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

• Autorzy: Piotr Dowbor
• Języki publikacji: EN

Abstrakty

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).

Słowa kluczowe

EN

Galois covering; tame; locally finite-dimensional module

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2000
• Tom: 83
• Numer: 2
• Strony: 231-265

Daty

wydano

2000

otrzymano

1998-10-12

poprawiono

1999-01-12

poprawiono

1999-10-25

Twórcy

autor

Piotr Dowbor

• Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Bibliografia

• [1] I. Bucur and A. Deleanu, Introduction to the Theory of Categories and Functors, Wiley, 1968.
• [2] K. Bongartz and P. Gabriel, Covering spaces in representationtheory, Invent. Math. 65 (1982), 331-378.
• [3] P. Dowbor, On modules of the second kind for Galoiscoverings, Fund. Math. 149 (1996), 31-54.
• [4] P. Dowbor, Galois covering reduction to stabilizers, Bull. Polish Acad. Sci. Math. 44 (1996), 341-352.
• [5] P. Dowbor, The pure projective ideal of amodule category, Colloq. Math. 71 (1996), 203-214.
• [6] P. Dowbor, On stabilizers ofG-atoms of representation-tame categories, Bull. Polish Acad. Sci. Math. 46 (1998), 304-315.
• [7] P. Dowbor and S. Kasjan, Galois covering technique andnon-simply connected posets of polynomial growth, J. Pure Appl. Algebra, to appear.
• [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.
• [9] P. Dowbor and A. Skowroński, On Galois coveringsof tame algebras, Arch. Math. (Basel) 44 (1985), 522-529.
• [10] P. Dowbor and A. Skowroński, Galois coverings ofrepresentation-infinite algebras, Comment. Math. Helv. 62 (1987), 311-337.
• [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).
• [12] G P. Gabriel, The universal cover of a representation-finitealgebra, in: Lecture Notes in Math. 903, Springer, 1982, 68-105.
• [13] C. Geiss and J. A. de la Pe na, An interesting family of algebras, Arch. Math. (Basel) 60 (1993), 25-35.
• [14] E. L. Green, Group-graded algebras and the zero relation problem, in: Lecture Notes in Math. 903, Springer, 1982, 106-115.
• [15] C. U. Jensen and H. Lenzing, Model Theoretic Algebra, Gordon and Breach, 1989.
• [16] M B. Mitchell, Rings with several objects,Adv. Math. 8 (1972), 1-162.
• [17] P Z. Pogorzały, Regularly biserial algebras, in: Topics in Algebra, Banach CenterPubl. 26, Part 2, PWN, Warszawa, 1990, 371-384.
• [18] R C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199-224.
• [19] D. Simson, Socle reduction and socle projective modules,J. Algebra 108 (1986), 18-68.
• [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.
• [21] D. Simson, Right peak algebras of two-separate stratifiedposets, their Galois coverings and socle projective modules, Comm. Algebra 20 (1992), 3541-3591.
• [22] D. Simson, On representation typesof module categories and orders, Bull. Polish Acad. Sci. Math. 41 (1993), 77-93.
• [23] A. Skowroński, Selfinjective algebras of polynomialgrowth, Math. Ann. 285 (1989) 177-193.
• [24] A. Skowroński, Criteria for polynomialgrowth of algebras, Bull. Polish Acad. Sci. Math. 42 (1994), 173-183.
• [25] A. Skowroński, Tame algebras with strongly simply connectedGalois coverings, Colloq. Math. 72 (1997), 335-351.