math.RT — Representation Theory

Reflexive equivalence and reflexive-minimal algebras

Contributed by Haruhisa Enomoto

Let $A$ be a finite-dimensional algebra over a field. A finite-dimensional algebra $B$ over the same field is called reflexively equivalent to $A$ if their categories of reflexive modules are equivalent. We prove that there is a basic algebra $A_{\min}$, unique up to isomorphism, such that $B$ is reflexively equivalent to $A$ if and only if $B\cong\operatorname{End}_{A_{\min}}(A_{\min}\oplus X)$ for some reflexive $A_{\min}$-module $X$. Moreover, we compute $A_{\min}$ explicitly. We extend these results to module-finite algebras over henselian local rings of dimension at most one under a dominant dimension condition at minimal primes. We also introduce reflexive modules over additive categories and prove that taking reflexive modules is idempotent: the reflexive modules over the category of reflexive modules form an equivalent category. As an application, we show that reflexive equivalence classes of algebras with finitely many indecomposable reflexive modules correspond bijectively to Morita equivalence classes of algebras whose reflexive modules are projective.