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Criteria for shellable multicomplexes. (English) Zbl 1199.13013

For a finitely generated module \(M\) over a noetherian ring \(R\), the prime filtration \[ \mathcal{F}: 0=M_0 \subset M_1 \subset \dots \subset M_{r-1} \subset M_r=M \] is called pretty clean if every \(M_i/M_{i-1} \cong R/P_i\) for some prime ideal \(P_i\) and for all \(i < j\) with \(P_i \subseteq P_j\) one must have \(P_i=P_j\). The module \(M\) is called pretty clean if it admits such a filtration, and the ring \(R\) is called pretty clean if it is pretty clean as an \(R\)-module. J. Herzog and D. Popescu [Manuscr. Math. 121, 385–410 (2006; Zbl 1107.13017)] have proved that a so-called multicomplex \(\Gamma\subseteq \mathbb{N}^n_{\infty}\) is shellable if and only if the ring \(K[X_1,\dots,X_n]/I(\Gamma)\) is pretty clean. In this paper the author gives an easy criterion for multigraded pretty cleanness as well as a criterion for the shellability of multicomplexes in terms of maximal facets. A direct proof for a result already established by J. Herzog and D. Popescu [op. cit.] that shows that the arithmetic degree of \(K[X_1,\dots,X_n]/I(\Gamma)\) is equal to the number of facets of \(\Gamma\) is also provided. Several interesting examples are also included.

MSC:

13C14 Cohen-Macaulay modules
16W70 Filtered associative rings; filtrational and graded techniques
13C13 Other special types of modules and ideals in commutative rings
05E99 Algebraic combinatorics

Citations:

Zbl 1107.13017
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