Properties of G-atoms and full Galois covering reduction to stabilizers

Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra ${\displaystyle En{d}_R\left(B\right)}$ of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective ${\displaystyle En{d}_R\left(B\right)}$-module ${\displaystyle {\left(En{d}_R\left(B\right)\right)}^{\cdot }}$ is indecomposable (Corollary 3.1) and the socle of the tensor product functor ${\displaystyle -{\otimes }_R{B}^{\cdot }}$ is simple (Theorem 4.4). The problem when the Galois covering reduction to stabilizers with respect to a set U of periodic G-atoms (defined by the functors ${\displaystyle {\Phi }^U:co\prod _{B\in U}modk{G}_B\to mod\left(\frac{R}{G}\right)}$ and ${\displaystyle {\Psi }^U:mod\left(\frac{R}{G}\right)\to \prod _{B\in U}modk{G}_B}$)is full (resp. strictly full) is studied (see Theorems A, B and 6.3).