When does the F-signature exist?^{∗}

Ian M. Aberbach^{†}  Florian Enescu^{‡}

Abstract. We show that the F -signature of an F -finite local ring R of characteristic p > 0 exists when R is either the localization of an N-graded ring at its irrelevant ideal or Q-Gorenstein on its punctured spectrum. This extends results by Huneke, Leuschke, Yao and Singh and proves the existence of the F -signature in the cases where weak F -regularity is known to be equivalent to strong F -regularity.

Résumé. Nous prouvons dans cet article l’existence de la F-signature d’un anneau local F-fini R, de caractéristique positive p, quand R est la localisation à l’unique idéal homogène maximal d’un anneau N-gradué ou quand R est Q-Gorenstein sur son spectre épointé. Ceci généralise les résultats de Huneke, Leuschke, Yao et Singh et prouve l’existence de la Fsignature dans les cas où faible et forte F-régularité sont équivalentes.

1 A sufficient condition for the existence of the Fsignature

Let (R, 𝔪, k) be a reduced, local F -finite ring of positive characteristic p > 0 and Krull dimension d. Let

R^{1/q} = R^{a_{q}} ⊕ M_{q}

be a direct sum decomposition of R^{1/q} such that M_{q} has no free direct summands. If R is complete, such a decomposition is unique up to isomorphism. Recent research has focused on the asymptotic growth rate of the numbers a_{q} as q → ∞. In particular, the F -signature (defined below) is studied in [7] and [3], and more generally the Frobenius splitting ratio is studied in [2].

For a local ring (R, 𝔪, k) , we set α(R) = log_{p}[k_{R} : k_{R}^{p}]. It is easy to see that, for an 𝔪-primary ideal I of R, λ(R^{1/q}/IR^{1/q}) = λ(R/I^{[q]})/q^{α(R)}, where λ(−) represents the length function over R.

We would like to first define the notion of F -signature as it appears in [3] and [7].

Definition 1.1. The F -signature of R is s(R) = lim_{q→∞} \frac{a_{q}}{q^{d+α(R)}} , if it exists.

∗2000 Mathematics Subject Classification: 13A35. The first author was partially supported by a grant from the NSA.

†Department of Mathematics, University of Missouri, Columbia, MO 65211; aberbach@math.missouri.edu

‡Department of Mathematics and Statistics, Georgia State University, Atlanta, 30303 and The Institute of Mathematics of the Romanian Academy, Romania; fenescu@mathstat.gsu.edu

The following result, due to Aberbach and Leuschke [3], holds:

Theorem 1.2. Let (R, 𝔪, k) be a reduced Noetherian ring of positive characteristic p. Then lim inf_{q→∞} a_{q}/q^{d+α(R)} > 0 if and only if lim sup_{q→∞} a_{q}/q^{d+α(R)} > 0 if and only if R is strongly F -regular.

The question of whether or not, in a strongly F -regular ring, s(R) exists, is open. We show in this paper that its existence is closely connected to the question of whether or not weak and strong F -regularity are equivalent.

Smith and Van den Bergh ([10]) have shown that the F -signature of R exists when R has finite Frobenius representation type (FFRT) type, that is, if only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules occur as direct summands of R^{1/q} for any q = p^{e}. Yao has proven that, under mild conditions, tight closure commutes with localization in a ring of FFRT type, [11]. Moreover, Huneke and Leuschke proved that if R is also Gorenstein, then the F -signature exists, [7]. Yao has recently extended this result to rings that are Gorenstein on their punctured spectrum, [12]. Singh has also shown that the F -signature exists for monomial rings, [9].

Let (R, 𝔪) be an approximately Gorenstein ring. This means that R has a sequence of 𝔪-primary irreducible ideals {I_{t}}_{t} cofinal with the powers of 𝔪. By taking a subsequence, we may assume that I_{t} ⊃ I_{t+1}. For each t, let u_{t} be an element of R which represents a socle element modulo I_{t}. Then there is, for each t, a homomorphism R/I_{t} ↪ R/I_{t+1} such that u_{t} + I_{t} ↦ u_{t+1} + I_{t+1}. The direct limit of the system will be the injective hull E = E_{R}(R/𝔪) and each u_{t} will map to the socle element of E, which we will denote by u. Hochster has shown that every excellent, reduced local ring is approximately Gorenstein ([5]).

