The algebraic formalism of soliton equations over arbitrary base fields

TL;DR

This paper develops an algebraic framework for infinite-dimensional Grassmannians and determinant bundles over arbitrary fields, enabling the construction of tau-functions and Baker-Akhiezer functions in a purely algebraic setting.

Contribution

It introduces an algebraic construction of infinite-dimensional Grassmannians and determinant bundles applicable to any base field, extending the geometric theory of soliton equations.

Findings

Constructed the functor of points for Grassmannians over arbitrary schemes.

Proved the representability of the Grassmannian functor by a separated scheme.

Established an algebraic approach to infinite determinants using determinant bundles.

Abstract

The aim of this paper is to offer an algebraic construction of infinite-dimensional Grassmannians and determinant bundles (and therefore valid for arbitrary base fields). As an application we construct the $\tau$-function and formal Baker-Akhiezer functions over arbitrary fields, by proving the existence of a ``formal geometry'' of local curves analogous to the geometry of global algebraic curves. We begin by defining the functor of points, $\fu{\gr}(V,V^+)$, of the Grassmannian of a $k$-vector space $V$ in such a way that its rational points are precisely the points of the Grassmannian defined by Segal-Wilson, although the points over an arbitrary $k$-scheme $S$ have been not previously considered. This definition of the functor $\fu{\gr}(V,V^+)$ allows us to prove that it is representable by a separated $k$-scheme $\gr(V,V^+)$. Using the theory of determinants of Knudsen and Mumford, the determinant bundle is constructed. This is one of the main results of the paper because it implies that we can define ``infinite determinants'' in a completely algebraic way.