arXiv:1612.00509v2 [] 1 May 2017

DETECTING FINITE FLAT DIMENSION OF MODULES VIA
ITERATES OF THE FROBENIUS ENDOMORPHISM
DOUGLAS J. DAILEY, SRIKANTH B. IYENGAR, AND THOMAS MARLEY
Abstract. It is proved that a module M over a Noetherian ring R of positive
characteristic p has finite flat dimension if there exists an integer t > 0 such
fe
that TorR
i (M, R) = 0 for t 6 i 6 t + dim R and infinitely many e. This
extends results of Herzog, who proved it when M is finitely generated. It is
also proved that when R is a Cohen-Macaulay local ring, it suffices that the
Tor vanishing holds for one e > logp e(R), where e(R) is the multiplicity of R.

1. Introduction
The Frobenius endomorphism f : R → R of a commutative Noetherian local ring
R of prime characteristic p is an effective tool for understanding the structure of
such rings and the homological properties of finitely generated modules over them.
A paradigm of this is a result of Kunz [12] that R is regular if and only f e is flat for
some (equivalently, every) integer e > 1. Our work is motivated by the following
module-theoretic version of Kunz’s result:
There exists an integer c such that for any finitely generated R-module M the
following statements are equivalent:
(1) The flat dimension of M is finite.
fe
(2) TorR
i (M, eR) = 0 for all positive integers i and e.
f
(3) TorR
i (M, eR) = 0 for all i > 0 and infinitely many e > 0.
R
f
(4) Tori (M, R) = 0 for depth R+1 consecutive values of i > 0 and some e > c.
Peskine and Szpiro [15] proved that (1)⇒(2), Herzog [10] proved that (3)⇒(1),
and Koh and Lee [11] proved that (4)⇒(1). Recently, the third author and M.
Webb [14, Theorem 4.2] proved the equivalence of conditions (1), (2), and (3) for
all R-modules, even infinitely generated ones. In their work, the argument for
(3)⇒(1) is quite technical and heavily dependent on results of Enochs and Xu [8]
concerning flat cotorsion modules and minimal flat resolutions.
In this work we give another proof of [14, Theorem 4.2] that circumvents [8];
more to the point, it yields a stronger result and sheds additional light also on the
finitely generated case. See [4] for the definition of the flat dimension of a complex.
Theorem 1.1. Let R be a Noetherian local ring of prime characteristic p and M
an R-complex with s := sup H∗ (M ) finite. The following conditions are equivalent:
(1) The flat dimension of M is finite.
Date: March 28, 2018.
2010 Mathematics Subject Classification. 13D05; 13D07, 13A35.
Key words and phrases. Frobenius map, flat dimension, homotopical Loewy length.
S.B.I. was partly supported by NSF grant DMS-1503044.
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D. J. DAILEY, S. B. IYENGAR, AND T. MARLEY

f
(2) TorR
i (M, R) = 0 for all i > s and e > 0.
fe
(3) There exists an integer t > s such that TorR
i (M, R) = 0 for t 6 i 6 t+dim R
and for infinitely many e.
Moreover, when R is Cohen-Macaulay of multiplicity e(R), it suffices that the vanishing in (3) holds for one e > logp e(R).
e

This result is proved in Section 3. The key element in our proofs is the use of
homotopical Loewy lengths of complexes, in much the same way as in the work of
the second author and Avramov and C. Miller [1, Section 4]. Using rather different
methods, Avramov and the second author [2] have proved that the last part of the
theorem above holds without the Cohen-Macaulay hypothesis, but with a different
lower bound on e.
2. Homotopical Loewy length
In this section we collect results on homotopical Loewy length, and their corollaries, needed in our proof of Theorem 1.1. Throughout (R, m, k) will be a local
(this includes commutative and Noetherian) ring, with maximal ideal m and residue
field k; there is no restriction on its characteristic. We adopt the terminology and
notation of [1, Section 2] regarding complexes and related constructs. In particular, given R-complexes M and N , the notation M ≃ N means that M and N are
isomorphic in D(R), the derived category of R-modules.
The Loewy length of an R-complex M is the number
ℓℓR (M ) := inf{n ∈ N | mn M = 0}.
Following [3, 6.2], the homotopical Loewy length of an R-complex M is the number
ℓℓD(R) (M ) := inf{ℓℓR (V ) | M ≃ V in D(R)}.
Given a finite sequence x ⊂ R and an R-complex M , we write K[x; M ] for the
Koszul complex on x, with coefficients in M . The result below extends, with an
identical proof, [1, Proposition 4.1] and [3, Theorem 6.2.2] that deal with the case
when x generates m.
Proposition 2.1. Let x be a finite sequence in R such that ℓR (R/xR) is finite.
For each R-complex M there are inequalities
ℓℓD(R) K[x; M ] ≤ ℓℓD(R) K[x; R] < ∞ .
Proof. Let I = (x) and K = K[x; R]. For each i, consider the subcomplex C i of K
· · · → I i−2 K2 → I i−1 K1 → I i K0 → 0.
Since I i annihilates K/C i and I is m-primary, it follows that ℓℓR (K/C i ) is finite for
each i. There exists an r such that C i is acyclic for all i > r; cf. [7, Proposition].
Thus, for i > r the natural map K → K/C i is a quasi-isomorphism, and hence the
homotopical Loewy length of K is finite. The inequality ℓℓD(R) (K ⊗R M ) 6 ℓℓD(R) K
can be verified exactly as in the proof of [1, Proposition 4.1].