Aberbach and Leuschke have shown that, for every q, there exists t_{0}(q), such that

a_{q}/(q^{d+α(R)}) = λ(R/(I_{t}^{[q]} : u_{t}^{q}))/q^{d},

for all t ≥ t_{0}(q) (see [3], p. 55).

The situation when t_{0}(q) can be chosen independently of q is of special interest.

Definition 1.3. We say that R satisfies Condition (A), if there exist a sequence of irreducible 𝔪-primary ideals {I_{t}} and a t_{0} such that, for all t ≥ t_{0} and all q

(I_{t}^{[q]} : u_{t}^{q}) = (I_{t_{0}}^{[q]} : u_{t_{0}}^{q}).

Proposition 1.4. Let (R, 𝔪, k) be a local reduced F -finite ring. If R satisfies Condition A, then the F -signature exists.

Proof. We know that R is approximately Gorenstein and hence we will use the notation fixed in the paragraph above.

As explained above, Condition A implies that there exists t_{0}, independent of q, such that

a_{q}/(q^{d+α(R)}) = λ(R/(I_{t_{0}}^{[q]} : u_{t_{0}}^{q}))/q^{d},

for all q.

But λ(R/(I_{t_{0}}^{[q]} : u_{t_{0}}^{q})) = λ(R/I_{t_{0}}^{[q]}) − λ(R/(I_{t_{0}} + u_{t_{0}}R)^{[q]}). Dividing by q^{d} and taking the limit as q → ∞ yields s(R) = e_{HK}(I_{t_{0}}, R) − e_{HK}(I_{t_{0}} + u_{t_{0}}R, R).

Now we would like to concentrate on another condition, Condition (B), that appeared first in the work of Yao. First we need to introduce some notation.

Assume that E is the injective hull of the residue field k. By R^{(e)} we denote the Rbialgebra whose underlying abelian group equals R and the left and right R-multiplication is given by a · r ∗ b = arb^{q}, for a, b ∈ R, r ∈ R^{(e)}.

Let k = Ru → E be the natural inclusion and consider the natural induced map ϕ_{e} : R^{(e)} ⊗_{R} E → R^{(e)} ⊗_{R} (E/k). Then a_{q}/q^{α(R)} = λ(ker(ϕ_{e})) (by Aberbach-Enescu, Corollary 2.8 in [2], see also Yao’s work [12]).

One can in fact see that

λ(ker(ϕ_{e})) = λ(R/(c ∈ R : c ⊗ u = 0 in R^{(e)} ⊗_{R} E)) = λ(R/ ∪_{t} (I_{t}^{[q]} : u_{t}^{q})).

Definition 1.5. We say that R satisfies Condition (B) if there exists a finite length submodule E′ ⊂ E such that, if ψ_{e} : R^{(e)} ⊗_{R} E′ → R^{(e)} ⊗_{R} E′/k, then λ(ker(ϕ_{e})) = λ(ker(ψ_{e})), for all e.

Yao [12] has shown that Condition (B) implies that the F -signature of R exists.

Proposition 1.6. Let (R, 𝔪, k) be a local reduced F -finite ring. Then Conditions (A) and (B) are equivalent.

Proof. Assume that Condition (A) holds. Then one can take E′ = R/I_{t_{0}} and then Condition (B) follows.

If Condition (B) holds, then take t_{0} large enough such that E′ ⊂ Im(R/I_{t_{0}} → E).

As noted above, one can compute the length of the kernel of ψ_{e} as the colength of {c ∈ R : c ⊗ u = 0 in R^{(e)} ⊗_{R} E′}. Since R/I_{t_{0}} injects into E we see that {c ∈ R : c ⊗ u = 0 in R^{(e)} ⊗_{R} E′} is a subset of {c ∈ R : c ⊗ u = 0 in R^{(e)} ⊗_{R} R/I_{t_{0}}} = (I_{t_{0}}^{[q]} : u_{t_{0}}^{q}).

Since (I_{t_{0}}^{[q]} : u_{t_{0}}^{q}) ⊂ (I_{t}^{[q]} : u_{t}^{q}) for all t ≥ t_{0}, we see that Condition (B) implies that (I_{t_{0}}^{[q]} : u_{t_{0}}^{q}) = (I_{t}^{[q]} : u_{t}^{q}) for all t ≥ t_{0}, which is Condition (A).

