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

Abstract

For a simplicial poset $P$, Stanley assigned the face ring $A_P$, which is the quotient of the polynomial ring $S:=K[t_x\mid x\in P\setminus \{\hat{0}\}]$ 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 ${}^{\ast }{E}_S(S/{\mathfrak{p}}_x)$, where ${\mathfrak{p}}_x$is the prime ideal associated with $x\in P$, and analyze their behavior in the graded dualizing complex.

20 pages. Abstract and Introduction revised