Localization in tame and wild coalgebras. (Q995593)

Let \(K\) be an~algebraically closed field and let \(Q\) be an~acyclic quiver (not necessarily finite). Denote by \(KQ\) the path coalgebra of \(Q\). For a quiver with relations \((Q,\Omega)\) denote by \(C(Q,\Omega)\) the path coalgebra of \((Q,\Omega)\) (i.e. \(C(Q,\Omega)=\{a\in KQ\); \(\langle a,\Omega\rangle=0\}\), where \(\langle -,-\rangle\colon KQ\times KQ\to K\) is the bilinear map defined by \(\langle v,w\rangle=\delta_{v,w}\) (Kronecker delta)). Let \(C'\) be an~admissible subcoalgebra of the path coalgebra \(KQ\) (i.e. \(C'\) contains a~subcoalgebra of \(KQ\) generated by all vertices and all arrows). The authors investigate the following problem. When is an~admissible coalgebra \(C'\) isomorphic to the path coalgebra \(C(Q,\Omega)\) of a~quiver with relations \((Q,\Omega)\)? The main aim of this paper is to prove the following Theorem. Let \(K\), \(Q\) and \(C'\) be as above. If \(C'\) is not the path coalgebra of a quiver with relations, then \(C'\) is of wild comodule type. As a~consequence the authors get that every tame admissible subcoalgebra \(C'\) of the path coalgebra \(KQ\) is isomorphic to the path coalgebra \(C(Q,\Omega)\) of a~quiver with relations \((Q,\Omega)\). In the proof of the main theorem the authors apply the theory of localization for tame and wild coalgebras.