nLab infinitesimal extension

Idea

An infinitesimal extension of a ring $R$ is quotient map $\widehat R \to R$ whose kernel is a nilpotent ideal, hence a nilpotent ring extension.

(Only if the map $\widehat R \to R$ is finite, such as for Weil algebras, then this may be cast as “epimorphism with nilpotent kernel”, by Stacks Project lemma 10.106.6.)

If already the product of any two elements in the kernel is zero, this is also called a square-zero extension. See also at tangent category and at Mod for more on this.

Examples

For a Noetherian ring $R$ (for instance a finitely generated ring), the projection from $R$ to its reduced ring $R_{red}$ is an example of an infinitesimal extension. In general, however, the kernel of $R\to R_{red}$ is not nilpotent.

Related concepts

• infinitesimal object, infinitesimally thickened point, Artin algebra

• deformation theory

• module, tangent category

• formally étale morphism

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

References

A category of infinitesimal extensions regarded as a site over which to characterize formally etale morphisms by means of an infinitesimal shape modality is considered in

• Maxim Kontsevich, Alexander Rosenberg, section 2.6 of Noncommutative spaces, preprint MPI-2004-35 (pdf, ps, dvi)

Discussion in the context of simplicial algebras is in

• Mauro Porta, Gabriele Vezzosi, Infinitesimal and square-zero extensions of simplicial algebras (arXiv:1310.3573)

Further discussion in the context of higher algebra is in

• Jacob Lurie, Deformation Theory

Infinitesimal neighborhoods of subvarieties play role in completion of subvarieties and various definitions of formal schemes, the ring case of course plays a role, cf.

• J. Illusie, Grothendieck theorem FGA explained