Modules over categories and Betti posets of monomial ideals

Tchernev, Alexandre; Varisco, Marco

doi:10.1090/proc/12643

Modules over categories and Betti posets of monomial ideals
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by Alexandre Tchernev and Marco Varisco PDF

Proc. Amer. Math. Soc. 143 (2015), 5113-5128 Request permission

Abstract:

We introduce to the context of multigraded modules the methods of modules over categories from algebraic topology and homotopy theory. We develop the basic theory quite generally, with a view toward future applications to a wide class of graded modules over graded rings. The main application in this paper is to study the Betti poset $\mathcal {B}=\mathcal {B}(I,\Bbbk )$ of a monomial ideal $I$ in the polynomial ring $R=\Bbbk [x_1,\dots ,x_m]$ over a field $\Bbbk$, which consists of all degrees in $\mathbb {Z}^m$ of the homogeneous basis elements of the free modules in the minimal free $\mathbb {Z}^m$-graded resolution of $I$ over $R$. We show that the order simplicial complex of $\mathcal {B}$ supports a free resolution of $I$ over $R$. We give a formula for the Betti numbers of $I$ in terms of Betti numbers of open intervals of $\mathcal {B}$, and we show that the isomorphism class of $\mathcal {B}$ completely determines the structure of the minimal free resolution of $I$, thus generalizing with new proofs the results of Gasharov, Peeva, and Welker in 1999. We also characterize the finite posets that are Betti posets of a monomial ideal.

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