symmetric monoidal (∞,1)-category of spectra
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.
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.
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 |
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.