Minimal Free Resolutions of Fiber Products

Abstract

We construct free resolutions for quotient rings $R/⟨{\mathcal{I}}^{\prime },\mathcal{I}\mathcal{J},{\mathcal{J}}^{\prime }⟩$, 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 Poincaré series.

14 pages