Auslander–Reiten annihilators

• Esentepe, Özgür ^[1]
  1. [1] Fraunhofer Institute for Algorithms and Scientific Computing

     Fraunhofer Institute for Algorithms and Scientific Computing

     Rhein-Sieg-Kreis, Alemania

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 77, Fasc. 3, 2026, págs. 603-622
• Idioma: inglés
• DOI: 10.1007/s13348-025-00482-y
• Texto completo no disponible (Saber más ...)
• Resumen

  • The Auslander–Reiten conjecture for commutative Noetherian rings predicts that a finitely generated module is projective when certain Ext-modules vanish. But what if those Ext-modules do not vanish? We study the annihilators of these Ext-modules and formulate a generalisation of the Auslander–Reiten conjecture. We prove this general version for high syzygies of modules over several classes of rings including analytically unramified Arf rings, 2-dimensional local normal domains with rational singularities, Gorenstein isolated singularities of Krull dimension at least 2 and more. We also prove results for the special case of the canonical module of a Cohen–Macaulay local ring. These results both generalise and also provide evidence for a version of the Tachikawa conjecture that was considered by Dao–Kobayashi–Takahashi.

• Referencias bibliográficas
  • Auslander, M., Bridger, M.: Stable module theory, vol. 94 of Mem. Am. Math. Soc. American Mathematical Society (AMS), Providence, RI (1969)
  • Avramov, L.L., Buchweitz, R.-O., Şega, L.M.: Extensions of a dualizing complex by its ring: commutative versions of a conjecture of Tachikawa....
  • Auslander, M., Ding, S., Solberg, Ø.: Liftings and weak liftings of modules. J. Algebra 156(2), 273–317 (1993)
  • Avramov, L.L., Iyengar, S.B., Nasseh, S., Sather-Wagstaff, K.: Persistence of homology over commutative Noetherian rings. J. Algebra 610,...
  • Auslander, M., Reiten, I.: On a generalized version of the Nakayama conjecture. Proc. Am. Math. Soc. 52, 69–74 (1975)
  • Araya, T.: The Auslander-Reiten conjecture for Gorenstein rings. Proc. Am. Math. Soc. 137(6), 1941–1944 (2009)
  • Araya, T., Yoshino, Y.: Remarks on a depth formula, a grade inequality and a conjecture of Auslander. Commun. Algebra 26(11), 3793–3806 (1998)
  • Bruns, W., Herzog, J.: Cohen-Macaulay rings, volume 39 of Camb. Stud. Adv. Math. Cambridge University Press, Cambridge (1993)
  • Buchweitz, R.: Maximal Cohen-Macaulay modules and Tate cohomology. With appendices by Luchezar L. Avramov, Benjamin Briggs, Srikanth B. Iyengar...
  • Christensen, L.W., Holm, H.: Vanishing of cohomology over Cohen-Macaulay rings. Manuscr. Math. 139(3–4), 535–544 (2012)
  • Celikbas, O., Iima, K., Sadeghi, A., Takahashi, R.: On the ideal case of a conjecture of Auslander and Reiten. Bull. Sci. Math. 142, 94–107...
  • Celikbas, O., Takahashi, R.: Auslander-Reiten conjecture and Auslander-Reiten duality. J. Algebra 382, 100–114 (2013)
  • Dao, H.: Reflexive modules, self-dual modules and ARF rings (2021). arXiv preprint : 2105.12240
  • Dey, S.: Trace ideal and annihilator of EXT and tor of regular fractional ideals, and some applications (2022). arxiv preprint : 2210.03891
  • Dey, S., Ghosh, D.: Complexity and rigidity of Ulrich modules, and some applications. Math. Scand. 129(2), 209–237 (2023)
  • Dao, H., Iyama, O., Takahashi, R., Vial, C.: Non-commutative resolutions and Grothendieck groups. J. Noncommut. Geom. 9(1), 21–34 (2015)
  • Diveris, K.: Finitistic extension degree. Algebr. Represent. Theory 17(2), 495–506 (2014)
  • Dey, S., Kumashiro, S., Sarkar, P.: On a generalized Auslander-Reiten conjecture (2023). arxiv preprint : 2209.12718
  • Dao, H., Kobayashi, T., Takahashi, R.: Trace ideals of canonical modules, annihilators of Ext modules, and classes of rings close to being...
  • Dugas, A.S., Leuschke, G.J.: Some extensions of theorems of Knörrer and Herzog-Popescu. J. Algebra 571, 94–120 (2021)