2 N-Graded Rings

Let (R, 𝔪) be a Noetherian N-graded ring R = ⊕_{n≥0}R_{n}, where R_{0} = k is an F -finite field of characteristic p > 0.

For any graded R-module M one can define a natural grading on R^{(e)} ⊗ M: the degree of any tensor monomial r ⊗ m equals deg(r) + q deg(m).

In what follows we will need the following important Lemma by Lyubeznik and Smith ([8], Theorem 3.2):

Lemma 2.1. Let R be an N-graded ring and M, N two graded R-modules. Then there exists an integer t depending only on R such that whenever

M → N is a degree preserving map which is bijective in degrees greater than s, then the induced map

R^{(e)} ⊗ M → R^{(e)} ⊗ N

is bijective in degrees greater than p^{e}(s + t).

Let E be the injective hull of R_{𝔪}. In fact, E is also the injective hull of R/𝔪 over R and as a result is naturally graded with socle in degree 0. We can write E = ⊕_{n≤0}E_{n}.

Let t be as in the Lemma 2.1, and let s ≤ −t − 1. Obviously the map E′ = ⊕_{s≤n≤0}E_{n} → E = ⊕_{n≤0}E_{n} is bijective in degrees greater than s. So by Lemma 2.1, the map R^{(e)} ⊗ E′ → R^{(e)} ⊗ E is bijective in degrees greater than p^{e}(s + t) ≤ −p^{e}.

Theorem 2.2. Let R be an N-graded reduced ring over an F -finite field k of positive characteristic. Then Condition (B) is satisfied by R and hence the F -signature of R exists.

Proof. Let E be the injective hull of k = R/𝔪 over R_{𝔪}. As above, E = ⊕_{n≤0}E_{n}, where 0 is the degree of the socle generator u of E.

Using the notation introduced above, we will let s = −t − 1 and E′ = ⊕_{s≤n≤n_{0}}E_{n} → E. So, R^{(e)} ⊗ E′ → R^{(e)} ⊗ E is bijective in degrees greater than −p^{e}. In particular it is bijective in degrees greater or equal to 0.

We have the following exact sequences:

0 → k = Ru → E → E/k → 0

and

0 → k = Ru → E′ → E′/k → 0.

After tensoring with R^{(e)}, we get the exact sequences

R^{(e)} ⊗ k = R^{(e)} ⊗ Ru → R^{(e)} ⊗ E →^{ϕ_{e}} R^{(e)} ⊗ E/k → 0

and

R^{(e)} ⊗ k = R^{(e)} ⊗ Ru → R^{(e)} ⊗ E′ →^{ψ_{e}} R^{(e)} ⊗ E′/k → 0.

One can easily see that ker(ϕ_{e}) and ker(ψ_{e}) are the submodules generated by 1 ⊗ u in R^{(e)} ⊗ E and R^{(e)} ⊗ E′, respectively.

The degree of 1⊗u is q·0 = 0 and we have noted that the natural map R^{(e)}⊗E′ → R^{(e)}⊗E is bijective in degrees greater than −p^{e}. This shows that ker(ϕ_{e}) ≃ ker(ψ_{e}) and hence Condition (B) is satisfied.

3 ℚ-Gorenstein Rings

We turn now to showing that Condition (A) holds in strongly F -regular local rings which are ℚ-Gorenstein on the punctured spectrum. Let (R, 𝔪, k) be such a ring of dimension d, and assume that R has a canonical module (e.g. R is complete). In this case R has an unmixed ideal of height 1, say J ⊆ R, which is a canonical ideal. We may pick an element a ∈ J which generates J at all minimal primes of J, and then an element x_{2} ∈ 𝔪 which is a parameter on R/J such that x_{2}J ⊆ aR. It is easy to see that then x^{n}J^{(n)} ⊆ a^{n}R for all n ≥ 1 (where J^{(n)} is the height one component of J^{n}). The condition that R is ℚ-Gorenstein on the punctured spectrum implies that there is an integer h and two sequences of elements x_{3}, …, x_{d} ∈ 𝔪 and a_{3}, …, a_{d} ∈ J^{(h)} such that x_{i}J^{(h)} ⊆ a_{i}R for 3 ≤ i ≤ d, and x_{2}, …, x_{d} is a s.o.p. on R/J. We may then pick x_{1} ∈ J such that x_{1}, …, x_{d} is an s.o.p. for R. See [1], section 2.2 for more detail. Then by [1], Lemma 2.2.3 we have that for any N ≥ 0 and any n ≥ 0,

