The University of Montana

Department of Mathematical Sciences

Technical report #7/2006

Grassmannians Of Two-Sided Vector Spaces

**Adam Nyman**

Department of Mathematical Sciences

University of Montana

**Abstract**

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

**Keywords:** Grassmannian, two-sided vector space, noncommutative
vector bundle, bimodule.

**AMS Subject Classification:** Primary 15A03, 14M15, 16D20;
Secondary 14A22.

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