Algebraic Combinatorics Seminar
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UNIVERSITY OF KENTUCKY **

ALGEBRAIC COMBINATORICS SEMINAR

945 PATTERSON OFFICE TOWER

SPRING 2005

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Combinatorics of Poincaré series for monomial rings

Professor Patricia Hersh

Indiana University

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Friday, April 8, 2005

4 pm, 845 Patterson Office Tower

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Abstract:

Backelin proved that the multigraded Poincaré series for
resolving a residue field over a polynomial ring modulo a monomial
ideal is a rational function. The numerator is simple and
well-understood, but recently Alex Berglund found a formula for the
denominator, expressed in terms of ranks of homology groups of certain
simplicial complexes. I'll discuss recent joint work with Jonah
Blasiak in which we express the denominator in terms of lower
intervals in intersection lattices of subspace arrangements that in
many cases have been studied before. Crapo's Closure Lemma and related
combinatorics allow us to simplify the denominator substantially in
some cases (such as monomial ideals generated in degree two), make it
more explicit in other cases, and relate Golodness to the
Cohen-Macaulay property for associated posets. I'll briefly review
the notions of minimal free resolution, Golodness, etc. along the way.