When does the <span class="mathjax-formula">$F$</span>-signature exist?

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

IANM. ABERBACH& FLORIANENESCU

When does theF-signature exist?

Tome XV, n^o2 (2006), p. 195-201.

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pp. 195–201

When does the F-signature exists?

Ian M. Aberbach⁽¹⁾, Florian Enescu⁽²⁾

ABSTRACT.— We show that theF-signature of anF-finite local ringRof characteristicp >0 exists whenRis either the localization of anN-graded ring at its irrelevant ideal orQ-Gorenstein on its punctured spectrum.

This extends results by Huneke, Leuschke, Yao and Singh and proves the existence of theF-signature in the cases where weakF-regularity is known to be equivalent to strongF-regularity.

R^´ESUM´E.— Nous prouvons dans cet article l’existence de laF-signature d’un anneau localF-finiR, de caract´eristique positivep, quandRest la localisation `a l’unique id´eal homog`ene maximal d’un anneau N-gradu´e ou quandRestQ-Gorenstein sur son spectre ´epoint´e. Ceci g´en´eralise les r´esultats de Huneke, Leuschke, Yao et Singh et prouve l’existence de la F-signature dans les cas o`u faible et forteF-r´egularit´e sont ´equivalentes.

1. A sufficient condition for the existence of the F-signature Let (R,m, k) be a reduced, localF-finite ring of positive characteristic p >0 and Krull dimensiond. Let

R^1/q=R^aq⊕Mq

be a direct sum decomposition ofR^1/qsuch thatMq has no free direct sum- mands. IfRis complete, such a decomposition is unique up to isomorphism.

Recent research has focused on the asymptotic growth rate of the numbers aq asq→ ∞. In particular, theF-signature (defined below) is studied in [7]

and [3], and more generally the Frobenius splitting ratio is studied in [2].

(∗) Re¸cu le 7 juin 2004, accept´e le 15 d´ecembre 2004

(1) Department of Mathematics, University of Missouri, Columbia, MO 65211.

E-mail: aberbach@math.missouri.edu

(2) Department of Mathematics and Statistics, Georgia State University, Atlanta, 30303 and The Institute of Mathematics of the Romanian Academy (Romania).

E-mail: fenescu@mathstat.gsu.edu

Aberbachwas partially supported by a grant from the NSA.

Ian M. Aberbach, Florian Enescu

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 anm-primary idealI ofR,λ(R^1/q/IR^1/q) =λ(R/I^[q])/q^α(R), whereλ(−) represents the length function over R.

We would like to first define the notion ofF-signature as it appears in [3]

and [7].

Definition 1.1. — TheF-signature ofRiss(R) = lim_q→∞ a_q q^d+α(R), if it exists.

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 ifR is stronglyF-regular.

The question of whether or not, in a stronglyF-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 strongF-regularity are equivalent.

Smith and Van den Bergh ([10]) have shown that theF-signature of R exists whenR 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 ofR^1/qfor anyq=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 punc- tured spectrum, [12]. Singh has also shown that theF-signature exists for monomial rings, [9].

Let (R,m) be an approximately Gorenstein ring. This means thatRhas a sequence ofm-primary irreducible ideals{I_t}tcofinal with the powers ofm.

By taking a subsequence, we may assume thatI_t⊃I_t+1. For eacht, letu_tbe an element ofRwhich represents a socle element moduloIt. Then there is, for eacht, a homomorphismR/It→R/It+1such thatut+It→ut+1+It+1. The direct limit of the system will be the injective hullE=ER(R/m) and each ut 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 everyq, there existst₀(q), such that

aq/(q^d+α(R)) =λ(R/(I_t^[q]:u^q_t))/q^d, for alltt₀(q) (see [3], p. 55).

The situation whent0(q) can be chosen independently ofqis of special interest.

Definition 1.3. — We say thatRsatisfies Condition(A), if there exist a sequence of irreducible m-primary ideals {It} and a t0 such that, for all tt0 and allq

(I_t^[q]:u^q_t) = (I_t^[q]₀ :u^q_t0).

Proposition 1.4. — Let(R,m, k) be a local reducedF-finite ring. IfR satisfies ConditionA, then theF-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, ConditionA implies that there existst0, indepen- dent ofq, such that

aq/(q^d+α(R)) =λ(R/(I_t^[q]₀ :u^q_t0))/q^d, for allq.

