On $$p$$p-adic lattices and Grassmannians

@article{Kreidl2014OnL,
  title={On \$\$p\$\$p-adic lattices and Grassmannians},
  author={Martin Kreidl},
  journal={Mathematische Zeitschrift},
  year={2014},
  volume={276},
  pages={859-888}
}
  • M. Kreidl
  • Published 2 October 2010
  • Mathematics
  • Mathematische Zeitschrift
It is well-known that the coset spaces $$G(k((z)))/G(k[[z]])$$G(k((z)))/G(k[[z]]), for a reductive group $$G$$G over a field $$k$$k, carry the geometric structure of an inductive limit of projective $$k$$k-schemes. This $$k$$k-ind-scheme is known as the affine Grassmannian for $$G$$G. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form $$\mathcal {G}(\mathbf {W}(k)[1/p])/\mathcal {G}(\mathbf {W}(k))$$G(W(k)[1/p… 
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References

SHOWING 1-10 OF 14 REFERENCES
Infinite dimensional algebraic geometry: algebraic structures on {$p$}-adic groups and their homogeneous spaces
Let $k$ denote the algebraic closure of the finite field, $\mathbb F_p,$ let $\mathcal O$ denote the Witt vectors of $k$ and let $K$ denote the fraction field of this ring. In the first part of this
Infinite-dimensional vector bundles in algebraic geometry (an introduction)
Raynaud and Gruson showed that there is a reasonable algebro-geometric notion of family of discrete (infinite-dimensional) vector spaces. The author introduces a notion of family of Tate spaces
On the flatness of models of certain Shimura varieties of PEL-type
Abstract. Consider a PEL-Shimura variety associated to a unitary group that splits over an unramified extension of $\Q_p$. Rapoport and Zink have defined a model of the Shimura variety over the ring
Conformal blocks and generalized theta functions
LetSUXr be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical
Affine Springer Fibers and Affine Deligne-Lusztig Varieties
We give a survey on the notion of affine Grassmannian, on affine Springer fibers and the purity conjecture of Goresky, Kottwitz, and MacPherson, and on affine Deligne-Lusztig varieties and results
Multigraded Hilbert schemes
We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely
Notes on Grothendieck topologies, fibered categories and descent theory
This is an introduction to Grothendieck's descent theory, with some stress on the general machinery of fibered categories and stacks.
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