Commutative Torsion Theory

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.