The following invariant plays an important role in what follows.
c(R) := inf{ℓℓD(R) K[x; R] | x is an s.o.p. for R}.
Proposition 2.1 yields that c(R) is finite for any R. For our purposes, we need a
uniform bound on c(Rp ), as p varies over the primes ideals in R. We have been

MODULES OF FINITE FLAT DIMENSION AND THE FROBENIUS ENDOMORPHISM

3

able to establish this only for Cohen-Macaulay rings; this is the content of the next
result, where e(R) denotes the multiplicity of R.
Lemma 2.2. Let (R, m, k) be local ring with k infinite. When R is Cohen-Macaulay,
there is an inequality c(Rp ) 6 e(R) for each p in Spec R.
Proof. By a result of Lech [13], one has e(R) > e(Rp ) for all p ∈ Spec R. Moreover,
it is easy to verify that since k is infinite, so is k(p) for each p. It thus suffices to
verify that c(R) 6 e(R).
Let x be a s.o.p. of R that is a minimal reduction of m; this exists because k is
infinite; see [16, Proposition 8.3.7]. Then there are inequalities
e(R) = ℓR (R/(x)) > ℓℓR (R/(x)) = ℓℓD(R) K[x; R] > c(R).
For the first equality see, for example, [16, Proposition 11.2.2], while holds the
second equality holds because K[x; R] ≃ R/(x); both need the hypothesis that R
is Cohen-Macaulay.

The preceding result bring up the following question; its import for the results
in this paper will become apparent in the proof of Theorem 1.1.
Question 2.3. Is sup{c(Rp ) | p ∈ Spec R} finite for any Noetherian ring R?
The proof of Lemma 2.2 is not likely to be of help in answering this question.
Remark 2.4. Let x be a finite sequence in a local ring (R, m, k). Since the ideal (x)
annihilates H∗ (K[x; R]), it is immediate from definitions that there is an inequality
ℓℓR (R/(x)) ≤ ℓℓD(R) K[x; R] .
Equality holds when x is a regular sequence, for then R/(x) ≃ K[x; R]; this is the
main reason for the Cohen-Macaulay hypothesis in Lemma 2.2. The inequality can
be strict in general.
For example, if (x) = m, then ℓℓR (R/m) = 1, whilst ℓℓD(R) K[x; R] = 1 exactly
when R is regular; see [3, Corollary 6.2.3].
Here is an example where the inequality is strict for x a s.o.p.
Let R := k[|x, y|]/(xn y, y 2 ), where n > 1 is an integer. The residue class of
x in R is a s.o.p., and the Loewy length of R/(x) equals 2. We claim that the
homotopical Loewy length of K[x; R] is n + 1.
Indeed, the following subcomplex of K[x; R]
A := 0 → (xn ) → (xn+1 ) → 0
≃

is acyclic so one has K[x; R] −
→ K[x; R]/A. Since
R
R
K[x; R]/A = 0 −→ n −→ n+1 −→ 0
(x )
(x
)
and the Loewy length of this complex is n+1, it follows that ℓℓD(R) (K[x; R]) 6 n+1.
On the other hand
H1 (K[x; R]) = (xn−1 y) ⊂ R .
Suppose K[x; R] ≃ V for some R-complex V . Since K[x; R] is a finite complex of
free R-modules, there must exist a morphism f : K[x; R] → V of R-complexes with
H∗ (f ) an isomorphism. The map f1 : K1 = R → V1 satisfies
0 6= f1 (xn−1 y) = xn−1 yf (1)
It follows that xn−1 y · V1 6= 0, and hence that ℓℓR V > ℓℓR (V1 ) > n + 1.

