Extension of an associative algebra

$ R $ over a commutative ring $ K $

A homomorphism $ \phi : S \rightarrow R $ of a $ K $- algebra $ S $ onto $ R $. If $ \mathop{\rm Ker} \phi = I $ is an algebra with zero multiplication, then the extension is called singular. In this case, $ I $ is an $ R $- module in a natural way. The set of all extensions of $ R $ with kernel $ I $ admits an equivalence relation (the same as for groups, modules, etc.), and the set of equivalence classes of extensions is denoted by $ F ( R, I) $. If the algebra $ R $ is $ K $- projective, then the algebra $ S $ splits into a direct sum of $ K $- modules, $ S = I + R $, and the elements of $ S $ can be written as pairs $ ( u , r) $, $ u \in I $, $ r \in R $, with multiplication given by

$$ ( u _ {1} , r _ {1} ) ( u _ {2} , r _ {2} ) = \ ( u _ {1} r _ {2} + r _ {1} u _ {2} + a ( r _ {1} , r _ {2} ),\ r _ {1} r _ {2} ), $$

where $ a: R \otimes R \rightarrow I $. The associativity of multiplication imposes restrictions on $ a $ that make it into a cocycle. Mapping an extension to its cocycle defines an isomorphism of $ K $- modules between $ F ( R, I) $ and the second cohomology group $ H ^ {2} ( R, I) $ of $ R $ with coefficients in $ I $.

In a completely different context, any algebra containing $ R $ is also called an extension of $ R $. Such extensions are often connected with a specific construction (polynomials over $ R $, localization of $ R $, ring of partial fractions of the algebra $ R $, etc.). See also Extension of a field.

References

[1]	S. MacLane, "Homology" , Springer (1963)
[2]	G. Hochschild, "On the cohomology groups of an associative algebra" Ann. of Math. , 46 (1945) pp. 58–67

Comments

The cohomology group $ H ^ {2} ( R , I ) $ is also known as the Hochschild cohomology (group) of $ R $ with values in $ I $.