Exal
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). 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 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 & Dieudonné 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 {\begin{aligned}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)\end{aligned}}}
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.
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