PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 6, June 1998, Pages 1637–1643 S 0002-9939(98)04228-2

A CHARACTERIZATION OF GORENSTEIN RINGS IN CHARACTERISTIC p (> 0)

S. P. DUTTA

(Communicated by Wolmer V. Vasconcelos)

Abstract. A new characterization of Gorenstein rings in characteristic p (> 0) is proved. It involves asymptotic behaviour of lengths of modules under the Frobenius map.

In this note we are going to study how the Gorenstein property for a local ring (R, m, k = R/m) of characteristic p (> 0), dimension d, is connected with the behaviour of lim_{n→∞} ℓ(F^{n}(M))/p^{nd} and lim_{n→∞} ℓ(F^{n}(M^{v}))/p^{nd}, when M is a module of finite length and finite projective dimension over R, M^{v} = Hom_{R}(M, E), E is the injective hull of k over R and F^{n}(M) is obtained from M by applying the Frobenius map n times ((M ⊗_{R}f_{R}^{n} [P-S]); Notation). It is well known that for a module M of finite length over any local ring R of dim d and ch. p (> 0) lim_{n→∞} ℓ(F^{n}(M))/p^{nd} is positive, and if R is Cohen-Macaulay and M = R/(x_{1}, …, x_{d}), where x_{1}, …, x_{d} is a system of parameters in R, the above limit is ℓ(R/(x_{1}, …, x_{d})). Recall that when R is Gorenstein M^{v} ≃ Ext_{R}^{d}(M, R) for any module M of finite length. Moreover when pd_{R}M < ∞, we have (F^{n}(M))^{v} ≃ Ext^{d}(F^{n}(M), R) ≃ F^{n}(Ext^{d}(M, R)) ≃ F^{n}(M^{v}). Thus ℓ(F^{n}(M)) = ℓ(F^{n}(M^{v})), and obviously the above limits are same. This leads us to raise the same question for Cohen-Macaulay rings which are not Gorenstein. Our main theorem shows that the answer, in this case, is in the negative even for cyclic modules of the form R/(x_{1}, …, x_{d}), where x_{1}, …, x_{d} is a system of parameters of R. In fact our main theorem is the following:

Theorem. A Cohen-Macaulay local ring R of dimension d and ch. p (> 0) is Gorenstein if and only if there exists a system of parameters x_{1}, …, x_{d} such that

lim_{n→∞} ℓ(F^{n}((R/\underline{x})^{v}))/p^{nd} = ℓ(R/\underline{x}).

Motivation for the study of the above result came from the study of the Strong Intersection Conjecture due to Peskine and Szpiro [P-S]. In this conjecture they assert that, given a pair of finitely generated modules M and N over a local ring R such that ℓ(M ⊗_{R} N) < ∞ and pd_{R}M < ∞, dimN ≤ grade M. While studying the conjecture over a Cohen-Macaulay ring of dimension d and positive characteristic, with a canonical module Ω (≠ R), this author was led to inquire whether lim_{n→∞} ℓ(H_{m}^{0}(F^{n}(Ω/(x_{2}, …, x_{d})Ω)))/p^{nd} is 0. Recall that, though Ω/(x_{2}, …, x_{d})Ω is a Cohen-Macaulay module of dimension 1, F^{n}(Ω/(x_{2}, …, x_{2})Ω)

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors March 15, 1996 and, in revised form, November 26, 1996.

1991 Mathematics Subject Classification. Primary 13D02, 13H10.

This research was partially supported by an NSF grant and an NSA grant.

is not necessarily Cohen-Macaulay. In the course of our proof of the main theorem, we show that the above limit is never zero. If the limit were zero, one could venture for the construction of a counterexample to the above conjecture.

I am thankful to the referee for comments and suggestions, some of which have been incorporated into the body of this paper.

