FROBENIUS POWERS OF COMPLETE INTERSECTIONS

LUCHEZAR L. AVRAMOV AND CLAUDIA MILLER

Introduction

Let R be a commutative noetherian local ring of characteristic p > 0, and let ϕ: R → R be the Frobenius endomorphism, ϕ(a) = a^{p}. Each iteration ϕ^{r} defines on R a new structure of R-module, denoted ^{ϕ^{r}}R, for which a · b = a^{p^{r}}b.

In 1969 Kunz [7, (3.3)] observed that if R is regular, then ^{ϕ^{r}}R is flat for all r ≥ 0 and he discovered that if ^{ϕ^{r}}R is flat for some r ≥ 1, then R is regular. Regularity is equivalent to the finiteness of the projective dimension of the R-module k = R/𝔪, where 𝔪 is the maximal ideal of R, so Kunz’s theorem connects the homological properties of k and those of ϕ. To summarize further results along these lines, we let c(R) denote the least integer s such that (y : 𝔪) ⊈ 𝔪^{s} for some maximal R-regular sequence y (such an s exists by the Krull Intersection Theorem).

For a finitely generated R-module M the following conditions are equivalent.

(i) M has finite projective dimension.

(ii) Tor_{n}^{R}(M,^{ϕ^{r}}R) = 0 for all n, r ≥ 1.

(iii) Tor_{n}^{R}(M,^{ϕ^{r}}R) = 0 for all n ≥ 1 and infinitely many r.

(iv) Tor_{n}^{R}(M,^{ϕ^{r}}R) = 0 for j ≤ n ≤ j + depth R + 1 where j, r are fixed integers

satisfying j ≥ 1 and r > log_{p}(c(R)). The implication (i) ⟹ (ii) is a fundamental theorem of Peskine and Szpiro [9,

(1.7)]. An early converse, (iii) ⟹ (i), was given by Herzog [4, (3.1)]. Recently, Koh and Lee [6, (2.6)] proved (iv) ⟹ (i) (but stated a weaker result).

The local ring R is a complete intersection if in some (equivalently, in each) Cohen presentation of its 𝔪-adic completion as a homomorphic image of a regular local ring, the defining ideal is generated by a regular sequence. When R has this property and the length ℓ_{R}(M) is finite, a sharpening of Herzog’s theorem is proved (but is not stated explicitly) in [8, (2.4)]: If n ≥ 1 and M has infinite projective dimension, then 0 < lim_{r→∞} (ℓ_{R}( Tor_{n}^{R}(M,^{ϕ^{r}}R))/p^{r dim R}) < ∞.

Our main result links, qualitatively and quantitatively, the homology of Frobenius powers of a complete intersection and the homology of the residue field.

Theorem. Let M be a module over a local complete intersection ring (R, 𝔪, k).

If Tor_{j}^{R}(M,^{ϕ^{r}}R) = 0 for some fixed j, r ≥ 1 then Tor_{n}^{R}(M,^{ϕ^{r}}R) = 0 for all n ≥ j; if, furthermore, M is finitely generated, then M has finite projective dimension.

If M has finite length and infinite projective dimension, then for each r ≥ 1 both

lim_{s→∞} \frac{ℓ_{R}( Tor_{2s}^{R}(M,^{ϕ^{r}}R))}{ℓ_{R}( Tor_{2s}^{R}(M, k))} and lim_{s→∞} \frac{ℓ_{R}( Tor_{2s+1}^{R}(M,^{ϕ^{r}}R))}{ℓ_{R}( Tor_{2s+1}^{R}(M, k))}

are rational numbers, and at least one of them is positive.

Date: November 19, 2000.

Both authors were partly supported by grants from the National Science Foundation.

It should be noted that none of the conclusions of the theorems requires that R be a complete intersection. While we do not know whether this hypothesis is necessary, it does play a major role in our proofs. We use it in Proposition 4 to prove that ^{ϕ^{r}}R is rigid, by refining techniques from [3], [8]. We invoke it again to apply results from [1], [2]: in Proposition 8 to deduce finite projective dimension from rigidity, and in Proposition 9 to study asymptotic behavior of Tor’s.

