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Simplicial complexes and minimal free resolution of monomial algebras. (English) Zbl 1195.13015
Let \(\Bbbk\) be a field, \(S\) a commutative semigroup generated by \(n_1,\ldots, n_r\) so that \(S\cap (-S)=\{0\}\) and \(R=\Bbbk[x_1,\ldots, x_r]\). The \(S\)-degree of \(x^a=x_1^{a_1}\cdots x_r^{a_r}\) is \(\sum_{i=1}^r a_i n_i\) and the \(S\)-graded Nakayama Lemma applies. If \(I\) is the toric ideal generated by all \(S\)-homogeneous binomials then \(R/I\) is \(S\)-graded and one can consider the minimal free \(S\)-graded resolution of \(R/I\). Denote by \(s_{j+1,m}\) the multigraded \(j+1\)-Betti number of \(R/I\) of degree \(m\). It is well known that \(s_{j+1,m}\) equals the rank of the \(j\)-reduced homology group of the simplicial complex \(\Delta_m=\{ F\subset [r]:\;m-\sum_{i\in F}n_i\in S\}\).
The authors use the simplicial complex \(\nabla_m=\{ F\subset C_m:\;\text{gcd}(F)\neq 1\}\) where \(C_m\) consists of all monomials of \(S\)-degree \(m\). They prove that \(s_{j+1,m}\) equals the rank of the \(j\)-reduced homology group of \(\nabla_m\). Given a monomial term order on \(R\), they proceed to fix a particular basis of \(\tilde{H}_j(\nabla_m)\) for \(j\geq 0\). For this basis they show how to get \(j+1\)-syzygies of \(R/I\) of degree \(m\) and produce part of the minimal free resolution of \(R/I\). The result on the Betti numbers was also proved independently by H. Charalambous and A. Thoma [see for example Contemp. Math., No. 502, 33–44 (2009; Zbl 1183.13017)].

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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