Annihilators of modules with a finite free resolution
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- by Wolmer V. Vasconcelos PDF
- Proc. Amer. Math. Soc. 29 (1971), 440-442 Request permission
Abstract:
Let A be a commutative ring and let E be an A-module with a finite free resolution (see below for definition). Extending results known previously for noetherian rings, it is shown that ${\text {ann}}(E)=\text {annihilator}$ of E is trivial if and only if the Euler characteristic of $E = \chi (E) > 0$; in addition, if $\chi (E) = 0,{\text {ann}}(E)$ is dense (i.e. ${\text {ann}}({\text {ann}}(E)) = 0$). Also, a local ring is constructed with its maximal ideal with a finite free resolution but consisting exclusively of zero-divisors and thus, contrary to the noetherian case, one does not always have a nonzero divisor in ${\text {ann}}(E)$ if $\chi (E) = 0$. Finally, if E has a finite resolution by (f.g.) projective modules it turns out that ${\text {ann}}({\text {ann}}(E))$ is generated by an idempotent element.References
- Maurice Auslander and David A. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625–657. MR 99978, DOI 10.2307/1970159
- Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
- Wolmer V. Vasconcelos, On finitely generated flat modules, Trans. Amer. Math. Soc. 138 (1969), 505–512. MR 238839, DOI 10.1090/S0002-9947-1969-0238839-5
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 440-442
- MSC: Primary 13.40
- DOI: https://doi.org/10.1090/S0002-9939-1971-0280478-9
- MathSciNet review: 0280478