1. Rigidity

Throughout our discussion, different module structures on the same abelian group will be induced by various homomorphisms of commutative rings. We start by describing notation that will keep track of the module structure in use.

If α: A → B is a homomorphism of commutative rings, then ^{α}B denotes the A-B-bimodule B with A acting through α and B acting through id_{B}, that is, a · b′ = α(a)b′ and b · b′ = bb′ for all a ∈ A, b′ ∈ ^{α}B, b ∈ B. For each A-module M the tensor product M ⊗_{A} ^{α}B is a B-module: b · (m ⊗ b′) = m ⊗ (bb′) for all b ∈ B, m ∈ M, b′ ∈ ^{α}B. Using a projective resolution of M to compute Tor’s, one endows Tor_{n}^{A}(M,^{α}B), for each n ≥ 0, with a B-module structure that is natural in M.

We fix a prime number p and an integer r > 0, and set q = p^{r}. We assume all rings to be commutative noetherian of characteristic p, and for any such ring A we use φ to denote the r’th iteration of the Frobenius endomorphism: φ(a) = a^{q} for all a ∈ A. For a = a_{1}, …, a_{l} with a_{i} ∈ A, we set a^{q} = a_{1}^{q}, …, a_{l}^{q}.

From now on R denotes a local ring with maximal ideal 𝔪, residue field k = R/𝔪, and 𝔪-adic completion R̂; note that the canonical map ι: R → R̂ satisfies ιφ = φι.

Standard use of the flatness of ι yields the isomorphisms below.

Remark 1. For each R-module M and all n ≥ 0 there are natural isomorphisms

Tor_{n}^{R}(M,^{φ}R) ⊗_{R} R̂ ≅ Tor_{n}^{R}(M,^{ιφ}R̂) = Tor_{n}^{R}(M,^{φι}R̂) ≅ TorR̂ ; n(M ⊗_{R} R̂,^{φ}R̂) .

Choose, by Cohen’s Structure Theorem, a surjective homomorphism Q → R̂ where Q is a ring of formal power series k[[t]] on indeterminates t = t_{1}, …, t_{e}.

Remark 2. Let −′ denote the functor (− ⊗_{Q} ^{φ}Q) from the category of Q-modules into itself; it is exact by Kunz’s theorem. On the category of R̂-modules the functors (−′ ⊗_{Q} R̂) and (− ⊗R̂ ^{φ}R̂) are isomorphic, by associativity of tensor products.

We further assume that Ker(Q → R̂) is generated by a Q-regular sequence x = x_{1}, …, x_{c}. The subquotients of the (x)-adic filtration of the Q-algebra S = Q/(x^{q}) are free R̂-modules, we can refine it to a filtration

0 = S_{q^{c}} ⊂ S_{q^{c}−1} ⊂ ⋯ ⊂ S_{1} ⊂ S_{0} = S

with subquotients isomorphic to R̂. It defines exact sequences

0 ⟶ S_{i+1} ⟶^{τ_{i}} S_{i} ⟶^{σ_{i}} R̂ ⟶ 0 for i = 0, …, q^{c} − 1 . (1_{i})

For each Q-module N and for i = 0, …, q^{c} − 1, set S_{i}(N) = Ker(N′ ⊗_{Q} τ_{i}). The idea for the proof of part (b) below comes from [3, (2.2)] and [8, (2.1)].

Lemma 3. If R = R̂, then for each R-module M the following hold.

(a) S_{i}(M) is a homomorphic image of S_{0}(M) for i = 1, …, q^{c} − 1 .

(b) S_{0}(M) ≅ Tor_{1}^{R}(M,^{φ}R) .

