infinitesimal extension



Formal geometry



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

(Only if the map R^R\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.


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

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization


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

Discussion in the context of simplicial algebras is in

Further discussion in the context of higher algebra is in

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.

Last revised on June 14, 2017 at 10:27:59. See the history of this page for a list of all contributions to it.