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D. J. DAILEY, S. B. IYENGAR, AND T. MARLEY

As in [1, Proposition 4.3(2)] one can apply Proposition 2.1 to local homomorphisms to obtain an isomorphism relating Koszul homologies.
Proposition 2.5. Let (S, n, l) be a local ring, y a finite sequence of elements in S
such that the ideal (y) is n-primary, and set c := ℓℓD(S) K[y; S].
If ϕ : (R, m, k) → (S, n, l) is a local homomorphism satisfying mS ⊆ nc , then for
each R-complex M , there exists an isomorphism of graded k-vector spaces
R
∼
TorR
∗ (M, K[y; S]) = Tor∗ (M, k) ⊗k H∗ (K[y; S]) .



We also need the following routine computation.
Lemma 2.6. Let ϕ : R → S be a homomorphism of rings, y = y1 , . . . , yd a sequence
of elements in S. Let M be an R-complex and t an integer such that TorR
i (M, S) = 0
for t 6 i 6 t + d. Then TorR

t+d (M, K[x; S]) = 0.
Applied to (an appropriate composition of the Frobenius endomorphism) the
next result yields an analogue of [11, Proposition 2.6] for complexes. The number
of consecutive vanishing of Tor required in the case of modules is not optimal
(dim R + 1 as compared to depth R + 1 in [11]), but the proof we give applies to
complexes whose homology need not be finitely generated.
Lemma 2.7. Let ϕ : (R, m, k) → (S, n, l) be a homomorphism of local rings such
that ϕ(m) ⊆ nc(S) . Let M be an R-complex.
If there is an integer t such that TorR
i (M, S) = 0 for t 6 i 6 t + dim S, then
TorR
t+dim S (M, k) = 0 .
If moreover the R-module H(M ) is finitely generated and t ≥ sup H(M ) − dim S,
the flat dimension of M is at most t + dim S.
Proof. Set d := dim S and let y be an s.o.p of S such that c(S) = ℓℓD(S) K[y; S].
The hypothesis on ϕ and Lemma 2.6 yield TorR
t+d (M, K[y; S]) = 0. It then follows
R
from Proposition 2.5 that Tort+d (M, k) = 0, since H0 (K[y; S]) 6= 0.
Given this, and the additional hypotheses on H(M ) and t, the desired result
follows from the existence of minimal resolutions; see [4, Proposition 5.5(F)].

3. Finite flat dimension
This section contains a proof of Theorem 1.1. In preparation, we recall that
an R-complex has finite flat dimension if it is isomorphic in D(R) to a bounded
complex of flat R-modules. The following result is [6, Theorem 4.1].
Remark 3.1. Let R be a Noetherian ring and M an R-complex. If there exists an
p
integer n > sup H∗ (M ) + dim R with TorR
n (Mp , k(p)) = 0 for all p ∈ Spec R then
the flat dimension of M is finite.
In what follows, given an endomorphism φ : R → R and an R-complex M , we
write φM for M viewed as an R-complex via φ.
Proof of Theorem 1.1. Recall that R is a Noetherian ring of prime characteristic p
and f : R → R is the Frobenius endomorphism.
fe
(1)⇒(2): Fix an integer e > 1 and set r := sup TorR
∗ (M, R). Since flat dimR M
is finite, r < ∞ holds. The desired result is that r 6 s.

MODULES OF FINITE FLAT DIMENSION AND THE FROBENIUS ENDOMORPHISM

5

f
Pick a prime ideal p associated to TorR
r (M, R). Since Frobenius commutes
with localization one has
e
e
TorR (M, f R)p ∼
= TorRp (Mp , f Rp )
e

r

r

as Rp -modules. Moreover flat dimRp Mp is finite. Thus replacing R and M by their
fe
localizations at p we get that the maximal ideal of R is associated to TorR
r (M, R);
that is to say, the depth of the latter module is zero.
The next step uses some results concerning depth for complexes; see [9]. Given
the conclusion of the last paragraph, [9, 2.7] yields the last equality below.
e

R
f
depth R − sup TorR
∗ (k, M ) = depthR ( R) − sup Tor∗ (k, M )
e

f
= depthR (M ⊗L
R R)

= −r
The second one is by [9, Theorem 2.4]. The same results also yield
depth R − sup TorR
∗ (k, M ) = depthR M > − sup H∗ (M ) = −s
It follows that −r > −s, that is to say, r ≤ s, which is the desired conclusion.
(3)⇒(1): Let d = dim R. By Remark 3.1, it suffices to verify that
(3.1)

