Minimal free resolutions of fiber products

Abstract

We consider a local (or standard graded) ring R R with ideals I ′ \mathcal {I}’ , I \mathcal {I} , J ′ \mathcal {J}’ , and J \mathcal {J} satisfying certain Tor-vanishing constraints. We construct free resolutions for quotient rings R / ⟨ I ′ , I J , J ′ ⟩ R/\langle \mathcal {I}’, \mathcal {I}\mathcal {J}, \mathcal {J}’\rangle , give conditions for the quotient to be realized as a fiber product, and give criteria for the construction to be minimal. We then specialize this result to fiber products over a field k k and recover explicit formulas for Betti numbers, graded Betti numbers, and Poincaré series.

Lower bounds for Betti numbers over fiber product rings

“…և S be surjective homomorphisms of rings. Recently, the study of the fiber product rings R × T S has been an effective topic of investigation by several authors in different subjects, for instance, structural results [1,11,8,24,26], homological problems [13,16,23,17,21,24,25], and their geometric aspects [14,15]. Actually, most of these works consider the case T = k, and this provides a good understanding of the structure of R × k S. When T = k, few results are known and this motivate us the investigation showed here.…”

Números de Betti: aplicações e problemas relacionados

“…As principais referências para este capítulo são Huneke (2012), Dugger (2000) e Geller (2022). O Teorema 5.1, que provamos na Seção 5.1, foi escrito utilizando a referência Huneke (2012).…”