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Modules over categories and Betti posets of monomial ideals


Authors: Alexandre Tchernev and Marco Varisco
Journal: Proc. Amer. Math. Soc. 143 (2015), 5113-5128
MSC (2010): Primary 13D02, 05E40, 06A11
Published electronically: May 22, 2015
MathSciNet review: 3411130
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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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Additional Information

Alexandre Tchernev
Affiliation: Department of Mathematics and Statistics, University at Albany, SUNY, Albany, New York 12222
Email: atchernev@albany.edu

Marco Varisco
Affiliation: Department of Mathematics and Statistics, University at Albany, SUNY, Albany, New York 12222
Email: mvarisco@albany.edu

DOI: https://doi.org/10.1090/proc/12643
Received by editor(s): April 7, 2014
Received by editor(s) in revised form: September 30, 2014
Published electronically: May 22, 2015
Communicated by: Irena Peeva
Article copyright: © Copyright 2015 American Mathematical Society