On the relative homology of cleft extensions of rings and abelian categories (Q1577504)

Let \(\Gamma\) be an associative ring. A cleft extension of \(\Gamma\) is a ring \(\Lambda\) together with ring homomorphisms \(\varepsilon :\Lambda\to\Gamma\), \(\mu:\Gamma\to\Lambda\) so that \(\mu\varepsilon =\text{Id}_\Gamma\). Such an extension can also be described as an extension of \(\Gamma\) by a \(\Gamma\)-\(\Gamma\)-bimodule \(M\) that has an associative bimodule homomorphism \(\vartheta: M\otimes_\Gamma M\to M\). As the author shows, cleft extensions encompass a wide variety of interesting rings. The properties of the category of modules over a cleft extension are abstracted to obtain a definition of cleft extensions of abelian categories. Such categories provide a natural setting for relative homology. The approach is two track: first, results are proved in the categorical set-up, and then translated for rings. Thus many diverse problems can be attacked in one framework. Topics covered include the following. Trivial extensions; triangular matrix rings, giving an exact formula for the global dimension; tilting and cotilting modules, leading to a proof of a conjecture of Auslander-Reiten, that the trivial extension of a Cohen-Macaulay Artin algebra by the dualizing bimodule is Gorenstein; perfect rings; free, symmetric, polynomial and exterior categories and rings; cleft extensions of small homological dimension.