The Multiplicity Conjecture for Barycentric Subdivisions
Abstract
For a simplicial complex $\Delta$ we study the effect of barycentric subdivision on ring theoretic invariants of its StanleyReisner ring. In particular, for StanleyReisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog & Srinivasan, that relates the multiplicity of a standard graded $k$algebra to the product of the maximal and minimal shifts in its minimal free resolution up to the height. On the way to proving the conjecture we develop new and list well known results on behavior of dimension, Hilbert series, multiplicity, local cohomology, depth and regularity when passing from the StanleyReisner ring of $\Delta$ to the one of its barycentric subdivision.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606274
 Bibcode:
 2006math......6274K
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Combinatorics;
 13D02;
 05E99
 EPrint:
 28, pages, lower bound and equality cases added