# Existence of non-trivial reflexive modules

Recall that a module $$M$$ over a ring $$R$$ is reflexive in case the natural evaluation map $$f_M:M \rightarrow M^{**}$$ (where $$M^{*}=Hom_R(M,R)$$) is an isomorphism, where $$f_M(m)=g$$ with $$g(h)=h(m)$$, when $$h \in Hom_R(M,R)$$. For example every projective module is reflexive. Assume that all modules in the following are finitely generated. Call a reflexive module non-trivial in case it is not projective.

Question: Does an arbitrary ring of finite global dimension have global dimension at most two if and only if there is no non-trivial reflexive module?

Here two similar questions restricted to finite dimensional algebras:

Does a finite dimensional algebra have a non-trivial reflexive module in case it has finitistic dimension at least 3?

Does a finite dimensional algebra of finite global dimension have global dimension at most two if and only if it has no non-trivial reflexive modules?

In general a non-trivial reflexive module implies that the global dimension of a ring with finite global dimension is at least three but I do not know about the other direction. The answer should be positive for QF-3 algebras.

• For Noetherian commutative rings, this follows from Auslander-Buchsbaum formula. Dec 17, 2018 at 14:42
• Probably you intend some finiteness condition? There are infinitely generated non-projective reflexive abelian groups. Dec 18, 2018 at 9:13
• @JeremyRickard Thanks, I was actually only thinking about finitely generated modules. I edited it.
– Mare
Dec 18, 2018 at 9:32

$$\require{AMScd}$$ $$\begin{CD} \bullet @>>>\bullet@>>>\bullet\\ @AAA&@AAA\\ \bullet@>>>\bullet \end{CD}$$
• @Mare Well, for a radical square zero quiver algebra, it's not hard to see that having no non-projective simple reflexive modules is equivalent to having no arrows $\alpha:i\to j$ such that $j$ is not a sink and $\alpha$ is the only arrow starting at $i$ and the only arrow ending at $j$. So then you only have to worry about indecomposable non-simple, non-projective modules; and this algebra only has three of those. Dec 24, 2018 at 12:46