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In http://s.web.umkc.edu/segal/papers/refl.pdf the authors gave the first example of a maximal Cohen-Macaulay module $M$ (that is a module over an algebra A with $Ext^{i}(M,A)=0$ for all $i \geq 1$) that is not Gorenstein projective for a finite dimensional algebra over a field $k$ . However, the authors had to assume that the field is infinite.

Question: Can such a thing also occur for finite field?

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    $\begingroup$ You should clarify your definition of maximal Cohen–Macaulay here, since there are many options which are not in general equivalent: I guess you mean in the Buchweitz sense, so an $R$-module $M$ is MCM if $\operatorname{Ext}^i_R(M,R)=0$ for all $i>0$. $\endgroup$
    – Matthew Pressland
    Commented Jul 26, 2017 at 8:48
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    $\begingroup$ @MatthewPressland ok, I added the definition. $\endgroup$
    – Mare
    Commented Jul 26, 2017 at 8:49

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