Grassmannians of two-sided vector spaces

Abstract

Let $k\subset K$ be an extension of fields, and let $A\subset M_n(K)$ be a $k$-algebra. We study parameter spaces of $m$-dimensional subspaces of $K^n$ which are invariant under $A$. The space ${\mathbb{F}}_A(m,n)$, whose $R$-rational points are $A$-invariant, free rank $m$ summands of $R^n$, is well known. We construct a distinct parameter space, ${\mathbb{G}}_A(m,n)$, which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection ${\mathbb{F}}_A(m,n)\cap {\mathbb{G}}_A(m,n)$, which we denote by ${\mathbb{H}}_A(m,n)$. Under suitable hypotheses on $A$, we construct affine open subschemes of ${\mathbb{F}}_A(m,n)$ and ${\mathbb{H}}_A(m,n)$ which cover their $K$-rational points. We conclude by using ${\mathbb{F}}_A(m,n)$, ${\mathbb{G}}_A(m,n)$, and ${\mathbb{H}}_A(m,n)$ to construct parameter spaces of two-sided subspaces of two-sided vector spaces.

23 pages, changed content, minor errors corrected