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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 .

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  • $\begingroup$ Isn't it just the maximal a-invariant + 1 over the local cohomology modules? At least that would be my first guess. $\endgroup$
    – lemiller
    Commented May 21, 2015 at 20:02
  • $\begingroup$ What is the case in multigraded situation? how do you define a-invariants in this case. $\endgroup$
    – Cusp
    Commented May 22, 2015 at 5:54
  • $\begingroup$ 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. $\endgroup$
    – lemiller
    Commented May 24, 2015 at 18:59

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Castelnuovo-Mumford regularity is defined in the general mutligraded case in this paper of Maclagan and Smith. They give the definition in terms of local cohomology and discuss bounding the regularity so this may be what you are looking for.

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