Length of Local Cohomology in Positive Characteristic and Ordinarity

Abstract

Let D be the ring of Grothendieck differential operators of the ring R of polynomials in d ě 3 variables with coefficients in a perfect field of characteristic p. We compute the D-module length of the first local cohomology module H 1 f pRq with respect to a polynomial f with an isolated singularity, for p large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the exceptional fibre of a resolution of the singularity. Our proof rests on a tight closure computation of Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero Dmodule whose length is expected to equal that above, for ordinary primes.

Local cohomology -- an invitation

“…In general, it is a difficult problem to calculate the length ℓ D pH j I pRqq, or even just ℓ D pR f q. Some results are in [Tor09] and [Bit20]. The following upper bounds were obtained in [KMSZ18].…”

“…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 , .…”

On -modules related to the -function and Hamiltonian flow

“…Let be the ring of -polynomials in variables, be the ring of Grothendieck differential operators on the affine space and be the reduction of modulo . We have the following result of the first author, see [Bit18].…”