Proof. Applying Tor^{Q}(−,^{φ}R) to each sequences (1_{i}) we obtain isomorphisms

(2_{i}) S_{i}(M) ≅ Coker ( Tor_{1}^{Q}(M′, σ_{i})) for i = 0, …, q^{c} − 1 .

(a) As S_{0} = S, for each exact sequence of S-modules (1_{i}) there exists a map π_{i}: S_{0} → S_{i} with σ_{0} = σ_{i}π_{i}. In view of (2_{i}), it yields a commutative diagram

[figure: Tor_{1}^{Q}(M′, S_{0}); Tor_{1}^{Q}(M′,σ_{0}); Tor_{1}^{Q}(M′,π_{i}); Tor_{1}^{Q}(M′, R); S_{0}(M); ϖ_{i}; 0; Tor_{1}^{Q}(M′, S_{i}); Tor_{1}^{Q}(M′,σ_{i}); Tor_{1}^{Q}(M′, R); S_{i}(M); 0]

where the rows are exact, and so the homomorphism ϖ_{i} is surjective.

(b) Choose an exact sequence 0 ⟶ K ⟶^{κ} L ⟶^{λ} M ⟶ 0 with a free Rmodule L, then apply (− ⊗_{Q} ^{φ}Q) to get an exact sequence of Q-modules

0 ⟶ K′ ⟶^{κ′} L′ ⟶^{λ′} M′ ⟶ 0 . (3)

Writing L = G ⊗_{Q} R with a free Q-module G, we obtain a commutative diagram

\xymatrix{ & G ⊗_{Q} Tor_{1}^{Q}(S, S) \ar[r]^{G⊗_{Q}Tor_{1}^{Q}(S,σ)} \ar[d]_{≅} & G ⊗_{Q} Tor_{1}^{Q}(S, R) \ar[d]^{≅} \\ Tor_{1}^{Q}(L, R) ⊗_{Q} ^{φ}Q \ar[r]^{≅} \ar[d]_{Tor_{1}^{Q}(λ,R)⊗_{Q}^{φ}Q} & Tor_{1}^{Q}(L′, S) \ar[r]^{Tor_{1}^{Q}(L′,σ)} \ar[d]^{Tor_{1}^{Q}(λ′,S)} & Tor_{1}^{Q}(L′, R) \ar[d]^{Tor_{1}^{Q}(λ′,R)} \\ Tor_{1}^{Q}(M, R) ⊗_{Q} ^{φ}Q \ar[r]^{≅} & Tor_{1}^{Q}(M′, S) \ar[r]^{Tor_{1}^{Q}(M′,σ)} & Tor_{1}^{Q}(M′, R)}

with isomorphisms due to the flatness of G and ^{φ}Q over Q, and the equality R′ = S.

The Koszul complex K(x, Q) is a free resolution of R over Q. For each R-module N the differential of the complex N ⊗_{Q}K(x, Q) is trivial, so there is an isomorphism Tor_{1}^{Q}(−, R) ≅ (− ⊗_{R} R^{c}) of functors on the category of R-modules. In particular, Tor_{1}^{Q}(λ, R) is surjective, hence so is Tor_{1}^{Q}(λ′, S). Similarly, the Koszul complex K(x^{q}, Q) resolves S over Q. The differential of K(x^{q}, Q) ⊗_{Q} N is trivial for each S-module N, so there is an isomorphism Tor_{1}^{Q}(S, −) ≅ (S^{c} ⊗_{S} −) of functors on the category of S-modules. Thus, Tor_{1}^{Q}(S, σ) is surjective, hence so is Tor_{1}^{Q}(L′, σ).

Formula (2_{0}) and the preceding computations yield isomorphisms

S_{0}(M) ≅ Coker ( Tor_{1}^{Q}(M′, σ)) ≅ Coker ( Tor_{1}^{Q}(λ′, R)) .

