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 Kn. We determine the structure of these schemes. We devote particular attention to lattices of fixed discriminant in the lattice, p^−r𝒪ⁿ, computing the Zariski tangent space to a lattice in this scheme and determining the singular points.