symmetric monoidal (∞,1)-category of spectra
An infinitesimal extension of a ring is quotient map whose kernel is a nilpotent ideal, hence a nilpotent ring extension.
(Only if the map 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 (for instance a finitely generated ring), the projection from to its reduced ring is an example of an infinitesimal extension. In general, however, the kernel of is not nilpotent.
Examples of sequences of local structures
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 March 27, 2020 at 08:24:40. See the history of this page for a list of all contributions to it.