Separable Algebra - Separable Extensions For Noncommutative Rings

Separable Extensions For Noncommutative Rings

Let R be an associative ring with unit 1, and S a subring of R containing 1. Notice that an R-R-bimodule (see module theory and homological algebra) restricts to an S-S-bimodule. The ring extension R over S is said to be a separable extension if all short exact sequences of R-R-bimodules that are split as R-S-bimodules also split as R-R-bimodules. For example, the multiplication mapping m : given by is an R-R-bimodule epimorphism, which is split as an R-S-bimodule epi by the right inverse mapping given by . If R is a separable extension over S, then the multiplication mapping is split as an R-R-bimodule epi, so there is a right inverse s of m satisfying for s(1) := e, re = er for all r in R, and m(e) = 1. Conversely, if such an element (called a separability element in the tensor-square) exists, one shows by a judicious use of this element (like Maschke, applying its components within and without the splitting maps) that R is a separable extension of S.

Equivalently, the relative Hochschild cohomology groups of (R,S) in any coefficient bimodule M is zero for n > 0. Examples of separable extensions are many including first separable algebras where R = separable algebra and S = 1 times the ground field. More interestingly, any ring R with elements a and b satisfying ab = 1, but ba different from 1, is a separable extension over the subring S generated by 1 and bRa.

An interesting theorem in the area is that of J. Cuadra that a separable Hopf-Galois extension R | S has finitely generated natural S-module R. A fundamental fact about a separable extension R | S is that it is left or right semisimple extension: a short exact sequence of left or right R-modules that is split as S-modules, is split as R-modules. In terms of G. Hochschild's relative homological algebra, one says that all R-modules are relative (R,S)-projective. Usually relative properties of subrings or ring extensions, such as the notion of separable extension, serve to promote theorems that say that the over-ring shares a property of the subring. For example, a separable extension R of a semisimple algebra S has R semisimple, which follows from the preceding discussion.

There is the celebrated Jans theorem that a finite group algebra A over a field of characteristic p is of finite representation type if and only if its Sylow p-subgroup is cyclic: the clearest proof is to note this fact for p-groups, then note that the group algebra is a separable extension of its Sylow p-subgroup algebra B as the index is coprime to the characteristic. The separability condition above will imply every finitely generated A-module M is isomorphic to a direct summand in its restricted, induced module. But if B has finite representation type, the restricted module is uniquely a direct sum of multiples of finitely many indecomposables, which induce to a finite number of constituent indecomposable modules of which M is a direct sum. Hence A is of finite representation type if B is. The converse if proven by a similar argument noting that every subgroup algebra B is a B-bimodule direct summand of a group algebra A.