[Paper Review] D-modules over rings with finite F-representation type

Shunsuke Takagi, Ryo Takahashi|ArXiv.org|Jun 26, 2007

Commutative Algebra and Its Applications25 references3 citations

TL;DR

This paper establishes two finiteness properties for D-modules over Noetherian graded rings with finite F-representation type in positive characteristic. First, it proves that localization at any non-zerodivisor is generated by 1/x as a D-module, generalizing a result for polynomial rings. Second, for Gorenstein rings with finite F-representation type, local cohomology modules have only finitely many associated primes, extending results for regular rings. The paper also proves the discreteness of F-jumping exponents for homogeneous ideals in such rings.

ABSTRACT

Smith and Van den Bergh introduced the notion of finite F-representation type as a characteristic $p$ analogue of the notion of finite representation type. In this paper, we prove two finiteness properties of rings with finite F-representation type. The first property states that if $R=\bigoplus_{n \ge 0}R_n$ is a Noetherian graded ring with finite (graded) F-representation type, then for every non-zerodivisor $x \in R$, $R_x$ is generated by $1/x$ as a $D_{R}$-module. The second one states that if $R$ is a Gorenstein ring with finite F-representation type, then $H_I^n(R)$ has only finitely many associated primes for any ideal $I$ of $R$ and any integer $n$. We also include a result on the discreteness of F-jumping exponents of ideals of rings with finite (graded) F-representation type as an appendix.

Motivation & Objective

• To extend finiteness results for D-modules to rings with finite F-representation type, a characteristic p analogue of finite representation type.
• To address the long-standing open problem of whether local cohomology modules have finitely many associated primes, under the assumption of finite F-representation type.
• To establish the discreteness of F-jumping exponents for ideals in strongly F-regular graded rings with finite F-representation type.
• To generalize known results on differential operators and test ideals in positive characteristic to a broader class of singular rings.

Proposed method

• Use of Frobenius pullbacks and the structure of D-modules over graded rings with finite F-representation type.
• Application of generalized test ideals and their compatibility with Frobenius powers to control the generation of D-modules.
• Leveraging the finite generation of graded components in the category of R-modules under Frobenius twist to bound degrees of generators.
• Employing descending chain arguments on finite-dimensional graded subspaces to prove stabilization of generalized test ideals.
• Reduction of the problem to finite-dimensional vector spaces over the base field to exploit finite generation and stabilization.
• Use of the fact that for large e, the generalized test ideal τ(𝔞^t) coincides with I_e(𝔞^{⌈tp^e⌉}, R), enabling control over the degree of generators.

Experimental results

Research questions

• RQ1 Does the localization R_x of a graded ring R with finite F-representation type admit a finite D_R/k-generating set, and if so, is 1/x sufficient?
• RQ2 Under what conditions does the local cohomology module H^n_I(R) have only finitely many associated primes when R has finite F-representation type?
• RQ3 Are the F-jumping exponents of a homogeneous ideal in a strongly F-regular graded ring with finite F-representation type discrete?
• RQ4 Can the rationality of F-jumping exponents be deduced from the discreteness and finite F-representation type under additional assumptions?

Key findings

• For any non-zerodivisor x in a Noetherian graded ring R with finite (graded) F-representation type, R_x is generated by 1/x as a D_R/k-module.
• If R is a Gorenstein ring with finite F-representation type, then H^n_I(R) has only finitely many associated primes for any ideal I and any integer n.
• The set of F-jumping exponents of a homogeneous ideal 𝔞 in a strongly F-regular graded ring R with finite (graded) F-representation type has no accumulation points.
• Generalized test ideals τ(𝔞^t) are generated by elements of bounded degree, depending only on R and the degree of 𝔞, for sufficiently large Frobenius powers.
• The descending chain of generalized test ideals τ(𝔞^{α_m}) stabilizes when restricted to a fixed degree, implying discreteness of the F-jumping exponents.
• The result on discreteness holds under the assumption that R has finite graded F-representation type, extending prior results in the regular case.