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.

We construct free resolutions for quotient rings R/ I ′ , IJ , J ′ , 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.

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