Beitraege zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 34 (1993), No. 1, 45-55.
A New Algebraic Criterion for Shellability
Andreas Dress
Abstract.
In this paper, various types of filtrations of modules
are explored, with emphasis on what are called ``clean'' filtrations. If
$M$ is a module over a commutative ring $R$, a filtration by submodules
$0=M_0\subset M_1\subset M_2\subset \ldots \subset M_n=M$ is said to be
``clean'' if $M_i/M_{i-1}\cong R/P_i$ with $P_i\in \min ({\rm Spec } (R/{\rm
Ann}(M)))$ for $i=1,\ldots,n$. A module is said to be ``clean'' if it has
a clean filtration. It will be shown
that a simplicial complex is shellable if and
only if its face ring is clean (as a module over itself).
Key words and phrases: Prime decomposition, ``clean'' filtrations
of modules, prime ideals associated with a module, simplicial complexes,
shelling, face ring, Cohen-Macaulay property.