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?
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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?
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> 
> 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?
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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.



Existence of non-trivial reflexive modules
The algebra with quiver


$\require{AMScd}$
\begin{CD}
\bullet @>>>\bullet@>>>\bullet\\
@AAA&@AAA\\
\bullet@>>>\bullet
\end{CD}


and radical square zero has global dimension three and (according to my calculations) no non-projective reflexive modules.