  • Dao, H., Takahashi, R.: The radius of a subcategory of modules. Algebra Number Theory 8(1), 141–172 (2014)
  • Dey, S., Takahashi, R.: Comparisons between annihilators of Tor and Ext. Acta Math. Vietnam. 47(1), 123–139 (2022)
  • Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc. 260(1),...
  • Esentepe, Ö.: The cohomology annihilator of a curve singularity. J. Algebra 541, 359–379 (2020)
  • Goto, S., Matsuoka, N., Phuong, T.T.: Almost Gorenstein rings. J. Algebra 379, 355–381 (2013)
  • Goto, S., Takahashi, R.: On the Auslander-Reiten conjecture for Cohen-Macaulay local rings. Proc. Am. Math. Soc. 145(8), 3289–3296 (2017)
  • Ghosh, D., Takahashi, R.: Auslander-Reiten conjecture and finite injective dimension of \operatorname{Hom}. Kyoto J. Math. 64(1), 229–243...
  • Herzog, J., Hibi, T., Stamate, D.I.: The trace of the canonical module. Isr. J. Math. 233(1), 133–165 (2019)
  • Herzog, J., Kumashiro, S., Stamate, D.I.: The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one. J. Algebra...
  • Huneke, C., Leuschke, G.J.: On a conjecture of Auslander and Reiten. J. Algebra 275(2), 781–790 (2004)
  • Huneke, C., Şega, L.M., Vraciu, A.N.: Vanishing of Ext and Tor over some Cohen-Macaulay local rings. Ill. J. Math. 48(1), 295–317 (2004)
  • Huneke, C., Vraciu, A.: Rings that are almost Gorenstein. Pac. J. Math. 225(1), 85–102 (2006)
  • Iyengar, S.B., Takahashi, R.: Annihilation of cohomology and decompositions of derived categories. Homology Homotopy Appl. 16(2), 231–237...
  • Iyengar, S.B., Takahashi, R.: Annihilation of cohomology and strong generation of module categories. Int. Math. Res. Not. 2016(2), 499–535...
  • Iyengar, S.B., Takahashi, R.: The Jacobian ideal of a commutative ring and annihilators of cohomology. J. Algebra 571, 280–296 (2021)
  • Iyama, O.: Auslander correspondence. Adv. Math. 210(1), 51–82 (2007)
  • Kimura, K.: The Auslander-Reiten conjecture for normal rings. In: Proceedings of the 55th symposium on ring theory and representation theory,...
  • Kimura, K.: Compactness of the Alexandrov topology of maximal Cohen-Macaulay modules (2024). arxiv preprint : 2401.04987
  • Kimura, K., Otake, Y., Takahashi, R.: Maximal Cohen-Macaulay tensor products and vanishing of Ext modules. Bull. Lond. Math. Soc. 54(6), 2456–2468...
  • Lindo, H.: On the nonrigidity of trace modules. J. Commut. Algebra 14(2), 277–283 (2022)
  • Liu, J.: Annihilators and dimensions of the singularity category. Nagoya Math. J. 250, 533–548 (2023)
  • Nakayama, T.: On algebras with complete homology. Abh. Math. Semin. Univ. Hamb. 22, 300–307 (1958)
  • Nasseh, S., Sather-Wagstaff, S.: Vanishing of Ext and Tor over fiber products. Proc. Am. Math. Soc. 145(11), 4661–4674 (2017)
  • Nasseh, S., Takahashi, R.: Local rings with quasi-decomposable maximal ideal. Math. Proc. Camb. Philos. Soc. 168(2), 305–322 (2020)
  • Nasseh, S., Yoshino, Y.: On Ext-indices of ring extensions. J. Pure Appl. Algebra 213(7), 1216–1223 (2009)
  • Ringel, C.M., Zhang, P.: Gorenstein-projective and semi-Gorenstein-projective modules. Algebra Number Theory 14(1), 1–36 (2020)
  • Sadeghi, A., Takahashi, R.: Two generalizations of Auslander-Reiten duality and applications. Ill. J. Math. 63(2), 335–351 (2019)
  • Stangle, J.: Gorenstein and totally reflexive orders. J. Algebra 477, 56–68 (2017)
  • Takahashi, R.: Classifying thick subcategories of the stable category of Cohen-Macaulay modules. Adv. Math. 225(4), 2076–2116 (2010)
  • Wang, H.-J.: On the fitting ideals in free resolutions. Mich. Math. J. 41(3), 587–608 (1994)
  • Wei, J.: Generalized Auslander-Reiten conjecture and tilting equivalences. Proc. Am. Math. Soc. 138(5), 1581–1585 (2010)
  • Weston, D.: Divisorial properties of the canonical module for invariant subrings. Commun. Algebra 19(9), 2641–2666 (1991)
  • Yoshino, Y.: Cohen-Macaulay modules over Cohen-Macaulay rings, volume 146 of Lond. Math. Soc. Lect. Note Ser. Cambridge University Press,...