R

p
(Mp , k(p)) = 0
Tort+d

for all p ∈ Spec R.

f
Fix p ∈ Spec R and choose e such that p > c(Rp ) and TorR
i (M, R) = 0 for
t 6 i 6 t + d; such an e exists by our hypothesis. As the Frobenius map commutes
with localization, one gets
e

e

R

e

Tori p (Mp , f (Rp )) = 0
e

for t 6 i 6 t + d.

c(Rp )

The choice of e ensures that f (pRp ) ⊆ p
Rp . Thus, Lemma 2.7 applied to the
Rp
(Mp , k(p)) = 0, as desired.
Frobenius endomorphism Rp → Rp yields Tort+d
Assume now that R is Cohen-Macaulay and that the vanishing in (3) holds
for some e > logp e(R). It is a routine exercise to verify that the hypotheses
remain unchanged, and that the desired conclusion can be verified, after passage to
faithfully flat extensions. One can thus assume that the residue field k is infinite;
see [5, IX.37]. Then, by the choice of e and Lemma 2.2, one gets that pe > c(Rp )
for each p in Spec R. Then one can argue as above to deduce that (3.1) holds, and
that yields the finiteness of the flat dimension of M .

References
1. L. L. Avramov, M. Hochster, S. B. Iyengar, Y. Yao, Homological invariants of modules
over contracting endomorphisms, Math. Ann. 353 (2012), 275–291.
2. L. L. Avramov, S. B. Iyengar, Ghost homomorphisms in local algebra, in preparation.
3. L. L. Avramov, S. Iyengar, C. Miller, Homology over local homomorphisms, Am. J. Math.
128 (2006), 23 – 90.
4. L. L. Avramov, H.-B. Foxby, Homological dimensions of unbounded complexes, J. Pure
Appl. Alg. 71 (1991), 129–155.
5. N. Bourbaki, Éléments de mathématique: algèbre commutative, chapitres 8 et 9, Springer,
Berlin, 2006. Reprint of the 1983 original.
6. L. W. Christensen, S. B. Iyengar, T. Marley, Rigidity of Ext and Tor with coefficients in residue fields of a commutative noetherian ring, preprint 2016; available at
arxiv.org/pdf/1611.03280v1.pdf
7. J. A. Eagon, M. Fraser, A note on the Koszul complex, Proc. Am. Math. Soc. 19 (1968),
251–252.

6

D. J. DAILEY, S. B. IYENGAR, AND T. MARLEY

8. E. Enochs, J. Xu, On invariants dual to the Bass numbers, Proc. Am. Math Soc. 125
(1997), 951–960.
9. H.-B. Foxby, S. Iyengar, Depth and amplitude of unbounded complexes, Commutative algebra and its interaction with algebraic geometry (Grenoble-Lyon 2001), Contemp. Math.
331, American Math. Soc. Providence, RI, 2003; pp. 119–137.
10. J. Herzog, Ringe der Charakteristik p und Frobeniusfunktoren, Math. Z. 140 (1974), 67–
78.
11. J. Koh, K. Lee, Some restrictions on the maps in minimal resolutions, J. Algebra 202
(1998), 671–689.
12. E. Kunz, Characterizations of regular local rings for characteristic p, Am. J. Math. 91
(1969), 772–784.
13. C. Lech, Inequalities related to certain couples of local rings, Acta Math. 112 ((1964),
69–89.
14. T. Marley, M. Webb, The acyclicity of the Frobenius functor for modules of finite flat
dimension, J. Pure Appl. Algebra 220 (2016), no. 8, 2886-2896.
15. C. Peskine, L. Szpiro, Dimension projective finie et cohomologie locale. Applications à la
démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes
Études Sci. Publ. Math. No. 42 (1973), 47–119.
16. I. Swanson, C. Huneke, Integral closures of ideals, ring, and modules, London Math. Soc.
LNS no. 336, Cambridge University Press, Cambridge, 2006.
D.J.D. University of Dallas, Irving, Texas 75062, U.S.A.
E-mail address: ddailey@udallas.edu
URL: http://www.udallas.edu
S.B.I. University of Utah, Salt Lake City, UT 84112, U.S.A.
E-mail address: iyengar@math.utah.edu
URL: http://www.math.utah.edu/~iyengar
T.M. University of Nebraska-Lincoln, Lincoln, NE 68588, U.S.A.
E-mail address: tmarley1@unl.edu
URL: http://www.math.unl.edu/~tmarley

