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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.)
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