Gorenstein flat precovers and Gorenstein injective preenvelopes in Grothendieck categories (Q1741623)

In Gorenstein homological algebra one replaces the usual injective, projective and flat $R$-modules ($R$ a ring) with the Gorenstein injective, Gorenstein projective, and Gorenstein flat $R$-modules. The usual theory of homological algebra starts with the ability to take resolutions to compute derived functors. The same is true for Gorenstein homological algebra. The existence of such resolutions boils down to the existence of Gorenstein injective pre-envelopes, and Gorenstein projective (or flat) pre-covers. This paper proves that Gorenstein flat pre-covers exist in very general Grothendieck categories. (For a general Grothendieck category, projectivity is replaced with an appropriate notion of flatness). It also proves that Gorenstein injective preenvelopes exist over locally noetherian categories. The paper also provides a section indicating a variety of applications at the end of the paper.