Applied Categorical Structures

, Volume 22, Issue 1, pp 169–210

Even More Spectra: Tensor Triangular Comparison Maps via Graded Commutative 2-rings

Article

Abstract

We initiate the theory of graded commutative 2-rings, a categorification of graded commutative rings. The goal is to provide a systematic generalization of Paul Balmer’s comparison maps between the spectrum of tensor-triangulated categories and the Zariski spectra of their central rings. By applying our constructions, we compute the spectrum of the derived category of perfect complexes over any graded commutative ring, and we associate to every scheme with an ample family of line bundles an embedding into the spectrum of an associated graded commutative 2-ring.

Keywords

Graded ring Symmetric monoidal category Derived category Ample family 

Mathematics Subject Classifications (2010)

13A02 18D10 13D09 

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References

  1. 1.
    Baez, J.C., Lauda, A.D.: Higher-dimensional algebra. V. 2-groups. Theory Appl. Categ. 12, 423–491 (2004). http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html MATHMathSciNetGoogle Scholar
  2. 2.
    Balmer, P.: The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math. 588, 149–168 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Balmer, P.: Spectra, spectra, spectra—tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebraic and Geometric Topology 10, 1521–1563 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Balmer, P.: Tensor triangular geometry. In: Proceedings of the International Congress of Mathematicians, vol. II, pp. 85–112. Hindustan Book Agency, New Delhi (2010). http://www.math.ucla.edu/~balmer/research/publications.html Google Scholar
  5. 5.
    Brenner, H., Schröer, S.: Ample families, multihomogeneous spectra, and algebraization of formal schemes. Pac. J. Math. 208(2), 209–230 (2003)CrossRefMATHGoogle Scholar
  6. 6.
    Buan, A.B., Krause, H., Solberg, Ø.: Support varieties: an ideal approach. Homology, Homotopy, and Applications 9(1), 45–74 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Dell’Ambrogio I., Stevenson G.: On the derived category of a graded commutative noetherian ring. J. Algebra 373, 356–376 (2013)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dupont, M.: Abelian categories in dimension 2. PhD thesis (2008). http://arxiv.org/abs/0809.1760
  9. 9.
    Hochster, M.: Prime ideal structure in commutative rings. Trans. Am. Math. Soc. 142, 43–60 (1969)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. J. Pure Appl. Algebra 19, 193–213 (1980)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Lewis, L.G. Jr., May, J.P., Steinberger, M., McClure, J.E.: Equivariant stable homotopy theory. In: McClure, J.E. (ed.) Lecture Notes in Mathematics, vol. 1213. Springer-Verlag, Berlin (1986)Google Scholar
  12. 12.
    Mac Lane, S.: Categories for the working mathematician, 2nd edn. Graduate Texts in Mathematics, vol. 5. Springer-Verlag, New York (1998)MATHGoogle Scholar
  13. 13.
    Thomason, R.W.: The classification of triangulated subcategories. Compos. Math. 105(1), 1–27 (1997)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Laboratoire Paul PainlevéUniversité de Lille 1Villeneuve d’Ascq CedexFrance
  2. 2.Fakultät für Mathematik, BIREP GruppeUniversität Bielefeld,BielefeldGermany

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