$D$-module and $F$-module length of local cohomology modules

Abstract

Abstract. Let R be a polynomial or power series ring over a field k. We study the length of local cohomology modules H j I (R) in the category of D-modules and Fmodules. We show that the D-module length of H j I (R) is bounded by a polynomial in the degree of the generators of I. In characteristic p > 0 we obtain upper and lower bounds on the F -module length in terms of the dimensions of Frobenius stable parts and the number of special primes of local cohomology modules of R/I. The obtained upper bound is sharp if R/I is an isolated singularity, and the lower bound is sharp when R/I is Gorenstein and F -pure. We also give an example of a local cohomology module that has different D-module and F -module lengths.

Local cohomology -- an invitation

“…It turns out that F -module length of local cohomology modules is closely related to singularities defined by the Frobenius, and Lyubeznik's functor H R,A is a useful tool for studying this length. To illustrate this, let R be a regular local ring of characteristic p. That H R,A sets up a link between the length of H htpIq I pRq and the singularities of A " R{I was first discovered in [Bli04]; this was later extended and strengthened in [KMSZ18] as follows, see also [Bit20] Theorem 3.51. Let R " krrx 1 , .…”

A prime-characteristic analogue of a theorem of Hartshorne-Polini

“…Theorem 1.2 calculates these D-module lengths when char(k) = 0 and X is smooth. In positive characteristic, Katzman, Ma, Smirnov, and the second author's [10,Theorem 4.3] gives an explicit formula for the F-module length of H c I (R) (where c is the codimension of X in P n k ) under the assumption that R/I has an isolated non-F -rational point at the origin. Our Theorem 4.1 gives some new information about the F-module length of H i I (R) in the case i = c.…”

“…This does not fully recover the prime-characteristic analogue of Theorem 1.2 in the case when i = c = n − dim(X); the length differs by one from the desired analogous result. The reason is that the simple F-submodule of H c I (R) does not admit any non-zero F-module morphism to E. However, it follows directly from [10,Theorem 4.3] that H c I (R) admits a simple F-submodule H 0 such that H c I (R)/H 0 ∼ = E λ d . This completes our analogue of the theorem of Hartshorne-Polini.…”

Strong F-regularity and generating morphisms of local cohomology modules

“…We also invoke the following observation: Next we furnish a result that will be extremely useful in the sequel, in particular for our Theorem 4.6 (where we furnish a criterion of strong F -regularity). Further details, even in more generality, can be found in [1, Subsection 3.4.2], [21,Subsection 5.3] and [25, Section 4], but we supply a proof herein for the reader's convenience.…”