The convertibility of $\textrm {Ext}^{n}_{R}(-, A)$

Abstract:

Let R be a commutative ring and $\operatorname {Mod} (R)$ the category of R-modules. Call a contravariant functor $F:\operatorname {Mod} (R) \to \operatorname {Mod} (R)$ convertible if for every direct system $\{ {X_\alpha }\}$ in $\operatorname {Mod} (R)$ there is a natural isomorphism $\gamma :F(\lim \limits _ \to {X_\alpha }) \to \lim \limits _ \leftarrow F({X_\alpha })$. If A is in $\operatorname {Mod} (R)$ and n is a positive integer then ${\text {Ext}}_R^n( - ,A)$ is not in general convertible. The purpose of this paper is to study the convertibility of Ext, and in so doing to find out more about Ext as well as the modules A that make ${\text {Ext}}_R^n( - ,A)$ convertible for all n. It is shown that ${\text {Ext}}_R^n( - ,A)$ is convertible for all A having finite length and all n. If R is Noetherian then A can be Artinian, and if R is semilocal Noetherian then A can be linearly compact in the discrete topology. Characterizations are studied and it is shown that if A is a finitely generated module over the semilocal Noetherian ring R, then ${\text {Ext}}_R^1( - ,A)$ is convertible if and only if A is complete in the J-adic topology where J is the Jacobson radical of R. Morita-duality is characterized by the convertibility of ${\text {Ext}}_R^1( - ,R)$ when R is a Noetherian ring, a reflexive ring or an almost maximal valuation ring. Applications to the vanishing of Ext are studied.