Bigraded components of F-finite F-modules

TL;DR

This paper investigates the structure of bigraded components of F-finite F-modules over regular rings in characteristic p, revealing properties like vanishing and rigidity, with applications to local cohomology modules.

Contribution

It applies Lyubeznik's theory to analyze bigraded components of F-modules, providing new insights into their algebraic properties and implications for local cohomology.

Findings

Bigraded components exhibit vanishing and rigidity properties.

Bass numbers and associated primes are characterized.

Vanishing of certain local cohomology components under specific conditions.

Abstract

Let $A$ be a regular ring containing a field of characteristic $p>0$ and let $R=A[x_1,\ldots,x_m,y_1,\ldots,y_n]$ be standard bigraded over $A$, i.e., $\operatorname{bideg}(A)=(0,0)$, $\operatorname{bideg}(x_i)=(1,0)$ and $\operatorname{bideg}(y_j)=(0,1)$ for all $i$ and $j$. Assume that $M=\bigoplus_{i,j} M_{(i,j)}$ is a bigraded $F_R$-finite, $F_R$-module. We use Lyubeznik's theory of $F$-finite, $F$-modules from \cite{Lyu-Fmod} to study the bigraded components of $M$. The properties we study include vanishing, rigidity, Bass numbers, associated primes, and injective dimension of the components of $M$. As an application we show that if $(A,\mathfrak{m})$ is regular local ring containing a field of characteristic $p>0$, $R/I$ is equidimensional, $\operatorname{Bproj}(R/I)$ is Cohen-Macaulay and non-empty, then $H^j_I(R)_{(m,n)}=0$ for all $(m,n)\geq (0,0)$ and all $j>\operatorname{height} I$.