Tohoku Mathematical Journal

Sato Grassmannians for generalized Tate spaces

Luigi Previdi

Full-text: Open access


We generalize the concept of Sato Grassmannians of locally linearly compact topological vector spaces (Tate spaces) to the Beilinson category of the “locally compact objects”, or Generalized Tate Spaces, of an exact category. This allows us to extend the Kapranov dimensional torsor Dim and determinantal gerbe Det to generalized Tate spaces and unify their treatment in the determinantal torsor. We then introduce a class of exact categories, that we call partially abelian exact, and prove that if the base category is so, then Dim and Det are multiplicative in admissible short exact sequences of generalized Tate spaces. We then give a cohomological interpretation of these results in terms of the Waldhausen K-theoretical space of the Beilinson category. Our approach can be iterated and we define analogous concepts for the successive categories of $n$-dimensional (generalized) Tate spaces. In particular we show that the category of Tate spaces is partially abelian exact, so we can extend the results for Dim and Det obtained for 1-Tate spaces to 2-Tate spaces, and provide a new interpretation in the context of algebraic $K$-theory of results of Kapranov, Arkhipov-Kremnizer and Frenkel-Zhu.

Article information

Tohoku Math. J. (2), Volume 64, Number 4 (2012), 489-538.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18E10: Exact categories, abelian categories
Secondary: 19D10: Algebraic $K$-theory of spaces

Grassmannian Tate space Waldhausen space loop space delooping torsor gerbe exact category


Previdi, Luigi. Sato Grassmannians for generalized Tate spaces. Tohoku Math. J. (2) 64 (2012), no. 4, 489--538. doi:10.2748/tmj/1356038976.

Export citation


  • S. Arkhipov and K. Kremnizer, 2-gerbes and 2-Tate spaces (English summary), Arithmetic and geometry around quantization, 23–35, Progr. Math. 279, Birkhäuser Boston, MA, 2010.
  • M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics 100, Springer-Verlag, Berlin-Heidelberg 1969.
  • A. Beilinson, How to glue perverse sheaves, $K$-theory, arithmetic and geometry (Moscow, 1984-1986), 42–51, Lecture Notes in Mathematics 1289, Springer-Verlag, New York, 1987.
  • P. Deligne, Le determinant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 93–177, Contemp. Math. 67, Amer. Math. Soc., Providence, RI, 1987.
  • V. Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, The Unity of Mathematics, 263–304, Progr. Math. 144, Birkhäuser Boston, MA, 2006.
  • E. Frenkel and X. Zhu, Gerbal representations of double loop groups, arXiv:0810.1487v3, 2008.
  • D. Gaitsgory and D. Kazhdan, Representations of algebraic groups over a 2-dimensional local field, Geom. Funct. Anal. 14 (2004), 535–574.
  • S. Gelfand and Yu. Manin, Methods of homological algebra, Second Edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin - Heidelberg, 2003.
  • P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Springer-Verlag, New York, 1967.
  • A. Grothendieck et al., Séminaire de géometrie algébrique IV: Théorie des topos et cohomologie étale des schemas, Lecture Notes in Mathematics 269, Springer-Verlag, Berlin-Heidelberg, 1972.
  • A. Grothendieck et al., Séminaire de géometrie algébrique VII: Groupes de Monodromie en Géometrie Algébrique, Lecture Notes in Mathematics 288, Springer-Verlag, Berlin-Heidelberg, 1972.
  • M. Kapranov, Semiinfinite symmetric powers.v1, 2001.
  • M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Functional analysis on the eve of the 21st century Vol. I (New Brunswick, NJ, 1993) 119–151, Progr. Math. 131 Birkhäuser, Boston, MA 1995.
  • K. Kato, Existence theorem for higher local fields, Geom. Topol. Monogr. 3 (Münster, 1999), 165–195, Geom. Topol. Publ., Coventry, 2000.
  • S. Lefschetz, Algebraic Topology, AMS Colloquium Publications 27, Amer. Math. Soc., New York, 1942.
  • S. Mac Lane, Categories for the working mathematician, Second Edition, Grad. Texts in Math. 5, Springer-Verlag, New York, 1998.
  • L. Previdi, Locally compact objects in exact categories, Internat. J. Math. 22 (2011), 1787–1821.
  • D. Quillen, Higher algebraic K-theory I, Higher $K$-theories (Proc. Conf., Battele Memorial Inst., Seattle, Wash., 1972), 85–147, Lecture Notes in Mathematics 341, Springer-Verlag, New York, 1973.
  • M. Sato, The KP hierarchy and infinite-dimensional Grassmann manifolds, Theta Functions, Bowdoin 1987, Part I (Brunswick, ME, 1987), 51–66, Proc. Sympos. Pure Math. 49, Amer. Math. Soc. Providence, RI, 1989.
  • R. Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987), 283–335.
  • F. Waldhausen, Algebraic K-Theory of generalized free products, Part I, Ann. of Math. (2) 108, (1978), 135–204.
  • F. Waldhausen, Algebraic K-theory of spaces, Algebraic and Geometric Topology (New Brunswick, NJ, 1983), 318–419, Lecture Notes in Math. 1126, Springer-Verlag, New York, 1985.