World Scientific
  • Search
  •   
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×
Our website is made possible by displaying certain online content using javascript.
In order to view the full content, please disable your ad blocker or whitelist our website www.worldscientific.com.

System Upgrade on Tue, Oct 25th, 2022 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.

On Auslander’s formula and cohereditary torsion pairs

    https://doi.org/10.1142/S0219199717500717Cited by:3 (Source: Crossref)

    For a small abelian category C, Auslander’s formula allows us to express C as a quotient of the category modC of coherent functors on C. We consider an abelian category with the added structure of a cohereditary torsion pair τ=(𝒯,). We prove versions of Auslander’s formula for the torsion-free class of C, for the derived torsion-free class Db() of the triangulated category Db(C) as well as the induced torsion-free class in the ind-category IndC of C. Further, for a given regular cardinal α, we also consider the category modαC of α-presentable objects in the functor category Fun(Cop,Ab). Then, under certain conditions, we show that the torsion-free class can be recovered as a subquotient of modαC.

    AMSC: 18E30, 18E40

    References

    • 1. J. Adámek and J. Rosický, Locally Presentable and Accessible Categories (Cambridge University Press, Cambridge, 1994). CrossrefGoogle Scholar
    • 2. M. Artin, Letter to Grothedieck, unpublished. Google Scholar
    • 3. M. Auslander, Coherent functors, in Proc. Conf. Categorical Algebra, La Jolla, CA, 1965 (Springer, New York, 1966), pp. 189–231. Google Scholar
    • 4. A. A. Beĭlinson, J. Bernstein and P. Deligne, Faisceaux pervers, in Analysis and Topology on Singular Spaces, I, Astérisque, Vol. 100 (Société Mathématique de France, Paris, 1982), pp. 5–171. Google Scholar
    • 5. A. Beligiannis and I. Reiten, Homological and Homotopical Aspects of Torsion Theories, Memoirs of the American Mathematical Society, Vol. 188, No. 883 (American Mathematical Society, Providence, RI, 2007), viii+207 pp. Google Scholar
    • 6. D. J. Benson, S. B. Iyengar and H. Krause, Stratifying modular representations of finite groups, Ann. of Math. (2) 174(3) (2011) 1643–1684. Crossref, ISIGoogle Scholar
    • 7. D. Bourn and M. Gran, Torsion theories in homological categories, J. Algebra 305(1) (2006) 18–47. Crossref, ISIGoogle Scholar
    • 8. M. M. Clementino, D. Dikranjan and W. Tholen, Torsion theories and radicals in normal categories, J. Algebra 305(1) (2006) 98–129. Crossref, ISIGoogle Scholar
    • 9. W. Crawley-Boevey, Locally finitely presented additive categories, Comm. Algebra 22(5) (1994) 1641–1674. Crossref, ISIGoogle Scholar
    • 10. P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962) 323–448. CrossrefGoogle Scholar
    • 11. A. Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957) 119–221. CrossrefGoogle Scholar
    • 12. A. Grothendieck and J.-L. Verdier, Séminaire de Géométrie Algébriques du Bois-Marie 1963–64 (SGA 4), Lecture Notes in Mathematics, Vol. 269 (Springer, Berlin, 1972). CrossrefGoogle Scholar
    • 13. D. Happel, I. Reiten and S. Smalø, Tilting in Abelian Categories and Quasitilted Algebras, Memoirs of the American Mathematical Society, Vol. 120, No. 575 (American Mathematical Society, Providence, RI, 1996), viii+88 pp. Google Scholar
    • 14. R. Hartshorne, Coherent functors, Adv. Math. 140(1) (1998) 44–94. Crossref, ISIGoogle Scholar
    • 15. R. Hartshorne, M. Martin-Deschamps and D. Perrin, Triades et familles de courbes gauches, Math. Ann. 315(3) (1999) 397–468. Crossref, ISIGoogle Scholar
    • 16. R. Hartshorne, M. Martin-Deschamps and D. Perrin, Construction de familles minimales de courbes gauches, Pacific J. Math. 194(1) (2000) 97–116. Crossref, ISIGoogle Scholar
    • 17. R. Hartshorne, M. Martin-Deschamps and D. Perrin, l Un théorème de Rao pour les familles de courbes gauches, J. Pure Appl. Algebra 155(1) (2001) 53–76. Crossref, ISIGoogle Scholar
    • 18. M. Hovey, Classifying subcategories of modules, Trans. Amer. Math. Soc. 353 (2001) 3181–3191. Crossref, ISIGoogle Scholar
    • 19. G. Janelidze and W. Tholen, Characterization of torsion theories in general categories, in Categories in Algebra, Geometry and Mathematical Physics, Contemporary Mathematics, Vol. 431 (American Mathematical Society, Providence, RI, 2007), pp. 249–256. CrossrefGoogle Scholar
    • 20. M. Kashiwara and P. Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften, Vol. 332 (Springer-Verlag, Berlin, 2006). CrossrefGoogle Scholar
    • 21. B. Keller and D. Vossieck, Aisles in Derived categories, Bull. Soc. Math. Belg. 40 (1988) 239–253. Google Scholar
    • 22. H. Krause, Functors on locally finitely presented additive categories, Colloq. Math. 75(1) (1998) 105–132. CrossrefGoogle Scholar
    • 23. H. Krause, Deriving Auslander’s formula, Doc. Math. 20 (2015) 669–688. Crossref, ISIGoogle Scholar
    • 24. H. Lenzing, Homological transfer from finitely presented to infinite modules, in Abelian Group Theory, Lecture Notes in Mathematics, Vol. 1006 (Springer, Berlin, 1983), pp. 734–761. CrossrefGoogle Scholar
    • 25. H. Lenzing, Auslander’s work on Artin algebras, in Algebras and Modules, I, CMS Conference Proceedings, Vol. 23 (American Mathematical Society, Providence, RI, 1998), pp. 83–105. Google Scholar
    • 26. A. Neeman, Triangulated Categories, Annals of Mathematics Studies, Vol. 148 (Princeton University Press, Princeton, NJ, 2001). CrossrefGoogle Scholar
    • 27. A. Neeman, The homotopy category of injectives, Algebra Number Theory 8(2) (2014) 429–456. Crossref, ISIGoogle Scholar
    • 28. J.-E. Roos, Locally Noetherian categories and generalized strictly linearly compact rings: Applications, in Category Theory, Homology Theory and Their Applications, II, Lecture Notes in Mathematics, Vol. 92 (Springer, Berlin, 1969), pp. 197–277. CrossrefGoogle Scholar
    • 29. D. Stanley and B. Wang, Classifying subcategories of finitely generated modules over a Noetherian ring, J. Pure Appl. Algebra 215(11) (2011) 2684–2693. Crossref, ISIGoogle Scholar
    • 30. H. Stauffer, Derived functors without injectives, in Category Theory, Homology Theory and Their Applications I, Lecture Notes in Mathematics, Vol. 86 (Springer, Berlin, 1969), pp. 159–166. CrossrefGoogle Scholar
    • 31. R. Takahashi, Classifying subcategories of modules over a commutative Noetherian ring, J. Lond. Math. Soc. (2) 78(3) (2008) 767–782. Crossref, ISIGoogle Scholar
    • 32. J.-L. Verdier, Des Catégories Dérivées des Catégories Abéliennes, Astérisque, Vol. 239 (Sociéte Mathématique de France, Paris, 1996), xii+253 pp. (1997). Google Scholar
    Remember to check out the Most Cited Articles!

    Be inspired by these NEW Mathematics books for inspirations & latest information in your research area!