On the vanishing of $\textrm {Ext}$
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- by Mark Ramras
- Proc. Amer. Math. Soc. 27 (1971), 457-462
- DOI: https://doi.org/10.1090/S0002-9939-1971-0284427-9
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Abstract:
In this paper we exhibit certain modules $A$ over a commutative noetherian local ring $(R,\mathfrak {M})$ which test projective dimension of finitely generated modules in the following sense: if ${\operatorname {Ext} ^j}(M,A) = 0$ for all $j \geqq i$, then pd $M < i$. We also show that the module $\mathfrak {M}$ tests in a stronger way: if ${\operatorname {Ext} ^i}(M,\mathfrak {M}) = 0$, then pd $M < i$. In conclusion we show that if $(R,\mathfrak {M})$ is artin, then $R$ is self-injective if and only if ${\operatorname {Ext} ^1}(R/{\mathfrak {M}^n},R) = 0$, where the index of nilpotence of $\mathfrak {M}$ is $n + 1$.References
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- Maurice Auslander and Oscar Goldman, Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 1β24. MR 117252, DOI 10.1090/S0002-9947-1960-0117252-7
- Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8β28. MR 153708, DOI 10.1007/BF01112819
- Mark Ramras, Maximal orders over regular local rings of dimension two, Trans. Amer. Math. Soc. 142 (1969), 457β479. MR 245572, DOI 10.1090/S0002-9947-1969-0245572-2
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 457-462
- MSC: Primary 13.40
- DOI: https://doi.org/10.1090/S0002-9939-1971-0284427-9
- MathSciNet review: 0284427