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Abstract
Let k ⊂ K be an extension of fields, and let A ⊂ 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 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, A (m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection A (m, n) ∩ A (m, n), which we denote by A (m, n). Under suitable hypotheses on A, we construct affine open subschemes of A (m, n) and A (m, n) which cover their K-rational points. We conclude by using A (m, n), A (m, n), and A (m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces.
2000 Mathematics Subject Classification:
ACKNOWLEDGMENTS
The author was partially supported by the University of Montana University Grant program and by National Security Agency grant H98230-05-1-0021. I thank W. Adams, B. Huisgen-Zimmermann, C. Pappacena, and N. Vonessen for helpful conversations, I thank A. Magidin for proving Lemma 6.5, and I thank S. P. Smith for a number of helpful comments regarding an earlier draft of this article.
Notes
Communicated by B. Huisen-Zimmermann.