(J^{(nh)}, x_{2}^{N}, …, x̂_{i}^{N}, …, x_{d}^{N}) : x_{i}^{∞} = (J^{(nh)}, x_{2}^{N}, …, x̂_{i}^{N}, …, x_{d}^{N}) : x_{i}^{n}.  (3.1)

Theorem 3.1. Let (R, 𝔪, k) be an F -finite strongly F -regular ring which is ℚ-Gorenstein on the punctured spectrum. Then R satisfies Condition (A). In particular the F -signature of R exists.

Proof. If R is not complete, we observe that, since R is excellent, R̂ is strongly F -regular and ℚ-Gorenstein on the punctured spectrum. If {I_{t}} is a sequence of ideal in R̂ showing condition (A) in R̂, then {I_{t} ∩ R} does so for R. Thus we will assume that R is complete.

Let J, h, and x_{1}, …, x_{d} be as discussed above. Let I_{t} = (x_{1}^{t−1}J, x_{2}^{t}, …, x_{d}^{t}). Since x_{1}^{n}J ≅ J as R-modules, the quotient R/x_{1}^{n}J is Gorenstein. The hypothesis that x_{2}, …, x_{d} are parameters on R/J and R/x_{1}R (hence on R/x_{1}^{n}J) then shows that I_{t} is irreducible (see [4], Proposition 3.3.18). The sequence {I_{t}} is then a sequence of 𝔪-primary irreducible ideals cofinal with the powers of 𝔪. If u_{1} represents the socle element of I_{1}, then we may take u_{t} = (x_{1} ⋯ x_{d})^{t−1}u_{1} to represent the socle element of I_{t}. We will show that t_{0} may be taken to be 3.

Suppose that c ∈ I_{t}^{[q]} : u_{t}^{q} for some q. We will show that c ∈ I_{3}^{[q]} : u_{3}^{q}. Raising to the q′th power we have c^{q′}u_{t}^{qq′} = c^{q′} ((x_{1} ⋯ x_{d})^{t−1}u_{1})^{qq′} ∈ I_{t}^{[qq′]} = (x_{1}^{t−1}J, x_{2}^{t}, …, x_{d}^{t})^{[qq′]}. Hence c^{q′} ((x_{2} ⋯ x_{d})^{t−1}u_{1})^{qq′} ∈ (x_{2}^{t}, …, x_{d}^{t})^{[qq′]} : x_{1}^{(t−1)qq′} + (J, x_{2}^{t}, …, x_{d}^{t})^{[qq′]} = (J, x_{2}^{t}, …, x_{d}^{t})^{[qq′]}.

Write qq′ = n_{q′}h + r_{q′} with 0 ≤ r_{q′} < h. Repeated application of equation 3.1 (using 1 rather than h for x_{2}) gives

c^{q′}((x_{2} ⋯ x_{d})u_{1})^{qq′} ∈ (J^{(n_{q′}h)}, x_{2}^{2qq′}, …, x_{d}^{2qq′}).  (3.2)

Let d ∈ J^{(h)} ⊆ J^{(r_{q′})}. Multiplying by x_{2}^{qq′} and using that x_{2}^{qq′}J^{(qq′)} ⊆ a^{qq′}R ⊆ J^{[qq′]} we have dc^{q′} ((x_{2} ⋯ x_{d})^{2}u_{1})^{qq′} ∈ (J, x_{2}^{3}, …, x_{d}^{3})^{[qq′]}. Multiplying by x_{1}^{2qq′} shows that dc^{q′}u_{3}^{qq′} = d(cu_{3}^{q})^{q′} ∈ (I_{3}^{[q]})^{[q′]}. Thus cu_{3}^{q} ∈ (I_{3}^{[q]})^{∗} = I_{3}^{[q]}, as desired.

References

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[11] Y. Yao, Modules with finite F -representation type, preprint 2002.

[12] —, Observations on the F -signature of local rings of characteristic p > 0, preprint 2003.