Minimal free resolutions of fiber products
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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.References
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Additional Information
- Hugh Geller
- Affiliation: Department of Mathematics, The University of the South, Sewanee, Tenessee 37383
- MR Author ID: 1335052
- ORCID: 0000-0002-4012-6404
- Email: hrgeller@sewanee.edu
- Received by editor(s): April 9, 2021
- Received by editor(s) in revised form: October 20, 2021
- Published electronically: June 30, 2022
- Additional Notes: This work was partially funded by a Clemson University Doctoral Dissertation Completion Grant.
- Communicated by: Claudia Polini
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 4159-4172
- MSC (2020): Primary 13D02; Secondary 13D07
- DOI: https://doi.org/10.1090/proc/15963
- MathSciNet review: 4470165