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

Dowbor, Piotr

Dowbor, Piotr

Colloquium Mathematicae, Tome 84/85 (2000), p. 231-265 / Harvested from The Polish Digital Mathematics Library

Résumé

Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra $E n d_{R} (B)$ of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective $E n d_{R} (B)$ -module ${(E n d_{R} (B))}^{*}$ 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 $Φ^{U} : ∐_{B \in U} m o d k G_{B} \to m o d (R / G)$ and $Ψ^{U} : m o d (R / G) \to \prod_{B \in U} m o d k G_{B}$ )is full (resp. strictly full) is studied (see Theorems A, B and 6.3).

Publié le : 2000-01-01

Zbl 1012.16017

EUDML-ID : urn:eudml:doc:210784

@article{bwmeta1.element.bwnjournal-article-cmv83i2p231bwm,
     author = {Piotr Dowbor},
     title = {Properties of G-atoms and full Galois covering reduction to stabilizers},
     journal = {Colloquium Mathematicae},
     volume = {84/85},
     year = {2000},
     pages = {231-265},
     zbl = {1012.16017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv83i2p231bwm}
}

Dowbor, Piotr. Properties of G-atoms and full Galois covering reduction to stabilizers. Colloquium Mathematicae, Tome 84/85 (2000) pp. 231-265. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv83i2p231bwm/

Bibliographie

[000] [1] I. Bucur and A. Deleanu, Introduction to the Theory of Categories and Functors, Wiley, 1968.

[001] [2] K. Bongartz and P. Gabriel, Covering spaces in representationtheory, Invent. Math. 65 (1982), 331-378. | Zbl 0482.16026

[002] [3] P. Dowbor, On modules of the second kind for Galoiscoverings, Fund. Math. 149 (1996), 31-54. | Zbl 0855.16018

[003] [4] P. Dowbor, Galois covering reduction to stabilizers, Bull. Polish Acad. Sci. Math. 44 (1996), 341-352. | Zbl 0889.16003

[004] [5] P. Dowbor, The pure projective ideal of amodule category, Colloq. Math. 71 (1996), 203-214. | Zbl 0880.16009

[005] [6] P. Dowbor, On stabilizers ofG-atoms of representation-tame categories, Bull. Polish Acad. Sci. Math. 46 (1998), 304-315. | Zbl 0921.16008

[006] [7] P. Dowbor and S. Kasjan, Galois covering technique andnon-simply connected posets of polynomial growth, J. Pure Appl. Algebra, to appear. | Zbl 0998.16010

[007] [8] P. Dowbor, H. Lenzing and A. Skowroński, Galois coveringsof algebras by locally support-finite categories, in: Lecture Notes in Math. 1177, Springer, 1986, 91-93. | Zbl 0626.16009

[008] [9] P. Dowbor and A. Skowroński, On Galois coveringsof tame algebras, Arch. Math. (Basel) 44 (1985), 522-529. | Zbl 0576.16029

[009] [10] P. Dowbor and A. Skowroński, Galois coverings ofrepresentation-infinite algebras, Comment. Math. Helv. 62 (1987), 311-337. | Zbl 0628.16019

[010] [11] Yu. A. Drozd, S. A. Ovsienko and B. Yu. Furchin, Categorical construction in representation theory, in:Algebraic Structures and their Applications, University of Kiev, Kiev UMK VO, 1988, 43-73 (in Russian).

[011] [12] G P. Gabriel, The universal cover of a representation-finitealgebra, in: Lecture Notes in Math. 903, Springer, 1982, 68-105.

[012] [13] C. Geiss and J. A. de la Pe na, An interesting family of algebras, Arch. Math. (Basel) 60 (1993), 25-35. | Zbl 0821.16013

[013] [14] E. L. Green, Group-graded algebras and the zero relation problem, in: Lecture Notes in Math. 903, Springer, 1982, 106-115.

[014] [15] C. U. Jensen and H. Lenzing, Model Theoretic Algebra, Gordon and Breach, 1989.

[015] [16] M B. Mitchell, Rings with several objects,Adv. Math. 8 (1972), 1-162. | Zbl 0232.18009

[016] [17] P Z. Pogorzały, Regularly biserial algebras, in: Topics in Algebra, Banach CenterPubl. 26, Part 2, PWN, Warszawa, 1990, 371-384.

[017] [18] R C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199-224.

[018] [19] D. Simson, Socle reduction and socle projective modules,J. Algebra 108 (1986), 18-68. | Zbl 0599.16014

[019] [20] D. Simson, Representations of bounded stratified posets,coverings and socle projective modules, in: Topics in Algebra, Banach CenterPubl. 26, Part 2, PWN, Warszawa, 1990, 499-533.

[020] [21] D. Simson, Right peak algebras of two-separate stratifiedposets, their Galois coverings and socle projective modules, Comm. Algebra 20 (1992), 3541-3591. | Zbl 0791.16011

[021] [22] D. Simson, On representation typesof module categories and orders, Bull. Polish Acad. Sci. Math. 41 (1993), 77-93. | Zbl 0805.16011

[022] [23] A. Skowroński, Selfinjective algebras of polynomialgrowth, Math. Ann. 285 (1989) 177-193. | Zbl 0653.16021

[023] [24] A. Skowroński, Criteria for polynomialgrowth of algebras, Bull. Polish Acad. Sci. Math. 42 (1994), 173-183. | Zbl 0865.16011

[024] [25] A. Skowroński, Tame algebras with strongly simply connectedGalois coverings, Colloq. Math. 72 (1997), 335-351. | Zbl 0876.16007