The cotorsion pair generated by the Gorenstein projective modules and $\lambda$-pure-injective modules

Abstract

We prove that, if $\textrm{GProj}$ is the class of all Gorenstein projective modules over a ring $R$, then $\mathfrak{GP}=(\textrm{GProj},\textrm{GProj}^\perp)$ is a cotorsion pair. Moreover, $\mathfrak{GP}$ is complete when all projective modules are $\lambda$-pure-injective for some infinite regular cardinal $\lambda$ (in particular, if $R$ is right $\Sigma$-pure-injective); the latter condition is shown to be consistent with the axioms of ZFC modulo the existence of strongly compact cardinals. We also thoroughly study $\lambda$-pure-injective modules for an arbitrary infinite regular cardinal $\lambda$, proving along the way that: any cosyzygy module in an injective coresolution of a $\lambda$-pure-injective module is $\lambda$-pure-injective; the cotorsion pair cogenerated by a class of $\lambda$-pure-injective modules is cogenerated by a set and, under an additional technical assumption, generated by a set. Finally, assuming the set-theoretic hypothesis that $0^\sharp$ does not exist, we prove that the category of right $R$-modules has enough $\lambda$-pure-injective objects if and only if the ring $R$ is right pure-semisimple. This, in turn, follows from a rather surprising result that $\lambda$-pure-injectivity amounts to pure-injectivity in the absence of $0^\sharp$.