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 $S:=K[t_x \mid x \in P \setminus \{\widehat{0} \}]$ by the ideal $I_P$. This is a generalization of Stanley-Reisner rings, but $S$ and $A_P$ are not standard graded, and $I_P$ is not a monomial ideal. To develop the theory on the local cohomology $H_{I_p}^i(S)$ and its injective resolution, this paper establishes the foundation. Specifically, 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 $x \in P$. We also analyze morphisms between them.

Theorems & Definitions (43)

• Example 2.1
• Example 2.2
• Remark 2.3
• Example 2.4
• Lemma 2.5
• proof
• Proposition 2.6
• proof
• Theorem 2.7
• proof