Infinite dimensional algebraic geometry;: Algebraic structures on p-adic groups and their homogeneous spaces

Let k denote the algebraic closure of the finite field, F-p, let 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)O(n), computing the Zariski tangent space to a lattice in this scheme and determining the singular points.