nLab formal neighbourhood

Idea

A formal neighbourhood is a neighbourhood in formal geometry, hence an “infinitesimal neighbourhood”. Also called a formal completion or formal disk, see there for more.

Examples

• The formal spectrum of the completion of a commutative ring in the $I$-adic topology for a maximal ideal $I$ is the formal neighbourhood of the point corresponding to that ideal in the spectrum of a commutative ring.

• perturbative quantum field theory (in particular realized as formal deformation quantization) describes the infinitesimal neighbourhood of classical free field theory in the space of all quantum field theories.

Related concepts

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization