K-injective complexes in commutative algebra

Abstract

It is an unfortunate fact that even over commutative noetherian rings, localization of K-injective complexes need not be K-injective. However, not all is lost. Given a complex of injective modules $J$ over a commutative noetherian ring $A$, we show that K-injectivity of $J$ can be detected locally: if $(A,\mathfrak{m})$ is a noetherian local ring, then $J$ is K-injective over $A$ if and only if $\operatorname{Hom}_A(\widehat{A},J)$ is K-injective over $\widehat{A}$, the $\mathfrak{m}$-adic completion of $A$; and for an arbitrary commutative noetherian ring $A$, $J$ is K-injective over $A$ if and only if $\operatorname{Hom}_A(A_{\mathfrak{m}},J)$ is K-injective over $A_{\mathfrak{m}}$ for every maximal ideal $\mathfrak{m}$ in the singular locus of $A$. The local result is deduced from a more general theorem on faithfully flat descent of K-injectivity along maps to (possibly noncommutative, possibly non-noetherian) rings. The global result is established via the local-to-global principle in tensor-triangulated geometry. As an application, we generalize a theorem of Grothendieck on faithfully flat descent of regularity, extending it to faithfully flat extensions by noncommutative left coherent left regular rings.