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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Commutative torsion theory


Author: Paul-Jean Cahen
Journal: Trans. Amer. Math. Soc. 184 (1973), 73-85
MSC: Primary 13C10
MathSciNet review: 0327735
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Abstract: This paper links several notions of torsion theory with commutative concepts. The notion of dominant dimension [H. H. Storrer, Torsion theories and dominant dimensions, Appendix to Lecture Notes in Math., vol. 177, Springer-Verlag, Berlin and New York, 1971. MR 44 #1685.] is shown to be very close to the notion of depth. For a commutative ring A and a torsion theory such that the primes of A, whose residue field is torsion-free, form an open set U of the spectrum of A, Spec A, a concrete interpretation of the module of quotients is given: if M is an A-module, its module of quotients $ Q(M)$ is isomorphic to the module of sections $ \tilde M(U)$, of the quasi-coherent module $ \tilde M$ canonically associated to M. In the last part it is proved that the (T)-condition of Goldman is satisfied [O. Goldman, Rings and modules of quotients, J. Algebra 13 (1969), 10-47. MR 39 #6914.] if and only if the set of primes, whose residue field is torsion-free, is an affine subset of Spec A, together with an extra condirion. The extra, more technical, condition is always satisfied over a Noetherian ring, in this case also it is classical that the (T)-condition of Goldman means that the localization functor Q is exact. This gives a new proof to Serre's theorem [J.-P. Serre, Sur la cohomologie des variétés algébriques, J. Math. Pures Appl. (9) 36 (1957), 1-16. MR 18,765.]. As an application, the affine open sets of a regular Noetherian ring are also characterized.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0327735-1
Keywords: Torsion theory, depth, dominant dimension, ring and module of quotients, localization, (T)-condition of Goldman, affine sets
Article copyright: © Copyright 1973 American Mathematical Society