 0 for i≥ r+1, we see that TorR(k,L )= 0 for all j and all i ≥ r+1. Hence,
i j i
from the long exact sequence above, we see that TorR(k,K) = 0 for all j. Now, we have an
j
exact sequence
K−→Extr+1(R/J,M)−→0.
R
Tensoringthissequencewithk andnotingthatExtr+1(R/J,M) ∼= Extr (k,M), weconclude
R R
that k⊗ Extr (k,M) = 0. Hence, µ (M) = 0.
R R r
(cid:3)
Corollary 3.2. Let R be a Noetherian ring and M an R-module such that fd M < ∞.
R
Then TorRp(k(p),M )= 0 for all p ∈ SpecR and i> depthR .
i p p
Proof. By localizing at p, we may assume R is local and p = m. Let k = R/m, E = E (k),
R
and let (−)v = Hom (−,E) be the Matlis dual functor. Since the Matlis dual is exact and
R
Tv is injective for any flat R-module T, we see that id Mv < ∞. By Proposition 3.1 and
R
[11, Corollary 11.54], we have TorR(k,M)v ∼= Exti (k,Mv) = 0 for i > depthR. Hence,
i R
TorR(k,M) =0 for i > depthR.
i
(cid:3)
We now prove the analogue of Proposition 3.1 for the Enochs-Xu numbers of a module
of finite flat dimension:
Proposition 3.3. Let R be a Noetherian ring and M an R-module such that fd M < ∞.
R
Then π (p,M) = 0 for all p ∈ SpecR and i> depthR .
i p
Proof. By Theorem 2.3(a), we may assume M is a cotorsion module. Let p ∈SpecR. Note
thatHom (R ,M)isacotorsion R -modulebyLemma2.4(a)andfd Hom (R ,M) < ∞
R p p Rp R p
by Lemmas 2.4(b) and 2.5(b). Using part (b) of Theorem 2.3, we have that π (p,M) =
i
THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION 7
π (pR ,Hom (R ,M)) for all i ≥ 0. Resetting notation, we can now assume R is a Noe-
i p R p
therian local ring and p = m, the maximal ideal of R. By Theorem 2.3(b) and Corollary
3.2, we have π (m,M) = dim TorR(R/m,M) = 0 for i> depthR. (cid:3)
i R/m i
4. An acyclicity theorem for finite flat resolutions
We begin by making a convention regarding depth for (possibly) non-finitely generated
modules. Let R be a Noetherian local ring with maximal ideal m and M an arbitrary
R-module. For i ≥ 0, let Hi (M) denote the ith local cohomology module with support in
m
m. We define the depth of M by
depthM := inf{i ∈ N | Hi (M) 6= 0}.
0 m
Notethatunderthisdefinition,depthM = ∞ifandonlyifHi (M) = 0foralli. Weremark
m
that if F is a flat R-module then Hi (F) ∼= Hi (R)⊗ F; hence, depthF ≥ depthR. We’ll
m m R
need the following result concerning the depths of certain cotorsion flat modules:
Lemma 4.1. Let ϕ: R−→S be a homomorphism of Noetherian local rings. Let m and n be
the maximal ideals of R and S, respectively, and assume ϕ(m) ⊆ n. Let F be a cotorsion flat
R-module such that π(m,F) = 0. Then for every S-module N we have depth N⊗ F = ∞.
S R
In particular, depthF = ∞.
Proof. We first remark that as F is a flat R-module, Hi(N ⊗ F) ∼= Hi(N)⊗ F for all i.
n R n R
Note that, as mS ⊆ n, Supp Hi(N) ⊆ {m} for all i. Thus, it suffices to prove that given
R n
any R-module M with Supp M ⊆ {m} then M ⊗ F = 0. As F is flat and cotorsion,
R R
we have F ∼= T(p) where T(p) is the pR -adic completion of a free R -module of
Qp∈SpecR p p
rank π(p,F). As π(m,F) = 0, we can write this decomposition as F ∼= T(p). Let
Qp6=m
M be an R-module of finite length. As R is Noetherian, M is finitely presented and hence
M ⊗ F ∼= (M ⊗ T(p)) = 0, since M ⊗ R = 0 for all p 6= m. Suppose now that
R Qp6=m R R p
M is an arbitrary R-module such that Supp M ⊆ {m}. Then M is the direct limit of the
R
direct system {M } consisting of thefinite length R-submodulesof M (with morphismsthe
α
inclusion maps). Thus
M ⊗ F ∼= (limM )⊗ F ∼= lim(M ⊗ F) = 0.
