The Kiev algorithm for bocses applies directly to representation-finite algebras (Q1262378)

Let K be an algebraically closed field. We recall that a pair \({\mathcal A}=(A,V)\) is a bocs if A is a skeletally small K-category and V is an A- coalgebra. The category R(\({\mathcal A})\) of representations of \({\mathcal A}\) has as objects the functors \(A^{op}\to mod(K)\), and if F, G are in R(\({\mathcal A})\), then a morphism from F to G is an A-module map \(V\otimes_ AG\to F\). The composition is defined in a natural way. In [\textit{W. W. Crawley-Boevey}, Proc. Lond. Math. Soc., III. Ser. 56, 451- 483 (1988; Zbl 0661.16026)] the notion of a layered bocs is introduced and some reductions for representations of layered bocses are presented. They are applied in the proof of Drozd's tame-wild Theorem. In the reviewed paper the reductions are extended to a more general class of bocses. Given an ideal I of A there is an induced bocs \({\mathcal A}_ I=(A/I,V')\) such that \(R({\mathcal A}_ I)\) consists of objects of \(R({\mathcal A})\) vanishing on I. The main result of the paper is the following. Let A be a layered bocs and let \(I\subset A\) be an ideal such that I is compatible with the grouplike of \({\mathcal A}\) and \({\mathcal A}_ I\) is representation- finite. Then there is a trivial category B and a functor \(h: A\to B\) such that the induced bocs \({\mathcal A}^ h\) is trivial and h induces an equivalence \((R({\mathcal A}^ h)_{I_ 0})\cong R({\mathcal A}_ I)\). Applying this to the principal bocs \({\mathcal A}=(\Lambda,\Lambda)\) associated with a representation-finite K-algebra \(\Lambda\) one gets a trivial bocs \({\mathcal A}'\) such that mod(\(\Lambda)\cong R({\mathcal A}')\).