Ir al contenido

Documat

Mutation and the Gabriel spectrum

  • Michal Hrbek [3] ; Sergio Pavon [1] ; Jorge Vitória [2]
    1. [1] University of Verona

      University of Verona

      Verona, Italia

    2. [2] University of Padua

      University of Padua

      Padova, Italia

    3. [3] Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic
    Mostrar afiliaciones +
  • Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 32, Nº. 2, 2026
  • Idioma: inglés
  • DOI: 10.1007/s00029-026-01127-z
  • Enlaces
  • Resumen

    • Mutations occur in multiple algebraic contexts, often enjoying good combinatorial properties. In this paper we study mutations of pure-injective cosilting objects in compactly generated triangulated categories from a topological point of view. We consider the topologies studied by Gabriel, Burke and Prest on the set of indecomposable injective objects in a Grothendieck abelian category, transfer them to associated cosilting subcategories, and show that, in that context, right mutation induces a homeomorphism on two complementary subspaces. We then improve this result in the context of the derived category of a commutative noetherian ring, showing that right mutation is an open bijection. We end the paper with a detailed analysis of a range of cosilting subcategories over commutative noetherian rings for which the topology is completely known. As a byproduct of this analysis, we obtain that the category of modules over a commutative noetherian ring is the unique locally noetherian Grothendieck category in its derived-equivalence class.