The exact sequence (3) induces the top row of the commutative diagram

\xymatrix{Tor_{1}^{Q}(L′, R) \ar[rr]^{Tor_{1}^{Q}(λ′,R)} &  & Tor_{1}^{Q}(M′, R) \ar[r] & K′ ⊗_{Q} R \ar[r]^{λ′⊗_{Q}R} \ar[d]_{≅} & L′ ⊗_{Q} R \ar[d]^{≅} \\  & 0 \ar[r] & Tor_{1}^{R}(M,^{φ}R) \ar[r] & K ⊗_{R} ^{φ}R \ar[r]^{λ⊗_{R}^{φ}R} & L ⊗_{R} ^{φ}R}

with isomorphisms from Remark 2. It gives isomorphisms that finish the proof:

Coker ( Tor_{1}^{Q}(λ′, R)) ≅ Ker(λ′ ⊗_{Q} R) ≅ Ker(λ ⊗_{R} ^{φ}R) ≅ Tor_{1}^{R}(M,^{φ}R) .  □

Proposition 4. If R is a complete intersection and M is an R-module such that Tor_{j}^{R}(M,^{φ}R) = 0 holds for some j > 0, then Tor_{n}^{R}(M,^{φ}R) = 0 for all n ≥ j.

Proof. In view of Remark 1 and the faithful flatness of ι: R → R̂, we may assume that R is complete. Obvious inductive considerations show that it suffices to establish the vanishing of Tor_{j+1}^{R}(M,^{φ}R). Replacing M by a (j − 1)st syzygy, and adjusting notation, we may change our hypothesis to read Tor_{1}^{R}(M,^{φ}R) = 0. Thus, the proposition will be proved once we show that this implies Tor_{2}^{R}(M,^{φ}R) = 0.

The exact sequences (1_{i}) and (3), and Remark 2, yield commutative diagrams

\xymatrix{ &  &  &  & 0 \ar[d] \\ 0 \ar[r] & S_{i}(K) \ar[r] & K′ ⊗_{Q} S_{i+1} \ar[r]^{K′⊗_{Q}τ_{i}} \ar[d]_{κ′⊗_{Q}S_{i+1}} & K′ ⊗_{Q} S_{i} \ar[r] \ar[d]^{κ′⊗_{Q}S_{i}} & K ⊗_{R} ^{φ}R \ar[r] \ar[d] & 0 \\  & 0 \ar[r] & L′ ⊗_{Q} S_{i+1} \ar[r] \ar[d] & L′ ⊗_{Q} S_{i} \ar[r] \ar[d] & L ⊗_{R} ^{φ}R \ar[r] \ar[d] & 0 \\  & 0 \ar[r] & M′ ⊗_{Q} S_{i+1} \ar[r] \ar[d] & M′ ⊗_{Q} S_{i} \ar[r] \ar[d] & M ⊗_{R} ^{φ}R \ar[r] \ar[d] & 0 \\  &  & 0 & 0 & 0}

for i = 0, …, q^{c} − 1. The rows are exact by Lemma 3. The columns are exact due to right exactness of tensor products and, for the rightmost one, to our hypothesis.

By decreasing induction on i we prove the labeled maps are injective. If i = q^{c}−1, then S_{i+1} = 0, so all modules in the left hand column are trivial, and our assertion is clear. If 0 ≤ i < q^{c} − 1, then κ′ ⊗_{Q} S_{i+1} is injective by the induction hypothesis. Applying the Snake Lemma to the two top rows we see that κ′ ⊗_{Q} S_{i} is injective, then applying it to the two columns on the left we conclude that K′⊗_{Q}τ_{i} is injective.

The injectivity of K′ ⊗_{Q} τ_{0} yields S_{0}(K) = 0. Lemma 3 shows that Tor_{1}^{R}(K,^{φ}R) vanishes. This module is isomorphic to Tor_{2}^{R}(M,^{φ}R), so we are done.

2. Complexity

Let M and N be finitely generated modules over a local complete intersection ring R. Following [1], we say that the pair (M, N) has complexity d, and write cx_{R}(M, N) = d, if d is the least non-negative integer with the property

ℓ_{R}(Ext_{R}^{n}(M, N) ⊗_{R} k) ≤ βn^{d−1}

for some β ∈ ℝ and all n ≫ 0. As noted in [1, (1.3)], a result of Gulliksen [5, (3.1)] implies cx_{R}(M, N) ≤ codim R, where codim R = ℓ_{R}(𝔪/𝔪^{2}) − dim R.

The number cx_{R}(M, k) is called the complexity of M and is denoted cx_{R} M. It measures the polynomial rate of growth of a minimal free resolution of M. In particular, cx_{R} M = 0 if and only if M has finite projective dimension.

Remark 5. Inequalities, proved in [1, (5.7)], link various complexities:

(4) cx_{R} M + cx_{R} N − codim R ≤ cx_{R}(M, N) ≤ min{cx_{R} M , cx_{R} N}.

If M is a Cohen-Macaulay R-module, then [1, (5.6.2), (5.6.10)] yield an equality

(5) cx_{R} M = cx_{R} ( Ext_{R}^{dim R−dim M}(M, R)) .

Identifying R̂ ⊗_{Q} ^{φ}Q and S = Q/(x^{q}), let σ: S → R̂ denote the canonical surjection, and let ρ be the composition of local homomorphisms

R ⟶^{ι} R̂ = R̂ ⊗_{Q} Q ⟶^{R̂⊗_{Q}φ} R̂ ⊗_{Q} ^{φ}Q = S .

To each R-module M we associate the S-module M′′ = M ⊗_{R} ^{ρ}S.

Remark 6. The homomorphism ρ is flat and satisfies φι = σρ.

For each R-module M and all n ≥ 0 there are natural isomorphisms

Tor_{n}^{R}(M,^{φ}R) ⊗_{R} ^{ι}R̂ ≅ Tor_{n}^{S}(M′′,^{σ}R̂) .

Indeed, ι is flat, and R̂ ⊗_{Q} φ is flat because φ: Q → Q is, so their composition ρ is flat. The equality φι = σρ is easily verified. The desired isomorphisms follow.

Lemma 7. The ring S is a complete intersection, with codim S = cx_{S} ^{σ}R̂. For each finitely generated S-module L the following equalities hold

cx_{S}(L,^{σ}R̂) = cx_{S} L = cx_{S}(^{σ}R̂, L) . (6)

If M is a finitely generated R-module, then

cx_{S} M′′ = cx_{R} M . (7)

Proof. The sequence x^{q} = x_{1}^{q}, …, x_{c}^{q} is Q-regular and contained in (t)^{2}, so S = Q/(x^{q}) is a complete intersection of codimension c. In view of the inclusion (x^{q}) ⊆ (t)(x), a result of Tate [10, Theorem 6] provides the first equality below:

∑_{n=0}^{∞} ℓ_{R}( Ext_{S}^{n}(^{σ}R̂, k))t^{n} = \frac{(1 + t)^{c}}{(1 − t^{2})^{c}} = \frac{1}{(1 − t)^{c}} = ∑_{n=0}^{∞} \binom{n + c − 1}{c − 1}t^{n} .

Thus, cx_{S} ^{σ}R̂ = c = codim S. From this expression and Remark 4 we get the equalities in (6). Equality (7) comes from the isomorphisms Ext_{S}^{n}(M′′, k) ≅ Ext_{R}^{n}(M, k) which hold for all n, due to the flatness of ρ and ι.

Proposition 8. If M is a finitely generated R-module and Tor_{j}^{R}(M,^{φ}R) = 0 for some j > 0, then M has finite projective dimension.

Proof. Remark 6 and Proposition 4 yield Tor_{n}^{S}(M′′,^{σ}R̂) = 0 for n ≥ j. By [1, Theorem III] this means Ext_{n}^{S}(M′′,^{σ}R̂) = 0 for n ≫ 0, so cx_{S}(M′′,^{σ}R̂) = 0. Formulas (6) and (7) now give cx_{R} M = 0, so M has finite projective dimension.