Notations. Throughout this work (R, m, k) denotes a local ring of characteristic p (> 0), dimension d, m its maximal ideal; k = R/m and k is perfect. The Frobenius map f : R → R, given by f(x) = x^{p} for all x ∈ R, is a ring homomorphism. We denote by f_{R}^{n} the bi-algebra R, having the structure of an R-algebra from the left by f^{n} and from the right by the identity map, i.e. if α ∈ R, x ∈ f_{R}^{n}, α · x = α^{p^{n}}x and x · α = xα. For any module M, we write F_{R}^{n}(M) for M ⊗_{R} f_{R}^{n} (we usually drop R from the notation when there is no chance of confusion) and when ℓ(M) (= length of M) < ∞, M^{v} for Hom_{R}(M, E), where E is the injective hull of k over R.

If pd_{R}M < ∞, χ_{i}(M, N) will denote ∑_{j=0}^{pdM}(−1)^{j}ℓ(Tor_{i+j}^{R}(M, N)). For any finitely

generated module M, μ(M) denotes the minimal number of generators of M over R. We write lim to denote lim_{n→∞} and Ω to denote a canonical module of R. For any system of parameters x_{1}, …, x_{d} of R and for any module M, \underline{x}_{i}M denotes the submodule (x_{i}, …, x_{d})M and \overline{M} denotes M/x_{1}M. If θ = (θ_{ij}) denotes an s × r matrix, then θ^{[p^{n}]} stands for the s × r matrix (θ_{ij}^{p^{n}}). Also for any module M, and for any element x ∈ R, (0 : x)M = {α ∈ M | xα = 0}. If I denotes the ideal (y_{1}, …, y_{t}) in R, then I^{[p^{n}]} denotes the ideal (y_{1}^{p^{n}}, …, y_{t}^{p^{n}}) in R.

Now we state our main theorem.

Theorem. Let R be a Cohen-Macaulay local ring of dimension d and characteristic p (> 0). Then R is Gorenstein if and only if there exists a system of parameters x_{1}, …, x_{d} such that

lim_{n→∞} ℓ(F^{n}((R/\underline{x})^{v}))/p^{nd} = ℓ(R/\underline{x}).

Proof. We note that without any loss of generality one can assume R is complete. Recall that for any system of parameters x_{1}, …, x_{d} of R

lim ℓ(F^{n}(R/\underline{x}))/p^{nd} = ℓ(R/\underline{x}).

Now let R be Gorenstein. Then we have

(F^{n}(R/\underline{x}))^{v} ≃ Ext_{R}^{d}(F^{n}(R/\underline{x}), R) ≃ F^{n}(Ext^{d}(R/\underline{x}, R)) ≃ F^{n}((R/\underline{x})^{v}).

Hence

lim ℓ(F^{n}((R/\underline{x})^{v}))/p^{nd} = lim ℓ((F^{n}(R/\underline{x}))^{v})/p^{nd} = lim ℓ(F^{n}(R/\underline{x}))/p^{nd} = ℓ(R/\underline{x}).

Next, let us assume that R is not Gorenstein. Our proof is completed in two parts.

In Part 1, we will show that it is possible to construct a system of parameters x_{1}, … x_{d} of R such that

lim ℓ(F^{n}((R/\underline{x})^{v}))/p^{nd} ≠ ℓ(R/\underline{x});

and in Part 2 we will prove the above result for any system of parameters.

Part 1. We first note a rather trivial fact in dimension 0, which will be used in our proof:

GORGENSTEIN RINGS IN CHARACTERISTIC p (> 0) 1639

Fact. Let R be a local ring of dimension zero. Let M be a module of finite length. Let μ(M) denote the number of minimal generators of M. Then ℓ(F^{n}(M)) = μ(M)ℓ(R) for n >> 0.

Proof of the Fact. Consider a minimal representation of M:

R^{r} ⟶^{θ} R^{s} → M → 0

where the entries of θ are in m. Applying ⊗f_{R}^{n}, we get an exact sequence

R^{r} ⟶^{θ^{[p^{n}]}} R^{s} → F^{n}(M) → 0.

