nLab adic residual

Idea

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

Properties

Explicit characterization

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

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 modality	symbol	interpretation
flat modality	$\flat$	formal completion at
shape modality	$ʃ$	torsion approximation
dR-shape modality	$ʃ_{dR}$	localization away
dR-flat modality	$\flat_{dR}$	adic residual

the differential cohomology hexagon/arithmetic fracture squares:

$$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$$

References