Skip to main content
SpringerLink
Log in

Operations on Categories of Modules are Given by Schur Functors

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Let k be a commutative \(\mathbb {Q}\)-algebra. We study families of functors between categories of finitely generated modules which are defined for all commutative k-algebras simultaneously and are compatible with base changes. These operations turn out to be Schur functors associated to k-linear representations of symmetric groups. This result is closely related to Macdonald’s classification of polynomial functors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyah, M., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, vol. 361. Addison-Wesley, Boston (1969)

    Google Scholar 

  2. Bouc, S.: Non-additive Exact Functors and Tensor Induction for Mackey Functors, volume 144 of Memoirs, vol. 683. A.M.S. (2000)

  3. Brandenburg, M.: Tensor categorical foundations of algebraic geometry, Ph.D. thesis, preprint, arXiv:1410.1716 (2014)

  4. Blechschmidt, I.: Using the internal language of toposes in algebraic geometry, draft https://github.com/iblech/internal-methods/blob/master/notes.pdf, retrieved on 27 Sep 2016

  5. Bénabou, J.: Introduction to bicategories, In: Reports of the Midwest Category Seminar, Lecture Notes in Mathematics, vol. 47. Springer, pp. 1–77 (1967)

  6. Brandenburg, M., Chirvasitu, A.: Tensor functors between categories of quasi-coherent sheaves. J. Algebra 399, 675–692 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bendel, C.P., Friedlander, E.M., Suslin, A.: Infinitesimal 1-parameter subgroups and cohomology. J. Am. Math. Soc. 10, 693–728 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Baez, J., Trimble, T., et al.: Schur functor, nLab article. https://ncatlab.org/nlab/show/Schur+functor, retrieved on 27 Sep 2016

  9. Chirvasitu, A., Johnson-Freyd, T.: The fundamental pro-groupoid of an affine 2-scheme. Appl. Categ. Struct. 21(5), 469–522 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Demazure, M., Gabriel, P.: Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam (1970)

  11. Eilenberg, S.: Abstract description of some basic functors. J. Indian Math. Soc. (New Ser.) 24, 231–234 (1960)

    MathSciNet  MATH  Google Scholar 

  12. Eisenbud, D.: Commutative Algebra—With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150. Springer, Berlin (1995)

    MATH  Google Scholar 

  13. Fulton, W.: Young Tableaux. With Applications to Representation Theory and Geometry, London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  14. Fulton, W., Harris, J.: Representation Theory. A First Course Graduate Texts in Mathematics, Readings in Mathematics, vol. 129. Springer, Berlin (1991)

    Google Scholar 

  15. Friedlander, E.M., Suslin, A.: Cohomology of finite group schemes over a field. Invent. Math. 127(2), 209–270 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grandis, M.: Finite sets and symmetric simplicial sets. Theory Appl. Categ. 8(8), 244–252 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Hall, J., Rydh, D.: Coherent tannaka duality and algebraicity of hom-stacks, preprint, arXiv:1405.7680 (2014)

  18. Harris, T., Köck, B., Taelman, L.: Exterior power operations on higher K-groups via binary complexes, preprint, arXiv:1607.01685v2 (2016)

  19. Joyal, A.: Foncteurs Analytiques et Espèces de Structures. In: Combinatoire énumérative (Montreal, Quebec, 1985), Lecture Notes in Mathematics, vol. 1234. Springer, pp. 126–159 (1986)

  20. Krause, H.: Koszul, Ringel and Serre duality for strict polynomial functors. Compos. Math. 149(6), 996–1018 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Loday, J.-L., Valette, B.: Algebraic operads, Grundlehren der mathematischen Wissenschaften, vol. 346. Springer, Berlin (2012)

    Google Scholar 

  22. Lorenz, M.: A Tour of Representation Theory, Preliminary Version (January 11, 2017): https://www.math.temple.edu/~lorenz/book.html, retrieved on 02 Apr 2017

  23. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1979)

    MATH  Google Scholar 

  24. Macdonald, I.G.: Polynomial functors and wreath products. J. Pure Appl. Algebra 18, 173–204 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mulvey, C.: Intuitionistic algebra and representations of rings. In: Recent Advances in the Representation Theory of Rings and C*-Algebras by Continuous Sections, vol. 148, pp. 3–57. Memoirs of the American Mathematical Society, Providence, RI (1974)

  26. Procesi, C.: Lie Groups: An Approach Through Invariants and Representations. Springer, Berlin (2006)

    MATH  Google Scholar 

  27. Roby, N.: Lois polynomes et lois formelles en théorie des modules. Ann. Sci. École Norm. Sup. (3) 80, 213–348 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  28. Touzé, A.: Foncteurs strictement polynomiaux et applications, Habilitation thesis, 2014, Université Paris 13, http://math.univ-lille1.fr/~touze/NotesRecherche/Habilitation_A_Touze.pdf, retrieved on 02 Apr 2017

  29. Vistoli, A.: Grothendieck topologies, fibered categories and descent theory. In: Fundamental Algebraic Geometry, Mathematical Surveys and Monographs, vol. 123, pp. 1–104. American Mathematical Society, Providence, RI (2005)

Download references

Acknowledgements

I am very grateful to Alexandru Chirvasitu, who made major contributions to this paper in its early stages, particularly to Sect. 4. I also would like to thank Ingo Blechschmidt for sharing his beautiful constructive proofs appearing in Sect. 6, Oskar Braun for his careful proofreading of the preprint, Bernhard Köck for drawing my attention to [18], as well as Antoine Touzé and Ivo Dell’Ambrogio for helpful discussions on the subject at the University of Lille. Finally I would like to thank the anonymous referee for making several suggestions for improvement.

Author information

Authors and Affiliations

  1. Münster, Germany

    Martin Brandenburg

Authors
  1. Martin Brandenburg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Brandenburg.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brandenburg, M. Operations on Categories of Modules are Given by Schur Functors. Appl Categor Struct 26, 287–308 (2018). https://doi.org/10.1007/s10485-017-9494-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-017-9494-y

Keywords

Search

Navigation