Since dim R = 0, for n >> 0, entries of θ^{[p^{n}]} are all 0. Hence the result.

We note that (R/\underline{x})^{v} ≃ Ext^{d}(R/\underline{x}, Ω) = Ω/\underline{x}Ω. So we need to establish that for a certain system of parameters x_{1}, …, x_{d}

ℓ(R/\underline{x})(= lim ℓ(F^{n}(R/\underline{x}))/p^{nd} = lim ℓ(R/\underline{x}^{p^{n}})/p^{nd}) ≠ lim ℓ(F^{n}(Ω/\underline{x}Ω))/p^{nd}.

(\underline{x}^{p^{n}} stands for (x_{1}^{p^{n}}, …, x_{d}^{p^{n}}) and \underline{x}_{i}^{p^{n}} stands for (x_{i}^{p^{n}}, …, x_{d}^{p^{n}})). Note that ℓ(Ω/\underline{x}Ω) = ℓ(R/\underline{x}) (for any module M of finite length, ℓ(M) = ℓ(M^{v})).

Step 0. dim R = 0. In this case ℓ(F^{n}(Ω)) = μ(Ω)ℓ(R) for n >> 0 (Fact). Thus the above limits are equal if and only if μ(Ω) = 1 i.e., R is Gorenstein. Now suppose x_{1}, …, x_{d} (d > 0) is a system of parameters of R such that

(1) lim ℓ(R/\underline{x}^{p^{n}})/p^{nd} = lim ℓ(F^{n}(Ω/\underline{x}Ω))/p^{nd}.

Note that the left-hand side is ℓ(R/\underline{x}). We also recall that F^{n}(M ⊗_{R} N) ≃ F^{n}(M) ⊗_{R} F^{n}(N). So F^{n}(Ω/\underline{x}Ω) ≃ F^{n}(Ω) ⊗ F^{n}(R/\underline{x}) = F^{n}(Ω) ⊗ R/\underline{x}^{p^{n}}. Since R is Cohen-Macaulay, pd_{R}R/\underline{x} < ∞.

Step 1. (1) holds if and only if Ω ⊂ R and χ_{1}(R/\underline{x}^{p^{n}}, F^{n}(Ω))/p^{nd} → 0 as n → ∞. Since support Ω = support R and F^{n}(Ω)_{p} = F_{R_{p}}^{n}(Ω_{p}) [P-S], we have

χ(R/\underline{x}^{p^{n}}, F^{n}(Ω)) = p^{nd}χ(R/\underline{x}, F^{n}(Ω))(χ is additive)

= p^{nd} ∑_{P ∈Ass(R)} ℓ(F_{R_{P}}^{n} (Ω_{p}))χ(R/\underline{x}, R/P).

We have

χ(R/\underline{x}^{p^{n}}, F^{n}(Ω)) = p^{nd} ∑_{P ∈Ass(R)} μ_{R_{p}}(Ω_{p})ℓ(R_{P})χ(R/\underline{x}, R/P)

for n >> 0 (Fact).

Recall ([L]) that for any finitely generated module T such that ℓ(T/\underline{x}T) < ∞, χ(R/\underline{x}, T) ≥ 0 and equality holds if and only if dim T < d. Moreover χ_{i}(R/\underline{x}, T) ≥ 0. Thus, if μ_{R_{p}}(Ω_{p}) > 1 for any prime P ∈ Ass(R), it follows from the above that

lim χ(R/\underline{x}^{p^{n}}, F^{n}(Ω))/p^{nd} > ∑_{P ∈Ass(R)} ℓ(R_{P})χ(R/\underline{x}, R/P).

