Coproducts and Some Universal Ring Constructions

Abstract

Let $R$ be an algebra over a field $k$, and $P,Q$ be two nonzero finitely generated projective $R$-modules. By adjoining further generators and relations to $R$, one can obtain an extension $S$ of $R$ having a universal isomorphism of modules, $i:P{ \otimes _R}S \cong Q{ \otimes _R}S$. We here study this and several similar constuctions, including (given a single finitely generated projective $R$-module $P$) the extension $S$ of $R$ having a universal idempotent module-endomorphism $e:P \otimes S \to P \otimes S$, and (given a positive integer $n$) the $k$-algebra $S$ with a universal $k$-algebra homomorphism of $R$ into its $n \times n$ matrix ring, $f:R \to {\mathfrak {m}_n}(S)$. A trick involving matrix rings allows us to reduce the study of each of these constructions to that of a coproduct of rings over a semisimple ring ${R_0}$ ($( = k \times k \times k,k \times k$, and $k$ respectively in the above cases), and hence to apply the theory of such coproducts. As in that theory, we find that the homological properties of the construction are extremely good: The global dimension of $S$ is the same as that of $R$ unless this is 0, in which case it can increase to 1, and the semigroup of isomorphism classes of finitely generated projective modules is changed only in the obvious fashion; e.g., in the first case mentioned, by the adjunction of the relation $[P] = [Q]$. These results allow one to construct a large number of unusual examples. We discuss the problem of obtaining similar results for some related constructions: the adjunction to $R$ of a universal inverse to a given homomorphism of finitely generated projective modules, $f:P \to Q$, and the formation of the factor-ring $R/{T_P}$ by the trace ideal of a given finitely generated projective $R$-module $P$ (in other words, setting $P = 0$). The idea for a category-theoretic generalization of the ideas of the paper is also sketched.