Commutative torsion theory

This paper links several notions of torsion theory with com- mutative 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 #1685.1 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 ii(U), of the quasi-coherent module V 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 quo- tients, 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, to- gether with an extra condition. 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 variete's algebriques, 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 character- ized. This paper is part of the author's doctoral dissertation defended at Queen's