But the right-hand side of the above inequality is nothing but χ(R/\underline{x}, R) = ℓ(R/\underline{x}). Since χ_{1}(R/\underline{x}^{pn}, F^{n}(Ω)) ≥ 0, when lim ℓ(F^{n}(Ω/\underline{x}Ω))/p^{nd} = ℓ(R/\underline{x}), the above inequality forces us to have

(2) μ_{R_{p}}(Ω_{p}) = 1, ∀P ∈ Ass(R) and lim χ_{1}(R/\underline{x}^{p^{n}}, F^{n}(Ω))/p^{nd} = 0.

This can easily be checked to be both necessary and sufficient in order that (1) may hold. Note that Ω_{p} is a canonical module of R_{p}, and thus (2) implies that R is generically Gorenstein. So, we can assume that Ω is an ideal of height 1 in R. Recall that in such a case R/Ω is Gorenstein.

Now we need to study what it means to have lim χ_{1}(R/\underline{x}^{p^{n}}, F^{n}(Ω))/p^{nd} = 0.

Step 2. We can choose x_{1} so that lim χ_{1}(R/\underline{x}^{p^{n}}, F^{n}(Ω))/p^{nd} = 0 if and only if

lim ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω)))/p^{nd} = 0.

From Step 1, Ω ⊂ R is a height one ideal of R. Take x_{1} ∈ Ω.

Consider the short exact sequence

0 → Ω → R → R/Ω → 0.

Applying ⊗f_{R}^{n}, we obtain the following short exact sequences:

0 → Tor_{1}(R/Ω, f_{R}^{n}) → F^{n}(Ω) → Ω^{[p^{n}]} → 0, 0 → Ω^{[p^{n}]} → R → R/Ω^{[p^{n}]} → 0.

Note that ℓ(R/(Ω + \underline{x}_{2})) < ∞ and hence ℓ(Tor_{i}(R/\underline{x}_{2}, F^{n}(Ω)) < ∞. Now consider the exact sequence

0 → R/\underline{x}_{2}^{p^{n}} ⟶^{x_{1}^{p^{n}}} R/\underline{x}_{2}^{p^{n}} → R/\underline{x}^{p^{n}} → 0.

Applying ⊗F^{n}(Ω) to the above sequence, from the long exact sequence of Tor’s we obtain

χ_{1}(F^{n}(Ω), R/\underline{x}^{p^{n}}) = ℓ((0 : x_{1}^{p^{n}})F^{n}(Ω/\underline{x}_{2}Ω)).

Again from the exact sequence

0 → Ω/\underline{x}_{2}Ω → R/\underline{x}_{2} → R/(Ω + \underline{x}_{2}) → 0,

applying ⊗f_{R}^{n}, we get the following exact sequence:

0 → Tor_{1}(R/(Ω + \underline{x}_{2}), f_{R}^{n}) → F^{n}(Ω/\underline{x}_{2}Ω) → R/\underline{x}_{2}^{p^{n}} → R/(Ω + \underline{x}_{2})^{[p^{n}]} → 0.

Since x_{1} ∈ Ω, it follows from the above sequence that

(0 : x_{1}^{p^{n}})F^{n}(Ω/\underline{x}_{2}Ω) = Tor_{1}(R/(Ω + \underline{x}_{2}), f_{R}^{n}) = H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω)).

Thus lim χ_{1}(R/\underline{x}^{pn}, F^{n}(Ω))/p^{nd} = 0 if and only if

lim ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω)))/p^{nd} = 0. (3)

We have now two cases to consider:

(a) ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))) ≤ Kp^{n(d−2)}, where K is a constant.

(b) ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))) = O(p^{n(d−1)}).

