Grassmannians of Two-sided Vector Spaces

Let k ⊂ K be an extension of fields, and let A ⊂ Mn(K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of Kn which are invariant under A. The space FA(m, n), whose R-rational points are A-invariant, free rank m summands of Rn, is well known. We construct a distinct parameter space, GA(m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection FA(m, n)∩GA(m, n), which we denote by HA(m, n). Under suitable hypotheses on A, we construct affine open subschemes of FA(m, n) and HA(m, n) which cover their K-rational points. We conclude by using FA(m, n), GA(m, n), and HA(m, n) to construct parameter spaces of two-sided subspaces of two-sided vector spaces.