Abstract
Let \((A,\mathfrak {m})\) be a commutative complete equi-characteristic Gorenstein isolated singularity of dimension d with \(k = A/\mathfrak {m}\) algebraically closed. Let Γ(A) be the AR (Auslander-Reiten) quiver of A. Let \(\mathcal {P}\) be a property of maximal Cohen-Macaulay A-modules. We show that some naturally defined properties \(\mathcal {P}\) define a union of connected components of Γ(A). So in this case if there is a maximal Cohen-Macaulay module satisfying \(\mathcal {P}\) and if A is not of finite representation type then there exists a family {Mn}n≥ 1 of maximal Cohen-Macaulay indecomposable modules satisfying \(\mathcal {P}\) with multiplicity e(Mn) > n. Let \(\underline {\Gamma }(A)\) be the stable quiver. We show that there are many symmetries in \(\underline {\Gamma }(A)\). As an application we show that if \((A,\mathfrak {m})\) is a two dimensional Gorenstein isolated singularity with multiplicity e(A) ≥ 3 then for all n ≥ 1 there exists an indecomposable self-dual maximal Cohen-Macaulay A-module of rank n.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Auslander, M.: Isolated singularities and existence of almost split sequences. In: Proc. ICRA IV, Springer Lecture Notes in Math., vol. 1178, pp 194–241.h (1986)
Auslander, M., Buchweitz, R.-O.: The homological theory of maximal Cohen-Macaulay approximations, Colloque en l’honneur de Pierre Samuel (Orsay, 1987). Mem. Soc. Math. France (N.S.) 38, 5–37 (1989)
Auslander, M., Reiten, I.: Representation Theory of Artin algebra V: Methods for computing almost split sequences and irreducible morphisms. Communications in Algebra 5(5), 519–554 (1977)
Avramov, L.L.: Modules of finite virtual projective dimension. Invent. Math. 96, 71–101 (1989)
Avramov, L.L.: Infinite free resolutions, Six lectures on commutative algebra (Bellaterra, 1996), 1118, Progr. Math., 166. Birkhäuser, Basel (1998)
Avramov, L.L., Gasharov, V.N., Peeva, I.V.: Complete intersection dimension. Inst. Hautes Études Sci. Publ. Math. (1997) 86, 67–114 (1998)
Avramov, L.L., Buchweitz, R.-O.: Support varieties and cohomology over complete intersections. Invent. Math. 142(2), 285–318 (2000)
Benson, D.J. Cambridge Studies in Advanced Mathematics: Representations and cohomology. I. Basic representation theory of finite groups and associative algebras, 2nd, vol. 30. Cambridge University Press, Cambridge (1998)
Bergh, P.A.: On support varieties for modules over complete intersections. Proc. Amer. Math. Soc. 135 (12), 3795–3803 (2007)
Brennan, J.P., Herzog, J., Ulrich, B.: Maximally generated Cohen-Macaulay modules. Math. Scand. 61(2), 181–203 (1987)
Buchweitz, R.-O., Greuel, G.-M., Schreyer, F.-O.: Cohen-Macaulay modules on hypersurface singularities. II. Invent. Math. 88(1), 165–182 (1987)
Croll, A.: Periodic modules over Gorenstein local rings. J. Algebra 395, 47–62 (2013)
Dieterich, E.: Reduction of isolated singularities. Comment. Math. Helv. 62, 654–676 (1987)
Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc. 260(1), 35–64 (1980)
Gasharov, V., Peeva, I.: Boundedness versus periodicity over commutative local rings. Trans. Amer. Math. Soc. 320 (2), 569–580 (1990)
Green, E.L., Zacharia, D.: Auslander-reiten components containing modules with bounded Betti numbers. Trans. Amer. Math. Soc 361(8), 4195–4214 (2009)
Gulliksen, T.H.: A change of ring theorem with applications to poincaré series and intersection multiplicity. Math. Scand. 34, 167–183 (1974)
Herzog, J., Ulrich, B., Backelin, J.: Linear maximal Cohen-Macaulay modules over strict complete intersections. J. Pure Appl. Algebra 71 (2-3), 187–202 (1991)
Huneke, C., Leuschke, G.: Two theorems about maximal Cohen-Macaulay modules. Math. Ann. 324(2), 391–404 (2002)
Martsinkovsky, A., Strooker, J.R.: Linkage of modules. J. Algebra 271(2), 587–626 (2004)
Puthenpurakal, T.J.: Hilbert coefficients of a Cohen-Macaulay module. J. Algebra 264, 82–97 (2003)
Puthenpurakal, T.J.: The Hilbert function of a maximal Cohen-Macaulay module. Math. Z. 251(3), 551–573 (2005)
Takahashi, R.: Direct summands of syzygy modules of the residue class field. Nagoya Math. J. 189, 1–25 (2008)
Sally, J.D.: Number of generators of ideals in local rings, Lect. Notes Pure Appl. Math. vol. 35, M. Dekker (1978)
Yoshino, Y.: London Mathematical Society Lecture Note Series: Cohen-Macaulay modules over Cohen-Macaulay rings, vol. 146. Cambridge University Press, Cambridge (1990)
Acknowledgements
I thank Dan Zacharia, Srikanth Iyengar and Lucho Avramov for some useful discussions. I also thank the referee for many pertinent comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Michel Van den Bergh
Rights and permissions
About this article
Cite this article
Puthenpurakal, T.J. Symmetries and Connected Components of the AR-quiver of a Gorenstein Local Ring. Algebr Represent Theor 22, 1261–1298 (2019). https://doi.org/10.1007/s10468-018-9820-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-018-9820-6
Keywords
- Artin-reiten quiver
- Hensel rings
- Indecomposable modules
- Ulrich modules
- Periodic modules
- Non-periodic modules with bounded betti numbers