Integrations and cohomology theory
This paper introduces integrations of one bimodule into another, a notion which extends that of derivation of a ring into a bimodule. Let \(\Lambda\) be a K-ring, i.e. a homomorphism \(i: K\to \Lambda\) of two associative rings with a unit. Let \(\Gamma\) be an associative ring. Given \(\Lambda\)- \(\Gamma\)-bimodules M and N, a left K-integration of M into N is a \(\Lambda\)-\(\Gamma\)-linear map \(f: \Lambda\) \(\otimes_ KM\to N\) satisfying \(f(\lambda_ 1\lambda_ 2x)=\lambda_ 1f(\lambda_ 2\otimes x)+f(\lambda_ 1\otimes \lambda_ 2x)\) for \(\lambda_ 1,\lambda_ 2\in \Lambda\), \(x\in M.\) It is shown that the left K-integrations of M into N correspond to K- derivations \(\Lambda \to Hom_{\Gamma}(M,N)\) under the natural isomorphism \(Hom_{K-\Gamma}(\Lambda \otimes_ KM,N)\to Hom_{K- K}(\Lambda,Hom_{\Gamma}(M,N))\). Thus given an integration f, \(f(\lambda \otimes x)=D(\lambda)(x)\) for some derivation D. Further it is shown that the relative bifunctor Ext for two bimodules, when suitably renumbered is the derived functor of derivations. This extends a result of Barr and Rinehart. Relatively projective and injective modules have a nice characterization in terms of integrations: a bimodule is relatively projective (injective) if and only if its universal (couniversal) integration is inner.
Relatively projective and injective modules
scientific article; zbMATH DE number 3908631
scientific article