nLab
adic residual

Contents

Idea

Forming the kernel of an adic completion map is sometimes called forming the adic residual, for instance the pp-residual for pp-adic completion.

Properties

Explicit characterization

For AA a commutative ring and 𝔞A\mathfrak{a} \subset A an ideal, and NN an AA-module, then the 𝔞\mathfrak{a}-adic residual of NN (hence the kernel of the map to the completion of a module NN 𝔞 N \longrightarrow N^\wedge_{\mathfrak{a}}) is equivalently the submodule of elements annihilated by 1+𝔞1 + \mathfrak{a}.

E.g. theorem 4.3.2. here: pdf

As a modality in arithmetic cohesion

For suitably well behaved ideals, forming the adic residual may be understood as the dR-flat modality in the cohesion of E-infinity arithmetic geometry:

cohesion in E-∞ arithmetic geometry:

cohesion modalitysymbolinterpretation
flat modality\flatformal completion at
shape modalityʃʃtorsion approximation
dR-shape modalityʃ dRʃ_{dR}localization away
dR-flat modality dR\flat_{dR}adic residual

the differential cohomology hexagon/arithmetic fracture squares:

localizationawayfrom𝔞 𝔞adicresidual Π 𝔞dR 𝔞X X Π 𝔞 𝔞dRX formalcompletionat𝔞 𝔞torsionapproximation, \array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,

References

Last revised on August 15, 2014 at 03:19:42. See the history of this page for a list of all contributions to it.