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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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##### References:
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