R −→ α R −→ α R
(cid:3)
Ournextresultistheacyclicity lemmaofPeskineandSzpiro[9],butstatedforcomplexes
whosemodulesarenotnecessarily finitelygenerated. Theproofislargely mutatis mutandis,
but we include it for the convenience of the reader.
Proposition 4.2. ([9, Lemme d’acyclicit´e 1.8]) Let R be a Noetherian local ring and con-
sider a bounded complex T of R-modules
T : 0−→T −f→s T −→··· −f→0 T −→0.
s s−1 0
Suppose the following two conditions hold for each i> 0:
(1) depthT ≥ i;
i
(2) depthH (T)= 0 or H (T) =0.
i i
Then H (T) = 0 for all i > 0.
i
8 THOMASMARLEYANDMARCUSWEBB
Proof. If s = 0 there is nothing to show, so we assume s > 0. Since H (T) is a submodule
s
of T and depthT ≥ s ≥ 1, we see that depthH (T) > 0. Thus, H (T) = 0. Suppose for
s s s s
some j ≥ 1 we have that H (T) = 0 for j < i ≤ s. It suffices to show that H (T) = 0. Let
i j
C = imf . Then the truncated complex
j j+1
0−→T −f→s T −→··· −f−j−+→2 T −→C −→0
s s−1 j+1 j
is exact. Using the inequalities depthT ≥ i and induction, one can show that depthC ≥
i j
j +1. Let K = kerf and consider the exact sequence
j j
0−→C −→K −→H (T)−→0.
j j j
Since K ⊆ T and depthT ≥ j ≥ 1, we conclude that depthK ≥ 1. Since depthC ≥ 2,
j j j j j
we see that depthH (T) ≥ 1. Thus, H (T) = 0. (cid:3)
j j
We now prove the main result of this section, which generalizes [9, Corollary 1.10] to
modules of finite flat dimension:
Theorem 4.3. Let ϕ :R−→S be a homomorphism of Noetherian rings such that for every
q ∈ SpecS and p = ϕ−1(q) one has that depthS ≥ depthR . Let M be an R-module of
q p
finite flat dimension. Then
(a) TorR(S,M) = 0 for all i > 0.
i
(b) fd S ⊗ M ≤ fd M.
S R R
(c) If k(p)⊗ S 6= 0 for all p ∈ SpecR, then fd S ⊗ M = fd M.
R S R R
Proof. We prove part (a) by way of contradiction. SupposeTorR(S,M) 6= 0 for some i≥ 1.
i
Let q ∈ SpecS be a prime minimal with respect to the property that TorR(S,M) 6= 0
i q
for some i ≥ 1 and let p := ϕ−1(q). By replacing R, S, and M with R , S , and M , we
p q p
may assume (R,m) and (S,n) are local rings, ϕ(m) ⊆ n, TorR(S,M) 6= 0 for some i ≥ 1,
i
and Supp TorR(S,M) ⊆ {n} for all i ≥ 1. Let F be a minimal flat resolution of M and
S i
L := S ⊗ F. Since H (L) ∼= TorR(S,M) and Supp TorR(S,M) ⊆ {n} for all i ≥ 1, we
R i i S i
have that depthH (L) = 0 or H (L)= 0 for all i≥ 1.
i i
We claim that depth L ≥ i for all i. Since L is a flat S-module, depth L ≥ depthS.
S i i S i
Suppose i > depthS. Then F is a cotorsion flat R-module and π(m,F ) = π (m,M) = 0
i i i
by Proposition 3.3 since i > depthR. Then depth L = ∞ by Lemma 4.1. This proves the
S i
claim. By Proposition 4.2, we obtain that H (L) = TorR(S,M) = 0 for all i ≥ 1, which is
i i
a contradiction. This proves part (a).
