Toward the theory on local cohomologies at the ideals given by simplicial posets

TL;DR

This paper develops a foundational theory of local cohomology for face rings of simplicial posets, providing explicit descriptions of injective envelopes and analyzing their behavior within the dualizing complex.

Contribution

It introduces explicit descriptions of graded injective envelopes for face rings of simplicial posets, extending local cohomology theory beyond Stanley-Reisner rings.

Findings

Explicit description of graded injective envelope ${}^* ext{E}_S(S/ ext{p}_x)$

Analysis of injective envelopes in the graded dualizing complex

Foundation for local cohomology theory on simplicial poset face rings

Abstract

For a simplicial poset $P$, Stanley assigned the face ring $A_P$, which is the quotient of the polynomial ring $S:=K[t_x \mid x \in P \setminus \{\widehat{0} \}]$ by the ideal $I_P$. This is a generalization of Stanley-Reisner rings, but $S$ and $A_P$ are not standard graded in this case, and $I_P$ is not a monomial ideal. To establish the foundation of the theory on local cohomology $H_{I_p}^i(S)$ and its injective resolution, we give an explicit description of the graded injective envelope ${}^*\! E_S(S/\mathfrak{p}_x)$, where $\mathfrak{p}_x$is the prime ideal associated with $x \in P$, and analyze their behavior in the graded dualizing complex.