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.