D-module structure of local cohomology modules of toric algebras

TL;DR

This paper investigates the structure of local cohomology modules over toric algebras, demonstrating their finite length over differential operators and computing their characteristic cycles, thus extending classical polynomial algebra results.

Contribution

It establishes the finite length property of local cohomology modules over D(S;K) for toric algebras, generalizing known results from polynomial rings.

Findings

Local cohomology modules have finite length over D(S;K).

Characteristic cycles of certain local cohomology modules are explicitly computed.

Generalizes classical polynomial algebra results to toric algebras.

Abstract

Let S be a toric algebra over a field K of characteristic 0 and let I be a monomial ideal of S. We show that the local cohomology modules H^i_I(S) are of finite length over the ring of differential operators D(S;K), generalizing the classical case of a polynomial algebra S. As an application, we compute the characteristic cycles of some local cohomology modules.