# Castelnuovo-Mumford regularity in multigraded case

Let $R=\oplus_{n\geq 0}R_n$ be a standard Noetherian commuative graded ring over a local ring $(A,m)$ where $R_0=A.$ Put $R_+=\oplus_{n\geq 1}R_n.$ Let $M$ be a finitely generated $\mathbb Z$-graded $R$-module. Then Castelnuovo-Mumford regularity of $M$ is defined here. My question is

What is the definition of Castelnuovo-Mumford regularity when we consider $R=\bigoplus\limits_{(r,s)\geq (0,0)}R_{(r,s)},$ $R_+=\bigoplus\limits_{(r,s)\geq (1,1)}R_{(r,s)}$ and $M=\bigoplus\limits_{(r,s)\in \mathbb Z^2}M_{(r,s)}$ a finitely generated $\mathbb Z^2$-graded $R$-module.

I have seen the algebraic geometric version of the definition of multigraded Castelnuovo-Mumford regularity. But I am interested to know the definition with respect to local cohomology and bound of the regularity .

• Isn't it just the maximal a-invariant + 1 over the local cohomology modules? At least that would be my first guess. – lemiller May 21 '15 at 20:02
• What is the case in multigraded situation? how do you define a-invariants in this case. – Cusp May 22 '15 at 5:54
• Sorry, the multigraded a-invariant in this case would be an element of Z^2 (see Def 2.2 of this paper), so maybe the regularity arises as the norm of this vector? I haven't tried to write it down, but it would be a first place to try. – lemiller May 24 '15 at 18:59