SOBRIFICATION OF MODULES WITH I-ADIC TOPOLOGY

Abstract

Abstract. Let R be a commutative ring with identity and M be an Rmodule. For any ideal I of R, the I-adic sobrification of M denoted sIM, consists of the closure of elements of M for the I-adic topology on M. This article presents an algebraic theory for I-adic sobrification of modules. For this purpose, we show that sIR admits naturally a topological ring structure which can be embedded in the I-adic completion bRI of R. Moreover, sIM admits naturally a sIR-module structure, and in particular, sI can be viewed as an additive covariant functor from the category of R-modules to the category of sIR-modules. Considering I a sequential ideal (as a new type of ideal) of a Noetherian ring R, it is shown that sI is naturally isomorphic to sIR⊗− on finitely generated R-modules. We also study the left derived functors {LnsI}n∈N of sI , and show that if R is an Artinian ring, then LnsI (M) is isomorphic to Hn(Mic(Xi ⊗M)) as the n-th homology of the microscope complex Mic(Xi ⊗M) with Xi a free resolution of R/Ii.