Endomorphism algebras over large domains

Abstract

The paper deals with realizations of R-algebras A as endomorphism algebras End G ∼ = A of suitable R-modules G over a commutative ring R. We are mainly interested in the case of R having "many prime ideals", such as R = R[x], the ring of real polynomials, or R a non-discrete valuation domain. Introduction.This work is based on a previous paper [3] on realization theorems. In [3] an R-module G over a commutative ring R with 1 = 0 is constructed such that the endomorphism algebra of G coincides with a given R-algebra A (in general modulo an ideal). There is a given countable multiplicatively closed subset S of R such that A and therefore G is S-torsion-free and S-reduced; recall that an R-module G is S-torsion-free if gs = 0 implies g = 0 for any s ∈ S, g ∈ G, and it is S-reduced if s∈S Gs = 0. Such constructions are by now standard, they are discussed in [3] and in some of the references given there. The desired module G can be constructed between a free A-module B and its S-adic completion B.However, it is clear that in many cases S must be uncountable in order to have s∈S As = 0; for example, if R is a valuation domain with a lattice of ideals not coinitial to ω and A = R, then s∈S As = 0 for all countable S ⊆ R \ {0} (see [6]). In this case a different technique is needed to realize a given algebra A as endomorphism algebra of some module G. The topological methods may not work any longer for |S| > ℵ 0 since the natural S-topology (generated by Gs (s ∈ S)) may not be metrizable; see Example 3.8 in [7]. However, if S is uncountable, which may be necessary as we have seen, a construction of the desired module G is given in [8]. This construction [8] is difficult and awaits simplification. A first simplification is given in [9]; but here R is restricted to be a Prüfer ring.