Classifying Betti Numbers of Fiber Products
Ela Celikbas, Hugh Geller, Tony Se
TL;DR
This paper develops new techniques to compute Betti numbers and Poincaré series for fiber products of certain local algebras, addressing limitations of existing methods.
Contribution
It introduces modified methods based on Geller's approach to determine Betti numbers for fiber products of complete local noetherian algebras.
Findings
Successfully computes Betti numbers for classes of fiber products
Extends Geller's approach to cases previously unresolved
Provides explicit formulas for Poincaré series
Abstract
We consider fiber products of complete, local, noetherian algebras over a fixed residue field. Some of these rings cannot be minimally resolved with a free resolution using the recent work of the second author. We develop techniques to modify Geller's approach in order to recover the Betti numbers and Poincar\'e series for these fiber products.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Commutative Algebra and Its Applications · Polynomial and algebraic computation