## Tohoku Mathematical Journal 2005 March SECOND SERIES VOL. 57, NO. 1

 Tohoku Math. J. 57 (2005), 65-117

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 ind-schemes 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.