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.
Citation
William J. Haboush. "Infinite dimensional algebraic geometry: algebraic structures on {$p$}-adic groups and their homogeneous spaces." Tohoku Math. J. (2) 57 (1) 65 - 117, 2005. https://doi.org/10.2748/tmj/1113234835
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