  Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field,
and $\mathcal{S}$ be a subset of the space $\mathcal{L}(U,V)$ of all linear
maps from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is called
range-compatible when it satisfies $F(s) \in \mathrm{im}(s)$ for all $s \in
\mathcal{S}$; it is called quasi-range-compatible when the condition is only
assumed to apply to the operators whose range does not include a fixed
$1$-dimensional linear subspace of $V$. Among the range-compatible maps are the
so-called local maps $s \mapsto s(x)$ for fixed $x \in U$.
  Recently, the range-compatible group homomorphisms on $\mathcal{S}$ were
classified when $\mathcal{S}$ is a linear subspace of small codimension in
$\mathcal{L}(U,V)$. In this work, we consider several variations of that
problem: we investigate range-compatible affine maps on affine subspaces of
linear operators; when $\mathcal{S}$ is a linear subspace, we give the optimal
bound on its codimension for all quasi-range-compatible homomorphisms on
$\mathcal{S}$ to be local. Finally, we give the optimal upper bound on the
codimension of an affine subspace $\mathcal{S}$ of $\mathcal{L}(U,V)$ for all
quasi-range-compatible affine maps on it to be local.

Rings and Algebras