Abstract
We determine the Krull-Gabriel dimension of the cycle-finite categories of finitely generated modules over artin algebras and derive some consequences.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Assem, I., Skowronski, A.: Algebras with cycle-finite derived categories. Math. Ann. 280, 441–463 (1988)
Assem, I., Skowronski, A.: Minimal representation-infinite coil algebras. Manuscripta Math. 67, 305–331 (1990)
Auslander, M.: Representation theory of artin algebras II. Comm. Algebra. 1, 269–310 (1974)
Auslander, M.: A functorial approach to representation theory. In: Representations of Algebras, Lecture Notes in Math., vol. 944, pp 105–179. Springer, Berlin (1982)
Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math 86, 111–152 (1991)
Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. In: Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)
Baer, D.: Wild hereditary artin algebras and linear methods. Manuscripta Math. 55, 69–82 (1986)
Bautista, R.: On algebras of strongly unbounded representation type. Comment. Math. Helv. 60, 392–399 (1985)
Bialkowski, J., Skowronski, A.: Cycles of modules and finite representation type. Preprint (2015)
Bialkowski, J., Skowronski, A., Skowyrski, A., Wisniewski, P.: Cycle-finite algebras of semiregular type. Colloq. Math. 129, 211–247 (2012)
Bongartz, K.: Indecomposables are standard. Comment. Math. Helv. 60, 400–410 (1985)
Burke, K., Prest, M.: The Ziegler and Zariski spectra of some domestic string algebras. Algebr. Represent. Theory 5, 211–234 (2002)
Coelho, F.U., Marcos, E.N., Merklen, H.A., Skowronski, A.: Module categories with infinite radical square zero are of finite type. Comm. Algebra 22, 4511–4517 (1994)
Crawley-Boevey, W.W.: On tame algebras and bocses. Proc. London Math. Soc. 56, 451–483 (1988)
Crawley-Boevey, W.: Tame algebras and generic modules. Proc. London Math. Soc. 63, 241–265 (1991)
Dowbor, P., Skowronski, A.: Galois coverings of representation-infinite algebras. Comment. Math. Helv 62, 311–337 (1987)
Drozd, J.A.: Tame and wild matrix problems. In: Representation Theory II, Lecture Notes in Math., vol. 832, pp 242–258. Springer, Berlin (1980)
Faith, C.: Algebra: Rings, Modules and Categories I. Springer, Berlin (1973)
Farnsteiner, R., Skowronski, A.: Classification of restricted Lie algebras with tame principal block. J. Reine Angew. Math. 546, 1–45 (2002)
Farnsteiner, R., Skowronski, A.: Galois actions and blocks of tame infinitesimal group schemes. Trans. Amer. Math. Soc. 359, 5867–5898 (2007)
Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962)
Geigle, W.: The Krull-Gabriel dimension of the representation theory of a tame hereditary artin algebra and applications to the structure of exact sequences. Manuscripta Math. 54, 83–106 (1985)
Geigle, W.: Krull dimension and artin algebras. In: Representation Theory I, Lecture Notes in Math., vol. 1177, pp 135–155. Springer, Berlin (1986)
Kerner, O.: Tilting wild algebras. J. London Math. Soc. 39, 29–47 (1989)
Kerner, O., Skowronski, A.: On module categories with nilpotent infinite radical. Compositio Math. 77, 313–333 (1991)
Krause, H.: Generic modules over artin algebras. Proc. London Math. Soc. 76, 276–306 (1998)
Krause, H.: The spectrum of a module category. Mem. Amer. Math. Soc. 149 (707), No. 707 (2001)
Lenzing, H., de la Peña, J.A.: Concealed-canonical algebras and separating tubular families. Proc. London Math. Soc. 78, 513–540 (1999)
Liu, S.: Degrees of irreducible maps and the shapes of Auslander-Reiten quivers. J. London Math. Soc. 45, 32–54 (1992)
Liu, S.: Semi-stable components of an Auslander-Reiten quiver. J. London Math. Soc. 47, 405–416 (1993)
Malicki, P.: Krull dimension of tame generalized multicoil algebras. Algebr. Represent. Theory 18, 881–894 (2015)
Malicki, P., de la Peña, J.A., Skowronski, A.: Cycle-finite module categories. In: Algebras, Quivers and Representations, Abel Symposium, vol. 8, pp 209–252. Springer, Heidelberg (2013)
Malicki, P., de la Peña, J.A., Skowronski, A.: On the number of terms in the middle of almost split sequences over cycle-finite artin algebras. Cent. Eur. J. Math. 12, 39–45 (2014)
Malicki, P., de la Peña, J.A., Skowronski, A.: Finite cycles of indecomposable modules. J. Pure Appl. Algebra 219, 1761–1799 (2015)
Malicki, P., Skowronski, A.: Almost cyclic coherent components of an Auslander-Reiten quiver. J. Algebra 229, 695–749 (2000)
Malicki, P., Skowronski, A.: Algebras with separating almost cyclic coherent Auslander-Reiten components. J. Algebra 291, 208–237 (2005)
