Gillespie’s questions and Grothendieck duality

Abstract

Gillespie posed two questions in [Front. Math. China 12 (2017) 97-115], one of which states that “for what rings $R$ do we have $\mathrm{K}(\mathrm{AC})=\mathrm{K}(R\text{-}\mathrm{Inj})$?”. We give an answer to such a question. As applications, we obtain a new homological approach that unifies some well-known conditions of rings such that Krause’s recollement holds, and give an example to show that there exists a Gorenstein injective module which is not Gorenstein AC-injective. We also improve Neeman’s angle of view to the Grothendieck duality for derived categories of modules from the case of left Noether and right coherent rings such that all flat left modules have finite projective dimension to the case of left and right coherent rings.