Universal localization of triangular matrix rings.pdf

a r X i v : m a t h / 0 4 0 7 4 9 7 v 2 [ m a t h .R A ] 9 J u l 2 0 0 5 Universal localization of triangular matrix rings Desmond Sheiham Abstract If R is a triangular 2 × 2 matrix ring, the columns, P and Q, are f.g. projective R-modules. We describe the universal localization of R which makes invertible an R-module morphism σ : P → Q, generalizing a theorem of A.Schofield. We also describe the universal localization of R-modules. 1 Introduction Suppose R is an associative ring (with 1) and σ : P → Q is a morphism between finitely generated projective R-modules. There is a universal way to localize R in such a way that σ becomes an isomorphism. More precisely there is a ring morphism R → σ?1R which is universal for the property that σ?1R?R P 1?σ ???→ σ?1R?R Q is an isomorphism (Cohn [7, 9, 8, 6], Bergman [4, 5], Schofield [17]). Al- though it is often difficult to understand universal localizations when R is non- commutative1 there are examples where elegant descriptions of σ?1R have been possible (e.g. Cohn and Dicks [10], Dicks and Sontag [11, Thm. 24], Farber and Vogel [12] Ara, Gonza?lez-Barroso, Goodearl and Pardo [1, Example 2.5]). The purpose of this note is to describe and to generalize some particularly interesting examples due to A.Schofield [17, Thm. 13.1] which have application in topology (e.g. Ranicki [16, Part 2]). We consider a triangular matrix ring R = ( A M 0 B ) where A and B are associative rings (with 1) and M is an (A,B)-bimodule. Multiplication in R is given by ( a m 0 b )( a′ m′ 0 b′ ) = ( aa′ am′ +mb′ 0 bb′ ) for all a, a′ ∈ A, m,m′ ∈ M and b, b′ ∈ B. The columns P = ( A 0 ) and Q = ( M B ) are f.g. projective left R-modules with P ⊕Q ～= R. The general theory of triangular matrix rings can be found in Haghany and Varadarajan [13, 14]. Desmond Sheiham died on March 25, 2005. This article was prepared for publication by Andrew Ranicki, with the assistance of Aidan Schofield. 1If R is commutative one obtains a ring of fractions; see