Step 3. Case (a). We consider the case when ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))) ≤ Kp^{n(d−2)}. We use induction on d (≥ 0). For d = 0, see \underline{Step 0}. Consider the exact sequence

0 → H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω)) → F^{n}(Ω/\underline{x}_{2}Ω) → N_{n} → 0. (4)

Because of (1), it follows that lim ℓ(N_{n}/x_{1}^{pn}N_{n})/p^{nd} = ℓ(R/\underline{x}). But x_{1} is not a zero-divisor on N_{n}, so we have lim ℓ(N_{n}/x_{1}N_{n})/p^{n(d−1)} = ℓ(R/\underline{x}). Tensoring (4) by R/x_{1}R, we get the following exact sequence:

0 → \overline{H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))} → F n ; \overline{R}(\overline{Ω}/\underline{x}_{2}\overline{Ω}) → \overline{N}_{n} → 0. (5)

(For any module M, we write \overline{M} to denote M/x_{1}M.) Since ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))) ≤ Kp^{n(d−2)}, lim ℓ(\overline{H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))})/p^{n(d−1)} = 0. This shows that

lim ℓ(F n ; \overline{R}(\overline{Ω}/\underline{x}_{2}\overline{Ω}))/p^{n(d−1)} = lim ℓ(\overline{N}_{n})/p^{n(d−1)} = ℓ(R/\underline{x}) = ℓ(\overline{R}/\underline{x}_{2}\overline{R}).

Since the dimension of \overline{R} is d − 1, we are done by induction on d = dim R (\overline{Ω} = Ω/x_{1}Ω is a canonical module for \overline{R} = R/x_{1}R).

Step 4. Case (b). Now we consider the case when ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))) is strictly O(p^{n(d−1)}). If ℓ(\overline{H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))}) is O(p^{n(d−2)}), we are done by Step 3. So assume that ℓ(\overline{H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))}) = O(p^{n(d−1)}). Recall from Step 2, that H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω)) = Tor_{1}(R/(Ω + \underline{x}_{2}), f_{R}^{n}). We have

ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))) ≤ c p^{n(d−1)}, (6)

where c is a constant. Suppose that

ℓ(\overline{H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))}) = c_{1}p^{n(d−1)} + O(p^{n(d−2)}) (7)

(the existence of c_{1} follows from (5)) and

ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω)) ⊗ R/x_{1}^{i}R) = c_{i}p^{n(d−1)} + O(p^{n(d−2)}).

Let t be an integer such that t > [c/c_{1}]. Applying ⊗R/x_{1}^{i}R to (4), we obtain the following exact sequence:

0 → H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω)) ⊗ R/x_{1}^{i}R → F^{n}(Ω/\underline{x}_{2}Ω) ⊗ R/x_{1}^{i}R → N_{n}/x_{1}^{i}N_{n} → 0. (8)

Since x_{1} is a non-zero-divisor on N_{n}, ℓ(N_{n}/x_{1}^{i}N_{n}) = iℓ(\overline{N}_{n}). Again, from the exact sequence

0 → R/x_{1}R → R/x_{1}^{2}R → R/x_{1}R → 0

we get

ℓ(F^{n}(Ω/x_{1}Ω) ⊗ R/x_{1}^{2}R) = 2ℓ(F^{n}(Ω/\underline{x}_{2}Ω) ⊗ R/x_{1}R) + O(p^{n(d−2)})

(by applying Proposition 1, [S]). Hence for any fixed i,

ℓ(F^{n}(Ω/x_{1}Ω) ⊗ R/x_{1}^{i}R) = iℓ(F n ; \overline{R}(\overline{Ω}/\underline{x}_{2}\overline{Ω})) + O(p^{n(d−2)}).

Now choose an i > t. It then follows from (8) and the above equality that, for n >> 0,

ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω)) ⊗ R/x_{1}^{i}R) = iℓ(F n ; \overline{R}(\overline{Ω}/\underline{x}_{2}\overline{Ω})) − iℓ(\overline{N}_{n}) + O(p^{n(d−2)})
= iℓ(\overline{H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))}) + O(p^{n(d−2)}) from (5)
> tℓ(\overline{H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))}) + O(p^{n(d−2)}).

