Spheres arising from multicomplexes

Abstract

In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex $\Delta$ on the vertex set $V$ with $\Delta \ne 2^V$, the deleted join of $\Delta$ with its Alexander dual $\Delta^\vee$ is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that these spheres are related to polarizations and Alexander duality for monomial ideals which appear in commutative algebra theory.Comment: 20 pages. Improve presentation. To appear in Journal of Combinatorial Theory, Series

Sliding Functor and Polarization Functor for Multigraded Modules

“…However the pair pol a and pol a is only implicit in [11], since a different convention is used there. This pair appears in a recent work of Murai [9], which we will discuss in §5.…”

“…Murai [9] showed that the geometric realization of BM a (I) is homeomorphic to a sphere of dimension |a| − 2.…”

“…From a positively a-determined monomial ideal I of S, Murai [9] constructed the squarefree monomial ideal BM a (I) of k[ x i,j | 1 ≤ i ≤ n, 1 ≤ j ≤ a i + 1 ], which is the Stanley-Reisner ideal of a simplicial sphere BM a (I) of dimension −2 + n i=1 a i . If I itself is squarefree, this construction is due to T. Bier (unpublished).…”

Polarizations of powers of graded maximal ideals

“…That the standard polarization of an Artinian monomial ideal is shellable seems first to have been shown by A.Soleyman Jahn in [21]. In [18] S.Murai uses this to conclude that the standard polarizations give simplicial balls. More generally it is shown that letterplace ideals define simplicial balls, [6], by showing that these simplicial complexes are shellable.…”

“…, x m ) 2 give simplicial balls, by showing that the Alexander duals of all such polarizations have linear quotients. Conjecture 2.4 is known for the standard polarization by S.Murai [18], and for letterplace ideals by the second author et.al. in [6].…”