Malicki, P., Skowronski, A.: Cycle-finite algebras with finitely many t-rigid indecomposable modules. Comm. Algebra. In press (2015)
de la Peña, J.A., Skowronski, A.: Algebras with cycle-finite Galois coverings. Trans. Amer. Math. Soc. 363, 4309–4336 (2011)
Peng, L.G., Xiao, J.: On the number of DTr-orbits containing directing modules. Proc. Amer. Math. Soc. 118, 753–756 (1993)
Popescu, N.: Abelian Categories with Applications to Rings and Modules. Academic, London (1973). London Mathematical Society Monographs, No. 3
Prest, M.: Model Theory and Modules. London Mathematical Society Lecture Note Series, vol. 130. Cambridge University Press, Cambridge (1988)
Prest, M.: Topological and geometric aspects of the Ziegler spectrum. In: Infinite Length Modules, Trends Math., pp 369–392. Birkhäuser, Basel (2000)
Prest, M.: Purity, Spectra and Localization. Encyclopedia Mathematics and Applications, vol. 121. Cambridge University Press, Cambridge (2009)
Prest, M., Puninski, G.: Krull-Gabriel dimension of 1-domestic string algebras. Algebr. Represent. Theory 9, 337–358 (2006)
Puninski, G.: Krull-Gabriel dimension and Cantor-Bendixson rank of 1-domestic string algebras. Colloq. Math. 127, 185–211 (2012)
Reiten, I., Skowronski, A.: Characterizations of algebras with small homological dimensions. Adv. Math. 179, 122–154 (2003)
Reiten, I., Skowronski, A.: Generalized double tilted algebras. J. Math. Soc. Japan 56, 269–288 (2004)
Ringel, C.M.: Tame algebras. In: Representation Theory I, Lecture Notes in Math., vol. 831, pp 137–287. Springer, Berlin (1980)
Ringel, C.M.: Tame Algebras and Integral Quadratic Forms, Lecture Notes in Mathematics, vol. 1099. Springer, Berlin (1984)
Ringel, C.M.: The canonical algebras. With an appendix by William Crawley-Boevey. In: Topics in Algebra, Part 1, Banach Center Publ., vol. 26, pp 407–432. PWN, Warsaw (1990)
Schröer, J.: On the infinite radical of a module category. Proc. London Math. Soc. 81, 651–674 (2000)
Schröer, J.: On the Krull-Gabriel dimension of an algebra. Math. Z 233, 287–303 (2000)
Schröer, J.: The Krull-Gabriel dimension of an algebra—open problems and conjectures. In: Infinite Length Modules, Trends Math., pp 419–424. Birkhäuser, Basel (2000)
Skowronski, A.: Selfinjective algebras of polynomial growth. Math. Ann 285, 177–199 (1989)
Skowronski, A.: Algebras of polynomial growth. In: Topics in Algebra, Part 1, Banach Center Publ., vol. 26, pp 535–568. PWN, Warsaw (1990)
Skowronski, A.: Regular Auslander-Reiten components containing directing modules. Proc. Amer. Math. Soc. 120, 19–26 (1994)
Skowronski, A.: Generalized standard Auslander-Reiten components. J. Math. Soc. Japan 46, 517–543 (1994)
Skowronski, A.: Minimal representation-infinite artin algebras. Math. Proc. Cambridge Philos. Soc. 116, 229–243 (1994)
Skowronski, A.: On the composition factors of periodic modules. J. London Math. Soc. 49, 477–492 (1994)
Skowronski, A.: Cycle-finite algebras. J. Pure Appl. Algebra 103, 105–116 (1995)
Skowronski, A.: Simply connected algebras of polynomial growth. Compositio Math 109, 99–133 (1997)
Skowronski, A.: Tame algebras with strongly simply connected Galois coverings. Colloq. Math. 72, 335–351 (1997)
Skowronski, A.: Tame quasi-tilted algebras. J. Algebra 203, 470–490 (1998)
Skowronski, A.: Selfinjective algebras: finite and tame type. In: Trends in Representation Theory of Algebras and Related Topics, Contemporary Mathematics, vol. 406, pp 169–238. Amer. Math. Soc., Providence (2006)
Skowronski, A., Wenderlich, M.: Artin algebras with directing indecomposable projective modules. J. Algebra 165, 507–530 (1994)
Skowronski, A., Yamagata, K.: Galois coverings of selfinjective algebras by repetitive algebras. Trans. Amer. Math. Soc. 351, 715–734 (1999)
Skowyrski, A.: Cycle-finite algebras with almost all indecomposable modules of projective or injective dimension at most one. Colloq. Math. 132, 239–270 (2013)
Skowyrski, A.: A characterization of cycle-finite generalized double tilted algebras. J. Algebra 416, 1–24 (2014)
Wenderlich, M.: Krull dimension of strongly simply connected algebras. Bull. Polish Acad. Sci. Math. 44, 473–480 (1996)
Zhang, Y.B.: The structure of stable components. Canad. J. Math. 43, 652–672 (1991)
Ziegler, M.: Model theory of modules. Ann. Pure Appl. Logic 26, 149–213 (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Jon F. Carlson.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Skowroński, A. The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras. Algebr Represent Theor 19, 215–233 (2016). https://doi.org/10.1007/s10468-015-9571-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-015-9571-6