Exalcomm

In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcomm_k(R,M) are isomorphism classes of commutative k-algebrasE with a homomorphismonto the k-algebra R whose kernel is the R-module M (with all pairs of elements in M having product 0). Note that some authors use Exal as the same functor. 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 & Dieudonné (1964, 18.4.2).

Exalcomm is one of the André–Quillen cohomologygroups and one of the Lichtenbaum–Schlessinger functors.

Given homomorphisms of commutative ringsA → B → C and a C-module L there is an exact sequence of A-modules (Grothendieck & Dieudonné 1964, 20.2.3.1)

${\displaystyle {\begin{aligned}0 ightarrow \;&\operatorname {Der} _{B}(C,L) ightarrow \operatorname {Der} _{A}(C,L) ightarrow \operatorname {Der} _{A}(B,L) ightarrow \\&\operatorname {Exalcomm} _{B}(C,L) ightarrow \operatorname {Exalcomm} _{A}(C,L) ightarrow \operatorname {Exalcomm} _{A}(B,L)\end{aligned}}}$

where Der_A(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.