Exalcomm
In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcommk(R,M) are isomorphism classes of commutative k-algebras E with a homomorphism onto the k-algebra R whose kernel is the R-module M (with all pairs of elements in M having product 0). There are similar functors Exal and Exan for non-commutative rings and algebras, and functors Exaltop, Exantop. and Exalcotop that take a topology into account.
"Exalcomm" is an abbreviation for "COMMutative ALgebra EXtension" (or rather for the corresponding French phrase). It was introduced by Grothendieck (1964, 18.4.2) .
Exalcomm is one of the André–Quillen cohomology groups and one of the Lichtenbaum–Schlessinger functors.
Given homomorphisms of commutative rings A → B → C and a C-module L there is an exact sequence of A-modules (Grothendieck 1964, 20.2.3.1)
where DerA(B,L) is the module of derivations of the A-algebra B with values in L. This sequence can be extended further to the right using André–Quillen cohomology.
References
- Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20: 65. doi:10.1007/bf02684747. MR 0173675.
- Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, ISBN 978-0-521-43500-0, ISBN 978-0-521-55987-4, MR1269324