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1
Content available remote

Non-orbicular modules for Galois coverings

100%
Dowbor P.
Colloquium Mathematicum
|
2001
|
tom 89
|
nr 2
241-310
EN
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 $Φ^{B} : Iₙ - spr(H) → mod(R/G)$, 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 stabilizer, then usually the subcategory of non-orbicular indecomposables in $mod_{B̃,B}(R/G)$ is wild (Theorem 4.1, also 4.5). The analogous problem for the case of different stabilizers is discussed in Theorem 5.5. It is also shown that if R is tame then B̃ ≃ B for any infinite G-atom B with $End_{R}(B)/J(End_{R}(B)) ≃ k$ (Theorem 7.1). For this purpose the techniques of neighbourhoods (Theorem 7.2) and extension embeddings for matrix rings (Theorem 6.3) are developed.
2
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The pure-projective ideal of a module category

100%
Dowbor P.
Colloquium Mathematicum
|
1996
|
tom 71
|
nr 2
203-215
3
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On the category of modules of second kind for Galois coverings

100%
Dowbor P.
Fundamenta Mathematicae
|
1996
|
tom 149
|
nr 1
31-54
EN
Let F: R → R/G be a Galois covering and $mod_1(R/G)$ (resp. $mod_2(R/G)$) 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 $(mod (R/G))/[mod_1(R/G)]$ and $mod_2(R/G)$ is given in terms of representation categories of stabilizers of weakly-G-periodic modules for some class of coverings.
4
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Properties of G-atoms and full Galois covering reduction to stabilizers

100%
Dowbor P.
Colloquium Mathematicum
|
2000
|
tom 83
|
nr 2
231-265
EN
Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra $End_R (B)$ of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective $End_R (B)$-module $(End_R (B))^*$ is indecomposable (Corollary 3.1) and the socle of the tensor product functor $- ⊗_R B^*$ 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 $Φ^U: \coprod_{B ∈ U} mod kG_B → mod(R/G)$ and $Ψ^U: mod(R/G) → \prod_{B ∈ U} mod kG_B$)is full (resp. strictly full) is studied (see Theorems A, B and 6.3).
5
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Galois coverings and splitting properties of the ideal generated by halflines

100%
Dowbor P.
Colloquium Mathematicum
|
2004
|
tom 101
|
nr 2
237-257
EN
Given a locally bounded k-category R and a group $G ⊆ Aut_{k}(R)$ 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).
6
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On Bernstein-Gelfand-Gelfand equivalence and tilting theory

75%
Dowbor P.
Banach Center Publications
|
1990
|
tom 26
|
nr 1
273-285
7
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The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach

64%
Dowbor P., Mróz A.
Colloquium Mathematicum
|
2007
|
tom 107
|
nr 2
221-261
EN
Given a module M over an algebra Λ and a complete set 𝓧 of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector $m(M) = (m_X)_{X∈ 𝓧} ∈ ℕ ^{𝓧}$ such that $M ≅ ⨁ _{X∈ 𝓧} X^{m_X}$ 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 $𝔸̃_{p,q}$. In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).
8
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On a separation of orbits in the module variety for domestic canonical algebras

64%
Dowbor P., Mróz A.
Colloquium Mathematicum
|
2008
|
tom 111
|
nr 2
283-295
EN
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.
9
Content available remote

The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras

64%
Dowbor P., Mróz A.
Colloquium Mathematicum
|
2008
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tom 111
|
nr 2
221-282
EN
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 $m(M) = (m_{x})_{x∈X} ∈ ℕ^{X}$ such that $M ≅ ⨁_{x∈X} X_{x}^{m_{x}}$ is studied. A precise formula for $dim_{k} Hom_{Λ}(M,X)$, 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. Math. 107 (2007) leads to algorithms with the complexity $𝒪((dim_{k} M)⁴)$. A precise description of algorithms determining the multiplicities $m(M)_{x}$ for postprojective roots x ∈ X is given (Algorithms 6.1, 6.2 and 6.3).
10
Content available remote

Two classes of almost Galois coverings for algebras

64%
Dowbor P., Hajduk A.
Colloquium Mathematicum
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2012
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tom 127
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nr 2
253-298
EN
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 $G = Π(Γ_A)$ is the fundamental group of $Γ_A$. 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 much more regularly than usual with respect to the standard properties of the pair $(F_λ, F_•)$ of adjoint functors associated to F.
11
Content available remote

On indecomposable representations of quivers with zero-relations

51%
Dembiński M., Dowbor P., Skowroński A.
Fundamenta Mathematicae
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1988
|
tom 130
|
nr 3
169-180
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