Resolving by a free action linear category and applications to Hochschild-Mitchell (co)homology

Abstract

Let G be a group acting on a small category C over a field k, that is C is a G-k-category. We first obtain an unexpected result: C is resolvable by a category which is G-k-equivalent to it, on which G acts freely on objects. This resolving category enables to show that if the coinvariants and the invariants functors are exact, then the coinvariants and invariants of the Hochschild-Mitchell (co)homology of C are isomorphic to the trivial component of the Hochschild-Mitchell (co)homology of the skew category C[G]. If the action of G is free on objects, then there is a canonical decomposition of the Hochschild-Mitchell (co)homology of the quotient category C/G along the conjugacy classes of G. This way we provide a general frame for monomorphisms which have been described previously in low degrees.