Minimal Free Resolutions of Fiber Products

TL;DR

This paper develops methods to construct and analyze minimal free resolutions of certain fiber product quotient rings, providing explicit formulas for their Betti numbers and Poincaré series.

Contribution

It introduces a new construction for free resolutions of fiber product rings and establishes criteria for their minimality, with explicit formulas for algebraic invariants.

Findings

Constructed free resolutions for specific quotient rings.

Provided conditions for the quotient to be a fiber product.

Derived explicit formulas for Betti numbers and Poincaré series.

Abstract

We construct free resolutions for quotient rings $R/\langle \mathcal{I}', \mathcal{I}\mathcal{J}, \mathcal{J}'\rangle$, 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 $k$ and recover explicit formulas for Betti numbers, graded Betti numbers, and Poincar\'{e} series.