Thus, (6), (7), (8) and the above inequality imply that

ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω))) ⊗ R/x_{1}^{i}R) > ℓ(H_{m}^{0}(F^{n}(Ω/\underline{x}_{2}Ω)))

for n >> 0, which is impossible. (The above arguments also show that c cannot be equal to c_{1}.) Hence, by Step 3 and Step 4, lim ℓ(F^{n}(H_{m}^{0}(Ω/\underline{x}_{2}Ω)))/p^{nd} cannot be 0. Thus for any system of parameters x_{1}, …, x_{d} of R, with x_{1} ∈ Ω,

ℓ(R/\underline{x}) = lim ℓ(F^{n}(R/\underline{x}))/p^{nd} ≠ lim ℓ(F^{n}((R/\underline{x})^{v}))/p^{nd}.

Part 2. If possible, let there be a system of parameters x_{1}, …, x_{d} such that

ℓ(R/\underline{x}) = lim ℓ(F^{n}(R/\underline{x}))/p^{nd} = lim ℓ(F^{n}((R/\underline{x})^{v}))/p^{nd}.

Then as in Part 1, we conclude that Ω is a height 1 ideal of R and

lim χ_{1}(R/\underline{x}^{pn}, F^{n}(Ω))/p^{nd} = 0.

If any x_{i} ∈ Ω, we arrive at a contradiction by Theorem 1. So we can assume that not a single x_{i} is in Ω. Now consider x_{1}^{t_{1}}, x_{2}^{t_{2}}, …, x_{d}^{t_{d}}, where t_{1}, …, t_{d} are positive integers. Then

χ_{1}(R/(x_{1}^{t_{1}p^{n}}, …, x_{d}^{t_{d}p^{n}}), F^{n}(Ω)) ≤ t_{1}t_{2}, ..., t_{d}χ_{1}(R/\underline{x}^{pn}, F^{n}(Ω)).

Hence

lim χ_{1}(R/(x_{1}^{t_{1}p^{n}}, …, x_{d}^{t_{d}p^{n}}), F^{n}(Ω))/p^{nd} = 0.

Thus x_{1}^{t}, …, x_{d}^{t} also satisfy the above equality, i.e.

ℓ(R/(x_{1}^{t_{1}}, …, x_{d}^{t_{d}})) = lim ℓ(F^{n}(R/(x_{1}^{t_{1}}, …, x_{d}^{t_{d}}))^{v})/p^{nd} (Step 1, Theorem 1).

Now, since ℓ(R/(Ω+\underline{x})) < ∞, we can reorganize the ideal (x_{1}, …, x_{d}) in such a way that x_{1}, …, x_{d−1} is a system of parameters on R/Ω (Theorem 124, [K]). Choose t so that x_{d}^{t} ∈ Ω + (x_{1}, …, x_{d−1}). Then, by our observation above, the system of parameters x_{1}, …, x_{d−1}, x_{d}^{t} also satisfies the following:

ℓ(R/(x_{1}, …, x_{d−1}, x_{d}^{t})) = lim ℓ(F^{n}(Ω/(x_{1}, …, x_{d−1}, x_{d}^{t})Ω))/p^{nd}.

But this contradicts the result proved in Part 1, since

(x_{1}, …, x_{d−1}, x_{d}^{t}) = (x_{1}, …, x_{d−1}, x′_{d}),

where x′_{d} ∈ Ω.

Remark. On a complete Cohen-Macaulay local ring R of dimension d, if M is a module of finite length and finite projective dimension, ℓ(M) may be different from ℓ(M̃), where M̃ = Ext^{d}(M, R). (See [R], for such an example.) Still, one can show, by using Theorem 1.5 of [D2], that

lim ℓ(F^{n}(M))/p^{nd} = lim ℓ(F^{n}(M̃))/p^{nd}.

Actually, a more general statement is valid: for any module M of finite length

lim ℓ(Ext^{d}(F^{n}(M), R))/p^{nd} = lim ℓ(F^{n}(M))/p^{nd}.

References

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Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801

   E-mail address: dutta@math.uiuc.edu