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It was proved by Beligiannis and Krause that over certain Artin algebras,there are Gorenstein flat modules which are not direct limits of finitelygenerated Gorenstein projective modules. That is, these algebras have noGorenstein analogue of the Govorov-Lazard Theorem.  We show that, in fact, there is a large class of rings without such ananalogue. Namely, let R be a commutative local noetherian ring. Then theanalogue fails for R if it has a dualizing complex, is henselian, notGorenstein, and has a finitely generated Gorenstein projective module which isnot free.  The proof is based on a theory of Gorenstein projective (pre)envelopes. Weshow, among other things, that the finitely generated Gorenstein projectivemodules form an enveloping class in mod R if and only if R is Gorenstein or hasthe property that each finitely generated Gorenstein projective module is free.  This is analogous to a recent result on covers by Christensen, Piepmeyer,Striuli, and Takahashi, and their methods are an important input to our work.

Rings without a Gorenstein analogue of the Govorov-Lazard Theorem