Minimal free resolutions of fiber products.

We consider a local (or standard graded) ring R with ideals \mathcal {I}', \mathcal {I}, \mathcal {J}', and \mathcal {J} satisfying certain Tor-vanishing constraints. 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 Poincaré series. [ABSTRACT FROM AUTHOR]