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Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra $En{d}_R\left(B\right)$ of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective $En{d}_R\left(B\right)$-module ${\left(En{d}_R\left(B\right)\right)}^{*}$ is indecomposable (Corollary 3.1) and the socle of the tensor product functor $-{\otimes }_R{B}^{*}$ 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 ${\Phi }^U:{\coprod }_{B\in U}modk{G}_B\to mod(R/G)$ and ${\Psi }^U:mod(R/G)\to {\prod }_{B\in U}modk{G}_B$)is full (resp. strictly full) is studied...

Let F: R → R/G be a Galois covering and $mo{d}_1(R/G)$ (resp. $mo{d}_2(R/G)$) be a full subcategory of the module category mod (R/G), consisting of all R/G-modules of first (resp. second) kind with respect to F. The structure of the categories $\left(mod(R/G)\right)/\left[mo{d}_1(R/G)\right]$ and $mo{d}_2(R/G)$ is given in terms of representation categories of stabilizers of weakly-G-periodic modules for some class of coverings.

Given a locally bounded k-category R and a group $G\subseteq Au{t}_k\left(R\right)$ acting freely on R we study the properties of the ideal generated by a class of indecomposable locally finite-dimensional modules called halflines (Theorem 3.3). They are applied to prove that under certain circumstances the Galois covering reduction to stabilizers, for the Galois covering F: R → R/G, is strictly full (Theorems 1.5 and 4.2).

Given a group G of k-linear automorphisms of a locally bounded k-category R, the problem of existence and construction of non-orbicular indecomposable R/G-modules is studied. For a suitable finite sequence B of G-atoms with a common stabilizer H, a representation embedding ${\Phi }^B:Iₙ-spr\left(H\right)\to mod(R/G)$, which yields large families of non-orbicular indecomposable R/G-modules, is constructed (Theorem 3.1). It is proved that if a G-atom B with infinite cyclic stabilizer admits a non-trivial left Kan extension B̃ with the same...

Given a pair M,M' of finite-dimensional modules over a domestic canonical algebra Λ, we give a fully verifiable criterion, in terms of a finite set of simple linear algebra invariants, deciding if M and M' lie in the same orbit in the module variety, or equivalently, if M and M' are isomorphic.

Given a module M over an algebra Λ and a complete set of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector $m\left(M\right)={\left(m_X\right)}_{X\in }\in {\mathbb{N}}^{}$ such that $M\cong {⨁}_{X\in }{X}^{m_X}$ is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type ${̃}_{p,q}$. In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).

We prove that for any representation-finite algebra A (in fact, finite locally bounded k-category), the universal covering F: Ã → A is either a Galois covering or an almost Galois covering of integral type, and F admits a degeneration to the standard Galois covering F̅: Ã→ Ã/G, where $G=\Pi \left({\Gamma }_A\right)$ is the fundamental group of ${\Gamma }_A$. It is shown that the class of almost Galois coverings F: R → R’ of integral type, containing the series of examples from our earlier paper [Bol. Soc. Mat. Mexicana 17 (2011)], behaves...

Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m\left(M\right)={\left(m_x\right)}_{x\in X}\in {\mathbb{N}}^X$ such that $M\cong {⨁}_{x\in X}{X}_x^{m_x}$ is studied. A precise formula for $di{m}_{k}Ho{m}_{\Lambda }(M,X)$, for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors in Colloq....