To see part (b), let F be a flat resolution of M of length n = fd M. Then S ⊗ F is
R R
an S-flat resolution of S ⊗ M, since TorR(S,M) = 0 for all i > 0 by part (a). Hence,
R i
fd S ⊗ M ≤ n.
S R
Toprovepart(c)itsufficesbypart(b)toshowthatfd S⊗ M ≥ fd M. Letn = fd M
S R R R
and J an ideal maximal with respect to TorR(R/J,M) 6= 0. Then by [1, Proposition 2.2],
n
J = p is prime and TorR(R/p,M) is a (nonzero) k(p)-module, where k(p) = R /pR . Let
n p p
(F,∂) be a flat resolution of M of length n. Then we have an exact sequence
0−→TorR(R/p,M)−→k(p)⊗ F −1−⊗−∂→n k(p)⊗ F .
n R n R n−1
Now, tensoring with S over R (which is the same as tensoring by S⊗ k(p) over k(p), which
R
is flat over k(p)), we have
0−→S ⊗ TorR(R/p,M)−→S ⊗ k(p)⊗ F −1−⊗−1−⊗−∂→n S ⊗ k(p)⊗ F
R n R R n R R n−1
THE ACYCLICITY OF THE FROBENIUS FUNCTOR FOR MODULES OF FINITE FLAT DIMENSION 9
is exact. Reassociating, we obtain an exact sequence
0−→S⊗ TorR(R/p,M)−→(k(p)⊗ S)⊗ (S⊗ F ) −(−1⊗−−1)−⊗−(1−⊗−∂−n→) (k(p)⊗ S)⊗ (S⊗ F ).
R n R S R n R S R n−1
Since S ⊗ F is a flat S-resolution of S ⊗ M, this sequence shows that
R R
TorS(k(p)⊗ S,S ⊗ M)∼= S ⊗ TorR(R/p,M) ∼= S ⊗ k(p)ℓ,
n R R R n R
which is nonzero, since l 6= 0 and S ⊗ k(p)6= 0. Hence, fd S ⊗ M ≥ n. (cid:3)
R S R
In the case the ring homomorphism ϕ :R−→S is finite, we have the following dual result
for modules of finite injective dimension:
Corollary 4.4. Let ϕ: R−→S be as in in Theorem 4.3 and assume S is a finitely generated
R-module. Let M be an R-module of finite injective dimension. Then
(a) Exti (S,M) = 0 for all i > 0.
R
(b) id Hom (S,M) ≤ id M.
S R R
(b) If k(p)⊗ S 6= 0 for all p ∈ SpecR, then id Hom (S,M) = id M.
R S R R
Proof. For part (a), it suffices to prove the statement locally at every prime ideal (as S is a
f.g. R-module). So assume R is local and let (−)v denote the Matlis dual functor for R. By
[11, Theorem 11.57], TorR(A,Mv) ∼= Exti (A,M)v for all i ≥ 0 and all finitely generated
i R
R-modules A. In particular, we have that fd Mv = id M < ∞. Applying part (a) of
R R
Theorem4.3, wehaveExti (S,M)v ∼= TorR(S,Mv) = 0foralli > 0. Thus,Exti (S,M) = 0
R i R
for all i > 0.
As in Theorem 4.3, part (b) follows readily from part (a).
To prove (c), it suffices to show that id Hom (S,M) ≥ id M. Let n = id M. Then
S R R R
there exists p ∈ SpecR such that Extn (k(p),M )6= 0. An argument analagous to the one
Rp p
in the proof of part (c) of Theorem 4.3 shows that
Extn (k(p)⊗ S,Hom (S,M) )∼= Hom (S ⊗ k(p),Extn (k(p),M )),
Sp R R p Rp R Rp p
whichisnonzerosincek(p)⊗ S 6= 0andExtn (k(p),M ) 6=0. Thus,id Hom (S,M) ≥ n.
R Rp p S R
(cid:3)
We now apply our results to the Frobenius map. The following corollary generalizes
Th´eor`ems 1.7 and 4.15 of [9], which were proved for finitely generated modules.
Corollary 4.5. Let R be a Noetherian ring of prime characteristic, M an R-module, and
e ≥ 1 an integer.
