## Tohoku Mathematical Journal

### Infinite dimensional algebraic geometry: algebraic structures on {$p$}-adic groups and their homogeneous spaces

William J. Haboush

#### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 57, Number 1 (2005), 65-117.

Dates
First available in Project Euclid: 11 April 2005

https://projecteuclid.org/euclid.tmj/1113234835

Digital Object Identifier
doi:10.2748/tmj/1113234835

Mathematical Reviews number (MathSciNet)
MR2113991

Zentralblatt MATH identifier
1119.14004

#### Citation

Haboush, William J. Infinite dimensional algebraic geometry: algebraic structures on {$p$}-adic groups and their homogeneous spaces. Tohoku Math. J. (2) 57 (2005), no. 1, 65--117. doi:10.2748/tmj/1113234835. https://projecteuclid.org/euclid.tmj/1113234835