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

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.

Noetherian schemes over abelian symmetric monoidal categories by:1 (Source: Crossref)

    In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let A be a commutative monoid object in an abelian symmetric monoidal category (C,,1) satisfying certain conditions and let (A)=HomAMod(A,A). If the subobjects of A satisfy a certain compactness property, we say that A is Noetherian. We study the localization of A with respect to any s(A) and define the quotient A/ of A with respect to any ideal (A). We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc.) for schemes over (C,,1). Our notion of a scheme over a symmetric monoidal category (C,,1) is that of Toën and Vaquié.

    AMSC: 14A15, 19D23


    • 1. J. Adámek and J. Rosický, Locally presentable and accessible categories, in London Mathematical Society Lecture Note Series, Vol. 189 (Cambridge University Press, Cambridge, 1994). CrossrefGoogle Scholar
    • 2. A. Banerjee, Centre of monoids, centralisers and localization, Comm. Algebra 40(11) (2012) 3975–3993. Crossref, ISIGoogle Scholar
    • 3. A. Banerjee, Derived Schemes and the field with one element, J. Math. Pures. Appl. 97(3) (2012) 189–203. Crossref, ISIGoogle Scholar
    • 4. A. Banerjee, Schémas sur les catégories abéliennes monoïdales symétriques et faisceaux quasi-cohérents. J. Algebra 423 (2015) 148–176. Crossref, ISIGoogle Scholar
    • 5. A. Banerjee, On Noetherian schemes over (𝒞,,1) and the category of quasi-coherent sheaves, preprint (2015), arXiv:1505.01307. Google Scholar
    • 6. A. Banerjee, On integral schemes over symmetric monoidal categories, preprint (2015), arXiv:1506.04890. Google Scholar
    • 7. A. Banerjee, Schemes over symmetric monoidal categories and torsion theories, J. Pure Appl. Algebra 220(9) (2016) 3017–3047. Crossref, ISIGoogle Scholar
    • 8. P. Deligne, Catégories tannakiennes, in The Grothendieck Festschrift, Vol. II, Progress in Mathematics, Vol. 87 (Birkhäuser Boston, 1990), pp. 111–195. Google Scholar
    • 9. G. A. Garkusha, Grothendieck categories, (in Russian) Algebra i Analiz 13(2) (2001) 1–68; St. Petersburg Math. J. 13(2) (2002) 149–200. Google Scholar
    • 10. M. Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64 (Springer-Verlag, Berlin-New York, 1972). CrossrefGoogle Scholar
    • 11. D. Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1969) 81–128. Crossref, ISIGoogle Scholar
    • 12. S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5 (Springer-Verlag, New York, 1971). Google Scholar
    • 13. F. Marty, Relative Zariski open objects, J. K. Theory 10 (2012) 9–39. Crossref, ISIGoogle Scholar
    • 14. J. P. May, Picard groups, Grothendieck rings, and Burnside rings of categories, Adv. Math. 163(1) (2001) 1–16. Crossref, ISIGoogle Scholar
    • 15. M. Prest, Definable additive categories: Purity and model theory, Mem. Amer. Math. Soc. 210(987) (2011). ISIGoogle Scholar
    • 16. M. Prest and A. Ralph, Locally finitely presented categories of sheaves of modules, Manchester Institute for Mathematical Sciences Preprint 2010.21, University of Manchester (2010). Google Scholar
    • 17. M. Prest and A. Ralph, On sheafification of modules, Manchester Institute of Mathematical Sciences Preprint 2010.22, University of Manchester (2010). Google Scholar
    • 18. B. Stenström, Rings of quotients: Introduction to Methods of Ring Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 217 (Springer-Verlag, New York, 1975). CrossrefGoogle Scholar
    • 19. B. Toën, Bertrand and M. Vaquié, Au-dessous de Spec(), J. K. Theory 3(3) (2009) 437–500. Crossref, ISIGoogle Scholar