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Let $K$ be a field, and let $E$ be the algebra of $n\times n$ matrices over $K$. Let $V_0$ and $V_1$ be the (left) $E$-modules of matrices of size $n\times n_0$ and $n\times n_1$. Let $W \subseteq V_0$ be a $K$-linear subspace, and define $$\overline W := \{ v\in V_0 \:|\: f(v)\in f(W) \mbox{ for all } f\in \mathrm{Hom}_E(V_0,V_1) \} $$ Is there a method to compute $\overline W$? More precisely:

Given a $K$-basis $w_1,\ldots,w_s$ of $W$, is there an algorithm that produces a $K$-basis of $\overline W$?

This problem arises in the computation of certain cohomology groups attached to abelian varieties over number fields. In practice, the relevant fields are $K = \mathbb Q$ and $K = \mathbb Q_\ell$. Here are a few loose remarks:

(0) The problem is only interesting if the inequalities $0 < n_1 < n_0$ hold.

(1) One can always assume that $W$ generates $V_0$ as an $E$-module. Instead of a basis of $\overline W$, one can also ask for a finite list of $E$-module homomorphisms $f_i:V_0\to V_1$ which cut $\overline W$ out. Such homomorphisms are given by $n_0\times n_1$-matrices.

(2) The problem depends on the field $K$, in the sense that the formation of $\overline W$ does not in general commute with base change to a field extension of $K$. Even in the case where $K$ is algebraically closed, I don't know of a solution to the problem.

(3) One can replace the matrix-algebra $E$ by any finite dimensional semisimple $K$-algebra, and the question makes still sense. However, this more general question boils down to the matrix case, at least in characteristic zero.

I have tagged the question "algebraic geometry" for the following reason: We can look at the set of $E$-linear homomorphisms $V_0\to V_1$ as the set of $K$-rational points of the $n_0n_1$-dimensional affine space over $K$. The homomorphisms that impose strong conditions on $\overline W$ are those for which $f(V_0)/f(W)$ is large. This leads to interesting subvarieties of the affine (and in fact projective) space over $K$, called determinantal varieties. These have usually many singularities and irreducible components. In view of remark (1), we are interested in rational points on these varieties.

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