  • Referencias bibliográficas
    • Tarrío, L.A., López, A.J., Saorín, M.: Compactly generated -structures on the derived category of a Noetherian ring. J. Algebra 324(3), 313–346...
    • Hügel, L.A., Laking, R., Šťovíček, J., Vitória, J.: Mutation and torsion pairs. J. Algebr. Number Theory 19(7), 1313–1368 (2025)
    • Hügel, L.A., Marks, F., Šťovíček, J., Takahashi, R., Vitória, J.: Flat ring epimorphisms and universal localizations of commutative rings....
    • Hügel, L.A., Marks, F., Vitória, J.: Torsion pairs in silting theory. Pacific J. Math. 291(2), 257–278 (2017)
    • Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188(883), viii+207 (2007)
    • Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and topology on singular spaces. I (Luminy, 1981), volume 100...
    • Bondal, A.I., Orlov, D.: Reconstruction of a variety from the derived category and groups of autoequivalences. Compositio Math. 125(3), 327–344...
    • Breaz, S., Hrbek, M., Modoi, G.C.: Silting, cosilting and extensions of commutative rings. Mathematical Proceedings of the Cambridge Philosophical...
    • Burke, K.: Some model-theoretic properties of functor categories for modules. PhD thesis, University of Manchester (1994)
    • Lars Winther Christensen and Henrik Holm: Ascent properties of Auslander categories. Can. J. Math. 61(1), 76–108 (2009)
    • Crawley-Boevey, W.: Locally finitely presented additive categories. Comm. Algebra 22(5), 1641–1674 (1994)
    • Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962)
    • Gordon, R., Robson, J.C.: The Gabriel dimension of a module. J. Algebra 29, 459–473 (1974)
    • Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Amer. Math. Soc. 120(575), viii+ 88...
    • Herzog, I.: The Ziegler spectrum of a locally coherent Grothendieck category. Proc. London Math. Soc. (3) 74(3), 503–558 (1997)
    • Hrbek, M., Hu, J., Zhu, R.: Gluing compactly generated t-structures over stalks of affine schemes. Israel Journal of Mathematics, pp. 1–41...
    • Hrbek, M., Martini, L.: Product-complete tilting complexes and Cohen-Macaulay hearts. Rev. Mat. Iberoam. 40(6), 2339–2369 (2024)
    • Hrbek, M., Nakamura, T.: Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings. J. Pure Appl. Algebra...
    • Hrbek, M., Nakamura, T., Šťovíček, J.: Tilting complexes and codimension functions over commutative Noetherian rings. Nagoya Math. J. 255,...
    • Hrbek, M., Pavon, S.: Singular equivalences to locally coherent hearts of commutative noetherian rings. J. Algebra 632, 117–153 (2023)
    • Jensen, C.U., Lenzing, H.: Model-theoretic algebra with particular emphasis on fields, rings, modules. Algebra, vol. 2. Logic and Applications....
    • Kawasaki, T.: On arithmetic Macaulayfication of Noetherian rings. Trans. Am. Math. Soc. 354(1), 123–149 (2002)
    • Kawasaki, T.: Finiteness of Cousin cohomologies. Trans. Am. Math. Soc. 360(5), 2709–2739 (2008) Article Google Scholar
    • Krause, H.: The spectrum of a locally coherent category. J. Pure Appl. Algebra 114(3), 259–271 (1997)
    • Krause, H.: Exactly definable categories. J. Algebra 201(2), 456–492 (1998)
    • Krause, H.: Smashing subcategories and the telescope conjecture–an algebraic approach. Invent. Math. 139(1), 99–133 (2000)
    • Krause, H.: Thick subcategories of modules over commutative Noetherian rings (with an appendix by Srikanth Iyengar). Math. Ann. 340(4), 733–747...
    • Laking, R.: Purity in compactly generated derivators and t-structures with Grothendieck hearts. Math. Z. 295(3–4), 1615–1641 (2020)
    • Marks, F., Vitória, J.: Silting and cosilting classes in derived categories. J. Algebra 501, 526–544 (2018)
    • Matlis, E.: Injective modules over Noetherian rings. Pacific J. Math. 8, 511–528 (1958)
    • Nagata, M.: An example of a normal local ring which is analytically reducible. Memoirs of the College of Science, University of Kyoto. Series...
    • Neeman, A.: The chromatic tower for . With an appendix by Marcel Bökstedt. Topology 31(3):519–532 (1992)
    • Neeman, A.: Triangulated Categories. Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton, NJ (2001)
    • Neeman, A.: The -structures generated by objects. Trans. Amer. Math. Soc. 374(11), 8161–8175 (2021)
    • Nicolás, P., Saorín, M., Zvonareva, A.: Silting theory in triangulated categories with coproducts. J. Pure Appl. Algebra 223(6), 2273–2319...
    • Parra, C.E., Saorín, M.: Direct limits in the heart of a t-structure: the case of a torsion pair. J. Pure Appl. Algebra 219(9), 4117–4143...
    • Pavon, S.: Torsion-simple objects in abelian categories. J. Pure Appl. Algebra 229(1), 107818 (2025)
    • Pavon, S., Vitória, J.: Hearts for commutative Noetherian rings: torsion pairs and derived equivalences. Doc. Math. 26, 829–871 (2021)
    • Popescu, N.: Abelian categories with applications to rings and modules. London Mathematical Society Monographs, No. 3. Academic Press, London-New...
    • Positselski, L., Šťovíček, J.: The tilting-cotilting correspondence. Int. Math. Res. Not. IMRN 1, 191–276 (2021)
    • Prest, M.: Purity, Spectra and Localisation. Encyclopedia of Mathematics and its Applications, vol. 121. Cambridge University Press, Cambridge...
    • Psaroudakis, C., Vitória, J.: Realisation functors in tilting theory. Math. Z. 288(3–4), 965–1028 (2018)
    • Riedtmann, C., Schofield, A.: On a simplicial complex associated with tilting modules. Comment. Math. Helv. 66(1), 70–78 (1991)
    • Saorín, M., Šťovíček, J., Virili, S.: -structures on stable derivators and Grothendieck hearts. Adv. Math. 429, 109139 (2023)
    • The Stacks project authors. The Stacks project. https://stacks.math.columbia.edu, (2023)
    • Stenström, B.: Rings of quotients. Die Grundlehren der mathematischen Wissenschaften, Band 217. Springer-Verlag, New York-Heidelberg (1975)
    • Šťovíček, J.: Flat epimorphisms and silting epimorphisms coincide for commutative rings. Preprint arXiv:2510.03440 (2025)
    • Takahashi, R.: Faltings’ annihilator theorem and t-structures of derived categories. Math. Z. 304(1), 10 (2023)
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno