Numerical invariants and projective modules

Abstract

In a recent paper [9], Fr6hlich gives criteria in terms of numerical invariants to insure the projectivity of modules of an order in a commutative separable algebra over the quotient field of a Dedekind domain. More precisely, Frijhlich first defines a numerical invariant, the module index (see [8] for the definition), and then considers the following context: let A be a Dedekind domain, K its quotient field, 2 a finite dimensional commutative separable K-algebra, r the unique maximal A-order in Z and A any other A-order of Z. Now if M is a A-module such that M @A K E LY’), for an integer I, then M is A-projective if and only if [FM : M] = [r : A]’ (I’M is taken to denote the smallest r-module containing M, and [P : Q] is the notation for the module index of A-lattices P and Q). For purposes of reference we call this result “Friihlich’s theorem.” The object of this paper is to extend and give analogues to FrBhlich’s results for the case that Z is a finite dimensional separable K-algebra. It will be seen, however, that in the fairly simple case of a finite dimensional matrix algebra over the quotient field of a (complete) discrete rank one valuation rank, no direct analogue of either the necessity or the sufficiency of Fr6hlich’s theorem can be given. Nevertheless, for any finite dimensional central simple algebra Z over the quotient field of a complete discrete rank one valuation ring, we can give results similar to those of Frijhlich for some special types of A-orders in 2. Further, we shall show that Fr6hlich’s results extend almost entirely when Z is a central division K-algebra and the base ring A is a complete discrete rank one valuation ring. FrGhlich also gave a criteria to determine the projectivity of fractional ideals I in A such that I BA K E 2. This can be stated as: I is A-projective