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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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What makes $\textrm {Tor}^ R_ 1(R/I,I)$ free?
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by Shiro Goto and Naoyoshi Suzuki PDF
Proc. Amer. Math. Soc. 112 (1991), 605-611 Request permission

Abstract:

Let $I$ be a nonprincipal ideal in a Noetherian local ring $R$ and let ${H_1}(I)$ be the first homology module of the Koszul complex $K.(I)$ associated with a minimal basis of $I$. Then $T: = {\text {Tor}}_1^R(R/I,I)$ is a free $R/I$-module if and only if both the $R/I$-modules $I/{I^2}$ and ${H_1}(I)$ are free. When this is 2 2 the case, we have a canonical decomposition $T \cong {\Lambda ^2}(I/{I^2}) \oplus {H_1}(I)$ as well as the equality ${\text {rank}_{R/I}}T = {\beta _2}(R/I)$. (Here ${\beta _2}(R/I)$ denotes the second Betti number of the $R$-module $R/I$.) Some consequences are discussed too.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 112 (1991), 605-611
  • MSC: Primary 13D02; Secondary 13D05
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1052871-0
  • MathSciNet review: 1052871