##### Wikipedia

**Exalcomm**

In algebra, **Exalcomm** is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcomm(*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 .

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

0 → Der(*C*, *L*) → Der(*C*, *L*) → Der(*B*, *L*) → Exalcomm(*C*, *L*) → Exalcomm(*C*, *L*) → Exalcomm(*B*, *L*),

where Der(*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.