Minimal free resolutions of fiber products

Geller, Hugh

doi:10.1090/proc/15963

Minimal free resolutions of fiber products
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Proc. Amer. Math. Soc. 150 (2022), 4159-4172 Request permission

Abstract:

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.

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