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Tohoku Mathematical Journal
2005
March
SECOND SERIES VOL. 57, NO. 1
Tohoku Math. J.
57 (2005), 65117

Title
INFINITE DIMENSIONAL ALGEBRAIC GEOMETRY; ALGEBRAIC STRUCTURES ON $p$ADIC GROUPS AND THEIR HOMOGENEOUS SPACES
Author
Willian J. Haboush
(Received June 2, 2003, revised November 12, 2003) 
Abstract.
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 paper we construct an algebraic theory of indschemes that allows us to represent finite $K$ schemes as infinite dimensional $k$schemes and we apply this to semisimple groups. In the second part we construct spaces of lattices of fixed discriminant in the vector space $K^n.$ We determine the structure of these schemes. We devote particular attention to lattices of fixed discriminant in the lattice, $p^{r}\mathcal{O}^n,$ computing the Zariski tangent space to a lattice in this scheme and determining the singular points.
2000 Mathematics Subject Classification.
Primary 20G25; Secondary 20G99, 14L15.
Key words and phrases.
Group schemes, Witt vectors, lattices, Hilbert class field.


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