Minimal Free Resolutions of Fiber Products

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.