Colocalization and cotilting for commutative noetherian rings

For a commutative noetherian ring R  , we investigate relations between tilting and cotilting modules in Mod–RMod–R and Mod–RmMod–Rm, where mm runs over the maximal spectrum of R  . For each n<ωn<ω, we construct a 1–11–1 correspondence between (equivalence classes of) n-cotilting R-modules C   and (equivalence classes of) compatible families FF of n  -cotilting RmRm-modules (m∈mSpec(R)m∈mSpec(R)). It is induced by the assignment C↦(Cm|m∈mSpec(R))C↦(Cm|m∈mSpec(R)), where Cm=HomR(Rm,C)Cm=HomR(Rm,C) is the colocalization of C   at mm, and its inverse F↦∏F∈FFF↦∏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.