But λ(R/(I_t^[q]₀ : u^q_t0)) = λ(R/I_t^[q]₀)−λ(R/(I_t0 +u_t0R)^[q]). Dividing by q^d and taking the limit asq → ∞yields s(R) = e_HK(I_t0, R)−e_HK(I_t0 + u_t0R, 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 theR-bialgebra whose underlying abelian group equalsRand the left and rightR-multiplication is given bya·r∗b=arb^q, fora, b∈R, r∈R^(e). Let k = Ru → E be the natural inclusion and consider the natural induced mapφe:R^(e)⊗RE→R^(e)⊗R(E/k). Thenaq/q^α(R)=λ(ker(φe)) (by Aberbach-Enescu, Corollary 2.8 in [2], see also Yao’s work [12]).

Ian M. Aberbach, Florian Enescu

One can in fact see that

λ(ker(φe)) =λ(R/(c∈R:c⊗u= 0 inR^(e)⊗RE)) =λ(R/∪t(I_t^[q] :u^q_t)).

Definition 1.5. — We say thatRsatisfies Condition(B)if there exists a finite length submoduleE ⊂E such that, if

ψ_e:R^(e)⊗RE→R^(e)⊗RE/k, then λ(ker(φ_e)) =λ(ker(ψ_e)), for alle.

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

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

Proof. — Assume that Condition (A) holds. Then one can take E=R/It0 and then Condition (B) follows.

If Condition (B) holds, then take t₀ large enough such that E⊂Im(R/I_t0 →E).

As noted above, one can compute the length of the kernel ofψ_e as the colength of{c∈R:c⊗u= 0 inR^(e)⊗RE}. SinceR/It0 injects intoEwe see that {c∈R :c⊗u= 0 inR^(e)⊗RE} is a subset of{c∈R: c⊗u= 0 inR^(e)⊗RR/It0}= (I_t^[q]₀ :u^q_t0).

Since (I_t^[q]₀ : u^q_t0)⊂(I_t^[q] :u^q_t) for all t t0, we see that Condition (B) implies that (I_t^[q]₀ :u^q_t0) = (I_t^[q] :u^q_t) for all t t0, which is Condition (A).

2. N-Graded rings

Let (R,m) be a NoetherianN-graded ringR=⊕n0R_n, whereR₀=k is anF-finite field of characteristicp >0.

For any graded R-module M one can define a natural grading on R^(e)⊗M: the degree of any tensor monomialr⊗mequals deg(r)+qdeg(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 thans, then the induced map

R^(e)⊗M →R^(e)⊗N is bijective in degrees greater thanp^e(s+t).

LetE be the injective hull of Rm. In fact,E is also the injective hull of R/m overR and as a result is naturally graded with socle in degree 0. We can writeE=⊕n0E_n.

Let t be as in Lemma 2.1, and let s −t−1. Obviously the map E =⊕sn0E_n →E =⊕n0E_n is bijective in degrees greater than s. So by Lemma 2.1, the mapR^(e)⊗E→R^(e)⊗Eis bijective in degrees greater thanp^e(s+t)−p^e.

Theorem 2.2. — Let R be an N-graded reduced ring over an F-finite field kof positive characteristic. Then Condition (B)is satisfied byR and hence theF-signature ofR exists.

Proof. — Let E be the injective hull of k = R/m over Rm. As above, E=⊕n0En, where 0 is the degree of the socle generatoruofE.

Using the notation introduced above, we will let s = −t −1 and E = ⊕snn0E_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 withR^(e), we get the exact sequences R^(e)⊗k=R^(e)⊗Ru→R^(e)⊗E→^φe R^(e)⊗E/k→0 and

R^(e)⊗k=R^(e)⊗Ru→R^(e)⊗E→^ψeR^(e)⊗E/k→0.

Ian M. Aberbach, Florian Enescu

One can easily see that ker(φ_e) and ker(ψ_e) are the submodules gener- ated by 1⊗uin R^(e)⊗E andR^(e)⊗E, respectively.

The degree of 1⊗uisq·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(φe)ker(ψe) and hence Condition (B) is satisfied.

3. Q-Gorenstein rings

We turn now to showing that Condition (A) holds in stronglyF-regular local rings which areQ-Gorenstein on the punctured spectrum. Let (R,m, k) be such a ring of dimensiond, 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 elementa ∈ J which generatesJ at all minimal primes ofJ, and then an elementx2∈mwhich is a parameter on R/J such that x2J ⊆ aR. It is easy to see that then xⁿJ⁽ⁿ⁾⊆aⁿRfor alln1 (whereJ⁽ⁿ⁾is the height one component ofJⁿ).