(a) If fd M < ∞ then TorR(R(e),M) = 0 for all i >0 and fd R(e)⊗ M = fd M.
R i R(e) R R
(b) If the Frobenius map of R is finite and id M < ∞ then Exti (R(e),M) = 0 for all
R R
i> 0 and id Hom (R(e),M) = id M.
R(e) R R
Proof. It suffices to prove both (a) and (b) in the case e = 1. If f : R−→R is the Frobenius
map, then for all q ∈ SpecR, f−1(q) =q and k(q)⊗ R(e) 6= 0for all e ≥ 1. Theconclusions
R
now follow from Theorem 4.3 and Corollary 4.4. (cid:3)
10 THOMASMARLEYANDMARCUSWEBB
5. Proof of the converse
In this section we prove part (b) of Theorem 1.1. Before doing so, we need to establish
some results on completions of free modules. Let (R,m) be a Noetherian local ring and M
an R-module. We let M denote the m-adic completion of M; i.e., M := limM/mnM. If M
←−
c c
is separated (i.e., ∩ mnM = 0), then the natural map M−→M is injective. We note that
n
for any n ≥ 1, M/mnM ∼= M\/mnM ∼= M/m\nM. In particuclar, if M is separated, then
\ c \
mnM ∩M = mnM. Finally, as mn is finitely generated, we have that mnM = mnM.
c
Lemma 5.1. Let (R,m) be a Noetherian local ring, F a free R-module, and N a finitely
generated R-module. Let t := t(R) be the least integer such that mt∩H0(R) = 0. (Such a
m
t exists by the Artin-Rees lemma.) Then the following hold:
(a) H0(F) = H0(F).
m m
(b) For abll n ≥ t, mnF ∩H0(F) = 0.
m
(c) F ⊗ N ∼= F\⊗ Nb. b
R R
b
Proof. Let X be a basis for F. Then F ∼= R , where R = R for all α. We write
Lα∈X α α
this as F = ⊕ R for short. Note that mnF = ⊕ mn for all n ≥ 1 and H0(F) = ⊕ H0(R).
α α m α m
In particular, F is separated. We also observe that the topology on H0(F) induced from
m
the m-adic topology on F coincides with the m-adic topology on H0(F). (One can check
m
this on each component.) Thus, F/\H0(F) ∼= F/H\0(F). Clearly, H\0(F) = H0(F) and
m m m m
b \
H0(F) ⊆ H0(F). Therefore, to prove (a) it suffices to show that H0(F/H0(F)) = 0.
m m m m
If R = H0(R),bthis is clear. Otherwise, we can replace R by R/H0(R) and assume
m m
H0(R) = 0. In this case, H0(F) = 0 and it suffices to prove that H0(F) = 0. Let x ∈ m
m m m
beanon-zero-divisoronR. Then0−→F −µ→x F isexact, whereµ ismultibplication byx. We
x
claim that 0−→F −µ→x F is exact. To see this, it suffices to show that the topology induced
by µ−1(mnF) onbF cobincides with the m-adic topology on F. As x is a non-zero-divisor
x
on R, there exists (by the Artin-Rees lemma) an integer t such that (mn : x) ⊆ mn−t for
R
all n ≥ t. Then µ−1(mnF) = ⊕ (mn : x) ⊆ mn−tF for all n ≥ t. Hence the topologies
x α R
coincide, and x is a non-zero-divisor on F. Thus, H0(F) = 0 and part (a) is proved.
m
Using part (a) and that mnF ∩F = mbnF, we have fbor all n ≥ t
b
mnF ∩H0(F)= mnF ∩H0(F)
m m
b b
= ⊕ (mn∩H0(R))
α m
= 0.
Thus, (b) is proved.
We now prove part (c). Let Rr −→ϕ Rs−→N−→0 be a presentation for N . Let K = imϕ.
By the Artin-Rees lemma, there exists an integer t ≥ 1 such that mnRs∩K ⊆ mn−tK for
all n ≥ t. As F is free, we have 0−→F ⊗ K−→F ⊗ Rs−→F ⊗ N−→0 is exact. Then
R R R