Finite filtrations of modules and shellable multicomplexes (Q863448)

In commutative algebra there is the elementary notion of a prime filtration of a finitely generated module \(M\) over a noetherian ring \(R\); it is a filtration \(0=M_0\subseteq \dots \subseteq M_r=M\) such that every \(M_i/M_{i-1}\) is isomorphic to \(R/p\) for some prime ideal \(p\) of \(R\). Among other things this paper deals with the existence of special prime filtrations; the notion of clean prime filtrations (meaning that all prime ideals \(p\) occuring above are minimal in the support of \(M\)) which was introduced by \textit{A. Dress} [Beitr. Algebr. Geom. 340, 45--55 (1993; Zbl 0780.52012)], is generalized to the new notion of pretty clean prime filtrations (a prime filtration \(0=M_0\subseteq \dots \subseteq M_r=M\) with \(M_i/M_{i-1}\cong R/p_i\) is called pretty clean if \(i<j\) and \(p_i\subseteq p_j\) together imply \(p_i=p_j\); \(M\) is called pretty clean if it has a pretty clean prime filtration). It was shown by Dress that being clean corresponds to shellability of simplicial complexes (in the sense of Björner and Wachs) via the Stanley-Reisner ring of the complex. On the other hand \textit{R. Stanley} showed [``Combinatorics and Commutative Algebra'', Prog. Math. 41 (1983; Zbl 0537.13009)] that if the simplicial complex \(\Delta \) is shellable then the Stanley-Reisner ring is sequentially Cohen-Macaulay (a notion introduced by \textit{P. Schenzel} [``Commutative algebra and algebraic geometry''. Lect. Notes Pure Appl. Math. 206, 245--264 (1999; Zbl 0942.13015)] and \textit{R. Stanley} [Zbl 0537.13009] that also depends on the existence of certain prime filtrations). The implication one can derive from the latter two results is generalized to the result that pretty clean modules are sequentially Cohen-Macaulay (under some mild assumptions). It is shown that the coefficient rings belonging to ideals of Borel type are pretty clean and, hence, sequentially Cohen-Macaulay. (Multi)graded versions of the notions of pretty cleanness are defined. Furthermore, multicomplexes are defined (they correspond to monomial ideals of polynomial rings over a field) and the notion of shellabilty is extended from simplicial complexes to multicomplexes (a variant of multicomplexes in the sense of Stanley). A main result is that for a monomial ideal \(I\subseteq k[x_1,\dots ,x_n]=:S\) (\(k\) a field) the ring \(S/I\) is multigraded pretty clean iff the corresponding multicomplex is shellable. This extends a result of Dress.