Proposition 9. If M has finite length and infinite projective dimension, then for each r ≥ 1 the limits below are rational numbers, and at least one of them is positive:

lim_{s→∞} \frac{ℓ_{R}( Tor_{2s}^{R}(M,^{φ}R))}{ℓ_{R}( Tor_{2s}^{R}(M, k))} and lim_{s→∞} \frac{ℓ_{R}( Tor_{2s+1}^{R}(M,^{φ}R))}{ℓ_{R}( Tor_{2s+1}^{R}(M, k))} .

Proof. Completion preserves length and projective dimension, we may assume R = R̂. Let 𝔫 denote the maximal ideal of the local ring S, and let E be an injective envelope of the S-module S/𝔫 ≅ k. The functor Hom_{S}(−, E) of Matlis duality is exact, and its restriction to the category of S-modules of finite length is isomorphic to the functor Ext_{S}^{dim S}(−, S). Setting N = Ext_{S}^{dim S}(M′′, S) we obtain for each n ≥ 0 isomorphisms of S-modules of finite length

Hom_{S} ( Tor_{n}^{S}(M′′,^{σ}R), E) ≅ Ext_{S}^{n} (^{σ}R, Hom_{S}(M′′, E)) ≅ Ext_{S}^{n}(^{σ}R, N)

Using Remark 6 and the fact that Matlis duality preserves lengths, we get

ℓ_{R}( Tor_{n}^{R}(M,^{φ}R)) = ℓ_{S}( Tor_{n}^{S}(M′′,^{σ}R)) = ℓ_{S}( Ext_{S}^{n}(^{σ}R, N)) .

Gulliksen [5, (3.1)] shows that N = Ext_{S}^{∗}(^{σ}R, N) is a finitely generated graded module over a polynomial ring S[χ] with indeterminates χ = χ_{1}, …, χ_{c} of degree 2. With N^{+} (respectively, N^{−}) denoting the submodule of N consisting of all elements whose components of odd (respectively, even) degree are equal to 0, we have a direct sum decomposition N = N^{+} ⊕ N^{−} of graded S[χ]-modules.

By the Hilbert-Serre Theorem, there exist polynomials h_{±}(t) ∈ ℚ[t] such that

ℓ_{S}( Ext_{S}^{n}(^{σ}R, N)) = \begin{cases}h_{+}(n) & for all n = 2s ≫ 0 ; \\ h_{−}(n) & for all n = 2s + 1 ≫ 0 ; \end{cases}

max{deg h_{+}(t) , deg h_{−}(t)} = dim_{S[χ]} N .

The S-module N has finite length, so it is annihilated by 𝔫^{m} for some m ≥ 1, hence (𝔫S[χ])^{m}N = 𝔫^{m}N = 0. This implies the first equality in the following sequence

dim_{S[χ]} N = dim_{S[χ]}(N /𝔫N) = cx_{S}(^{σ}R, N) = cx_{S} N = cx_{S} M′′ = cx_{R} M

where the second comes from dimension theory, the rest from formulas (6), (5), (7). On the other hand, by [2, (8.1)] there exist polynomials b_{±}(t) ∈ ℚ[t] such that

ℓ_{R}( Tor_{n}^{R}(M, k)) = \begin{cases}b_{+}(n) & for all n = 2s ≫ 0 ; \\ b_{−}(n) & for all n = 2s + 1 ≫ 0 ; \end{cases}

deg b_{+}(t) = deg b_{−}(t) = cx_{R} M .

The formulas displayed above clearly imply the desired assertions.

References

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Department of Mathematics, Purdue University, West Lafayette, IN 47907

   E-mail address: avramov@math.purdue.edu

Department of Mathematics, Michigan State University, East Lansing, MI 48824

   E-mail address: clamille@math.msu.edu