Colocalization and cotilting for commutative noetherian rings

Abstract

For a commutative noetherian ring R, we investigate relations between tilting and cotilting modules in Mod–R and Mod–Rm, where m runs over the maximal spectrum of R. For each n<ω, we construct a 1–1 correspondence between (equivalence classes of) n-cotilting R-modules C and (equivalence classes of) compatible families F of n-cotilting Rm-modules (m∈mSpec(R)). It is induced by the assignment C↦(Cm|m∈mSpec(R)), where Cm=HomR(Rm,C) is the colocalization of C at m, and its inverse F↦∏F∈FF. We construct a similar correspondence for n-tilting modules using compatible families of localizations; however, there is no explicit formula for the inverse.