On the existence of Gorenstein injective covers

Abstract

For a locally finitely generated Grothendieck category, we characterise the Gorenstein injective covering property by several equivalent conditions, including closure under coproducts. To complete this characterisation, we construct a countable commutative Noetherian Cohen--Macaulay local ring of dimension one for which this property fails, answering negatively Iacob's question Question 1 whether character modules of Gorenstein injective modules over left Noetherian rings are Gorenstein flat. Using this characterisation, we prove that for Noetherian schemes the covering property is Zariski-local. Moreover, assuming that $\mathcal{GI}(\mathrm{QCoh}(X))$ is a covering class, we establish compatible adjunctions and recollements for homotopy categories of injectives and totally acyclic injective complexes under open restriction. However, this property is not stalk-local: we construct a commutative Noetherian ring $S$ such that $\mathcal{GI}(S_{\mathfrak{p}})$ forms a covering class for all prime ideals $\mathfrak{p}$, but $\mathcal{GI}(S)$ do not form a covering class.