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arXiv · 1512.06040

The Cohen-Macaulayness of the bounded complex of an affine oriented matroid

Ryota Okazaki·Kohji Yanagawa
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Abstract

An affine oriented matroid is a combinatorial abstraction of an affine hyperplane arrangement. From it, Novik, Postnikov and Sturmfels constructed a squarefree monomial ideal in a polynomial ring, called an oriented matroid ideal, and got beautiful results. Developing their theory, we will show the following. (1) If an oriented matroid ideal is Cohen-Macaulay, then the bounded complex (a regular CW complex associated with it) of the corresponding affine oriented matroid is a contractible homology manifold with boundary. This is closely related to Dong's theorem, which used to be "Zaslavsky's conjecture". (2) We characterize the affine oriented matroid whose corresponding ideal is Cohen-Macaulay. (3) In the Cohen-Macaulay case, we give a description of the canonical module of the residue class ring by an oriented matroid ideal.

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Ryota Okazaki, Kohji Yanagawa. 2017-11-24. The Cohen-Macaulayness of the bounded complex of an affine oriented matroid. https://arxiv.org/abs/1512.06040

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