The condition that R is Q-Gorenstein on the punctured spectrum implies that there is an integerhand two sequences of elementsx3, . . . , xd∈mand a3, . . . , ad ∈J^(h)such that xiJ^(h)⊆aiR for 3id, andx2, . . . , xd is a s.o.p. onR/J. We may then pickx₁∈J such thatx₁, . . . , x_dis 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 anyN0 and anyn0,

(J^(nh), x^N₂, . . . ,x^N_i , . . . , x^N_d) :x^∞_i = (J^(nh), x^N₂, . . . ,x^N_i , . . . , x^N_d) :xⁿ_i. (3.1)

Theorem 3.1. — Let (R,m, k) be an F-finite strongly F-regular ring which isQ-Gorenstein on the punctured spectrum. ThenR satisfies Condi- tion(A). In particular theF-signature ofR exists.

Proof. — If R is not complete, we observe that, since R is excellent, R is stronglyF-regular andQ-Gorenstein on the punctured spectrum. If{It} is a sequence of ideal inR showing condition (A) inR, then {I_t∩R}does so forR. Thus we will assume thatRis complete.

LetJ,h, andx1, . . . , xdbe as discussed above. LetIt= (x^t−1₁ J, x^t₂, . . . , x^t_d).

Since xⁿ₁J ∼= J as R-modules, the quotient R/xⁿ₁J is Gorenstein. The hypothesis that x2, . . . , xd are parameters on R/J and R/x1R (hence on R/xⁿ₁J) then shows that It is irreducible (see [4], Proposition 3.3.18). The sequence{It}is then a sequence ofm-primary irreducible ideals cofinal with

the powers ofm. Ifu₁represents the socle element ofI₁, then we may take u_t= (x₁· · ·x_d)^t−1u₁to represent the socle element ofI_t. We will show that t₀may be taken to be 3.

Suppose thatc ∈ I_t^[q] : u^q_t for some q. We will show that c ∈I₃^[q] : u^q₃. Raising to theqth power we havec^qu^qq_t =c^q

(x1· · ·x_d)^t−1u1

qq

∈I_t^[qq]

= (x^t₁⁻¹J, x^t₂, . . . , x^t_d)^[qq]. Hencec^q

(x₂· · ·x_d)^t−1u₁qq

∈(x^t₂, . . . , x^t_d)^[qq] : x^(t₁^−1)qq+ (J, x^t₂, . . . , x^t_d)^[qq]= (J, x^t₂, . . . , x^t_d)^[qq].

Writeqq =n_qh+r_qwith 0r_q < h. Repeated application of equation 3.1 (using 1 rather thanhforx₂) gives

c^q((x₂· · ·x_d)u₁)^qq ∈(J^(nqh), x^2qq₂ , . . . , x^2qq_d ). (3.2) Let d ∈ J^(h) ⊆ J^(rq). Multiplying by x^qq₂ and using that x^qq₂J^(qq) ⊆ a^qqR ⊆ J^[qq] we have dc^q

(x₂· · ·x_d)²u₁qq

∈ (J, x³₂, . . . , x³_d)^[qq]. Multi- plying by x^2qq₁ shows that dc^qu^qq₃ = d(cu^q₃)^q ∈ (I₃^[q])^[q]. Thus cu^q₃ ∈ (I₃^[q])^∗=I₃^[q], as desired.

Bibliography

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[2] Aberbach (I. M.), Enescu (F.). — The structure ofF-pure rings, Math. Zeit., to appear.

[3] Aberbach (I. M.), Leuschke (G.). — TheF-signature and strongF-regularity, Math. Res. Lett. 10, 51-56 (2003).

[4] Bruns (W.), Herzog (J.). — Cohen-Macaulay Rings, Cambridge University Press, Cambridge (1993).

[5] Hochster (M.). — Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231, no. 2, 463-488 (1977).

[6] Hochster (M.) and Huneke (C.). — Tight closure and strong F-regularity, Me ´moire no. 38, Soc. Math. France, 119-133 (1989).

[7] Huneke (C.), Leuschke (G.). — Two theorems about maximal Cohen-Macaulay modules, Math. Ann. 324, 391-404 (2002).

[8] Lyubeznik (G.), Smith (K.E.). — Strong and weakF-regularity are equivalent for graded rings, Amer. J. Math. 121, 1279-1290 (1999).

[9] Singh (A.K.). — TheF-signature of an affine semigroup ring, J. Pure Appl. Al- gebra, 196, 313-321 (2005).

[10] Smith (K.E.), Van den Bergh (M.). — Simplicity of rings of differential opera- tors in prime characteristic, Proc. London. Math. Soc. (3) 75, no. 1, 32-62 (1997).

[11] Yao (Y.). — Modules withfiniteF-representation type, Jour. London Math. Soc., to appear.

[12] —, Observations on theF-signature of local rings of characteristicp >0, preprint 2003.