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Ext and local cohomology modules of face rings of simplicial posets. (English) Zbl 1395.13023
Let $$P$$ be a simplicial poset with vertex set $$V=\{x_1,\dots,x_n\}$$ and let $$\mathrm{supp}(z)=\{i: x_i\preceq z\}$$ denote the support of an element $$z\in P$$. Define $$A=k[x_1,\dots, x_n]$$, let $$\mathfrak{m}$$ be the irrelevant ideal $$(x_1,\dots, x_n)$$, and let $${\mathfrak{m}}_{\ell}$$ be the ideal $$(x_1^{\ell},\dots, x_n^{\ell})$$. Lastly, let $$A_P$$ be the face ring of $$P$$; consider $$A_P$$ as a $$\mathbb{Z}^n$$-graded $$A$$-module.
The first result is an extension of M. Miyazaki’s calculation of the graded pieces of the Ext-modules of a Stanley-Reisner ring [Manuscr. Math. 63, No. 2, 245–254 (1989; Zbl 0671.13014)]:
Theorem 3.1. Let $$P$$ be a simplicial poset with vertex set $$V$$ , and let $$\alpha\in\mathbb{Z}^n$$. Set $$B=\{i: -\ell<{\alpha}_i< 0\}, C=\{i : {\alpha}_i=-\ell\}$$, and $$D=\{i: {\alpha}_i>0\}$$. If $$-\ell\leq{\alpha}_i$$ for all $$i$$, then $\mathrm{Ext}^i_A(A/{\mathfrak{m}}_{\ell},A_P )_{\alpha}\cong\bigoplus_{\mathrm{supp}(z)=B\cup D}\tilde{H}^{i-|B|-|C|-1}([\hat{0},z_D]\times Ik_P(z)_{V\setminus C})$ and $$\mathrm{Ext}^i_A(A/{\mathfrak{m}}_{\ell},A_P)_{\alpha}=0$$ otherwise. In particular, if $$D\neq\emptyset$$ then $$\mathrm{Ext}^i_A(A/{\mathfrak{m}}_{\ell},A_P)_{\alpha}=0$$.
The second main contribution is:
Theorem 4.4. Let $$\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb{Z}^n$$. Then $H_{\mathfrak{m}}^i(A_P)_{\alpha}\cong\bigoplus_{supp(w)=\{j: \alpha_j\neq 0\}}H^{i-1}(P,\mathrm{cost}_P(w))$ if $$\alpha\in\mathbb{Z}_{\leq 0}^n$$ and $$H_{\mathfrak{m}}^i(A_P)_{\alpha}=0$$ otherwise. Under these isomorphisms, the $$A$$-module structure of $$H_{\mathfrak{m}}^i(A_P)$$ is given as follows. Let $$\gamma=\alpha+\deg(x_j)$$. If $$\alpha_j<-1$$, then $$\cdot x_j: H_{\mathfrak{m}}^i(A_P)_{\alpha}\longrightarrow H_{\mathfrak{m}}^i(A_P)_{\gamma}$$ corresponds to the direct sum of identity maps
$\bigoplus_{\mathrm{supp}(w)=\{j: \alpha_j\neq 0\}}H^{i-1}(P,\mathrm{cost}_P(w))\longrightarrow \bigoplus_{\mathrm{supp}(w)=\{j: \alpha_j\neq 0\}}H^{i-1}(P,\mathrm{cost}_P(w)).$ If $$\alpha_j=-1$$, then $$\cdot x_j$$ corresponds to the direct sum of maps $\bigoplus_{\mathrm{supp}(w)=\{j: \alpha_j\neq 0\}}H^{i-1}(P,\mathrm{cost}_P(w))\longrightarrow \bigoplus_{\mathrm{supp}(z)=\{j: \gamma_j\neq 0\}}H^{i-1}(P,\mathrm{cost}_P(z))$ induced by the inclusions of pairs $$(P,\mathrm{cost}_P(w\setminus\{x_j\}))\longrightarrow(P,\mathrm{cost}_P(w))$$. If $$\alpha_j\geq 0$$, then $$\cdot x_j$$ is the zero map.

##### MSC:
 13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes 13D45 Local cohomology and commutative rings 13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
##### Keywords:
simplicial posets; ext modules; local cohomology
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