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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)* * * * 0 * → * * Der * * B * * * * ( * C * , * L * ) * → * * Der * * A * * * * ( * C * , * L * ) * → * * Der * * A * * * * ( * B * , * L * ) * → * * Exalcomm * * B * * * * ( * C * , * L * ) * → * * Exalcomm * * A * * * * ( * C * , * L * ) * → * * Exalcomm * * A * * * * ( * B * , * L * ) * , * * * {\displaystyle 0\rightarrow \operatorname {Der} _{B}(C,L)\rightarrow \operatorname {Der} _{A}(C,L)\rightarrow \operatorname {Der} _{A}(B,L)\rightarrow \operatorname {Exalcomm} _{B}(C,L)\rightarrow \operatorname {Exalcomm} _{A}(C,L)\rightarrow \operatorname {Exalcomm} _{A}(B,L),} * 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. |