Higman's criterion revisited.
The paper under review gives a self-contained introduction into certain aspects of the representation theory of symmetric algebras over a commutative ring \(R\). Recall that an associative \(R\)-algebra \(A\) is called symmetric if the \(A\)-\(A\)-bimodule \(A^*=\Hom_R(A,R)\) is isomorphic to \(A\). Important examples are the group algebras of finite groups and the Iwahori-Hecke algebras of finite Coxeter groups. In the representation theory of finite groups, Higman's criterion gives a useful characterization of projective and, more generally, relatively projective modules with respect to subgroups, in terms of a (relative) trace map. In the paper, this criterion is generalized to the setting of \(R\)-linear triangulated or Abelian categories \(\mathfrak A\) and \(\mathfrak B\). Suppose that \(M\colon\mathfrak B\to\mathfrak A\) and \(N\colon\mathfrak A\to\mathfrak B\) are exact functors forming a bi-adjoint pair. Then it is shown that an object \(X\) in \(\mathfrak A\) is isomorphic to a direct summand of \(MN(X)\) if and only if the identity morphism \(\text{id}_X\) is contained in the image of \(\text{Tr}^N_M(X)\). Here \(\text{Tr}^M_N(X)\colon\text{End}_{\mathfrak B}(NX)\to\text{End}_{\mathfrak A}(X)\) is a relative trace map defined in terms of the units and counits of the relevant adjunctions. The author illustrates his result by several applications, having to do with Casimir and Schur elements, with parabolic subalgebras and Heller operators, with blocks of finite groups and with stable equivalences of Morita type. Generalizations of Schanuel's Lemma and Shapiro's Lemma are also presented.
scientific article; zbMATH DE number 5566076
scientific article