Extension closed reflexive modules (Q1111660)

For a left and right noetherian ring \(R\), we show that both the class of the finitely generated torsionless left \(R\)-modules and the class of the finitely generated reflexive left \(R\)-modules are closed under extensions if and only if for \(i=1,2\) the functor \(\mathrm{Ext}_ R^{i-1}(\mathrm{Ext}^ i_ R(-,R),R)\) vanishes on the finitely generated right \(R\)-modules. As a particular case, we get the following: A commutative noetherian local domain \(R\) of Krull dimension two is a Cohen-Macaulay ring whenever the class of the finitely generated reflexive \(R\)-modules is closed under extensions.