2
$\begingroup$

This must be probably only reference request since I am inclined to believe that I am asking about something well known but just cannot pin down appropriate keywords for searching.

The starting point: in the exterior algebra $\Lambda^*V$, for a vector space $V$ over a field $k$, a decomposable multivector $v_1\wedge\cdots\wedge v_n$ corresponds to the $n$-dimensional subspace of $V$ spanned by $v_1,...,v_n$; the Plücker embedding $\operatorname{Grassmannian}_n(V)\hookrightarrow\mathbb P(\Lambda^nV)$ is based on this.

We might thus view a general element of $\Lambda^nV$ as a linear combination, with coefficients in $k$, of $n$-dimensional linear subspaces of $V$.

Question zero: is actually $\Lambda^nV$ a $k$-vector space spanned by the $n$-th Grassmannian of $V$ in some appropriate category? That is, is there a category $\mathscr C$, with the Grassmannian as one of its objects, and $\Lambda^nV$ as an object of $k$-$\mathrm{Vect}(\mathscr C)$ of the category of internal $k$-vector spaces in $\mathscr C$ which is universal for the Grassmannian with respect to the forgetful functor $k$-$\mathrm{Vect}(\mathscr C)\to\mathscr C$?

The rank of a multivector $x\in\Lambda^nV$ is the minimal number of "actual" subspaces such that $x$ is their linear combination. For $n=2$ it is well known how to relate it to the "usual" rank (of skew-symmetric matrices), but for larger $n$ as far as I know it is very difficult to compute in general. For example, I learned from an old paper by Elashvili and Vinberg that a generic trivector in a 9-dimensional space is a sum of four 3-dimensional subspaces.

From yet another point of view, $n$-dimensional subspaces of $V$ are particular $n$-dimensional subvarieties of $V$ considered as a $k$-variety. Thus $\Lambda^nV$ is also a subspace in something like $C_n(V;k)$ - $n$-cycles in the variety $V$, with coefficients in $k$.

Questions: are $k$-vector spaces of cycles in $k$-varieties with coefficients in $k$ considered in the literature? Do they have some distinguished features as opposed to cycles with more general coefficients? Is there a notion of rank of a cycle in this context, i. e. minimal number of "actual" subvarieties whose linear combination is the given cycle?

What is the name (if any) of the part of algebraic geometry that studies this kind of cycle spaces, with coefficients in the base field?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.