Journal of the Mathematical Society of Japan

A construction of non-regularly orbicular modules for Galois coverings

Piotr DOWBOR

Full-text: Open access

Abstract

For a given finite dimensional k -algebra A which admits a presentation in the form R / G , where G is an infinite group of k -linear automorphisms of a locally bounded k -category R , a class of modules lying out of the image of the "push-down" functor associated with the Galois covering R R / G , is studied. Namely, the problem of existence and construction of the so called non-regularly orbicular indecomposable R / G -modules is discussed. For a G -atom B (with a stabilizer G B ), whose endomorphism algebra has a suitable structure,a representation embedding Φ B ( f , s ) | : I n - s p r l ( s ) ( k G B ) m o d ( R / G ) , which yields large families of non-regularly orbicular indecomposable R / G -modules,is constructed (Theorem 2.2). An important role in consideration is played by a result interpreting some class of R / G -modules in terms of Cohen-Macaulay modules over certain skew grup algebra (Theorem 3.3). Also, Theorems 4.5 and 5.4, adapting the generalized tensor product construction and Galois covering scheme, respectively, for Cohen-Macaulay modules context, are proved and intensively used.

Article information

Source
J. Math. Soc. Japan, Volume 57, Number 4 (2005), 1077-1127.

Dates
First available in Project Euclid: 14 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1150287305

Digital Object Identifier
doi:10.2969/jmsj/1150287305

Mathematical Reviews number (MathSciNet)
MR2183585

Zentralblatt MATH identifier
1121.16013

Subjects
Primary: 16G60: Representation type (finite, tame, wild, etc.)

Keywords
Galois covering Cohen-Macaulay module generalized tensor product

Citation

DOWBOR, Piotr. A construction of non-regularly orbicular modules for Galois coverings. J. Math. Soc. Japan 57 (2005), no. 4, 1077--1127. doi:10.2969/jmsj/1150287305. https://projecteuclid.org/euclid.jmsj/1150287305


Export citation

References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Grad. Texts in Math., 13, Springer, 1992.
  • M. Auslander and I. Reiten, The Cohen-Macaulay type for Cohen-Macaulay rings, Adv. Math., 73 (1989), 1–23.
  • K. Bongartz and P. Gabriel, Covering spaces in representation theory, Invent. Math., 65 (1982), 331–378.
  • P. Dowbor, On modules of the second kind for Galois coverings, Fund. Math., 149 (1996), 31–54.
  • P. Dowbor, Galois covering reduction to stabilizers, Bull. Polish Acad. Sci. Math., 44 (1996), 341–352.
  • P. Dowbor, The pure projective ideal of a module category, Colloq. Math., 71 (1996), 203–214.
  • P. Dowbor, Properties of $G$-atoms and full Galois covering reduction to stabilizers, Colloq. Math., 83 (2000), 231–265.
  • P. Dowbor, Stabilizer conjecture for representation-tame Galois coverings of algebras, J. Algebra, 239 (2001), 112–149.
  • P. Dowbor, Non-orbicular modules for Galois coverings, Colloq. Math., 89 (2001), 241–310.
  • P. Dowbor and S. Kasjan, Galois covering technique and non-simply connected posets of polynomial growth, J. Pure Appl. Algebra, 147 (2000), 1–24.
  • P. Dowbor, H. Lenzing and A. Skowroński, Galois coverings of algebras by locally support-finite categories, Lecture Notes in Math., 1177, Springer, 1986, 91–93.
  • P. Dowbor and A. Skowroński, On Galois coverings of tame algebras, Arch. Math. (Basel), 44 (1985), 522–529.
  • P. Dowbor and A. Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv., 62 (1987), 311–337.
  • Yu. A. Drozd and S. A. Ovsienko, Coverings of tame boxes, preprint, Max-Planck Institute, Bonn, 2000, p.,43.
  • Yu. A. Drozd, S. A. Ovsienko and B. Yu. Furcin, Categorical constructions in representation theory, In: Algebraic Structures and their Applications, University of Kiev, Kiev UMK VO, 1988, 43–73.
  • K. Erdmann, Algebras and quaternion defect groups I, Math. Ann., 281 (1988), 545–560.
  • K. Erdmann, Algebras and quaternion defect groups II, Math. Ann., 281 (1988), 561–582.
  • K. Erdmann, On a class of tame symmetric algebras having only periodic modules, In: Topics in algebra, Banach Center Publ., 26, part 1, PWN, Warszawa, 1990, 287–302.
  • P. Gabriel, The universal cover of a representation-finite algebra, Lecture Notes in Math., 903, Springer, 1981, 68–105.
  • Ch. Geiss and J. A. de la Peña, An interesting family of algebras, Arch. Math., 60 (1993), 25–35.
  • E. L. Green, Group-graded algebras and the zero relation problem, Lecture Notes in Math., 903, Springer, 1981, 106–115.
  • Z. Leszczyński and A. Skowroński, Tame triangular matrix algebras, Colloq. Math., 86 (2000), 259–303.
  • S. Mac Lane, Categories for the Working Mathematician, Grad. Texts in Math., 5, Springer, 1971.
  • R. Martinez and J. A. de la Peña, Automrphisms of representation-finite algebras, Invent. Math., 72 (1983), 359–362.
  • B. Mitchell, Rings with several objects, Adv. Math., 8 (1972), 1–162.
  • I. Reiten and Ch. Riedtmann, Skew group algebras in the representation theory of artin algebras, J. Algebra, 92 (1985), 224–282.
  • Ch. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv., 55 (1980), 199–224.
  • D. Simson, Socle reduction and socle projective modules, J. Algebra, 108 (1986), 18–68.
  • D. Simson, Representations of bounded stratified posets, coverings and socle projective modules, In: Topics in Algebra, Banach Center Publ., 26, part 1, PWN, Warszawa, 1990, 499–533.
  • D. Simson, Right peak algebras of two-separate stratified posets, their Galois coverings and socle projective modules, Comm. Algebra, 20 (1992), 3541–3591.
  • D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, In: Algebra, Logic and Applications, 4, Gordon & Breach Science Publishers, 1992.
  • D. Simson, On representation types of module subcategories and orders, Bull. Polish Acad. Sci. Math., 41 (1993), 77–93.
  • D. Simson, Chain categories of modules and subprojective representations of posets over uniserial rings, Rocky Mountains J. Math., 33 (2003), 1627–1650.
  • A. Skowroński, Tame triangular matrix algebras over Nakayama algebras, J. London Math. Soc., 34 (1986), 245–264.
  • A. Skowroński, Selfinjective algebras of polynomial growth, Math. Ann., 285 (1989), 177–193.
  • A. Skowroński, Criteria for polynomial growth of algebras, Bull. Polish Acad. Sci. Math., 42 (1994), 173–183.
  • A. Skowroński, Tame algebras with strongly simply connected Galois coverings, Colloq. Math., 72 (1997), 335–351.
  • A. Skowroński and K. Yamagata, Galois coverings of selfinjective algebras by repetitive algebras, Trans. Amer. Math. Soc., 351 (1999), 715–734.
  • A. Skowroński and K. Yamagata, Stable equivalence of selfinjective algebras of tilted type, Arch. Math., 70 (1998), 341–350.
  • A. Skowroński and K. Yamagata, Socle deformations of self-injective algebras, Proc. Lond. Math. Soc., 72 (1996), 545–566.