  Let $L$ be the function field of a projective space ${\mathbb P}^n_k$ over an
algebraically closed field $k$ of characteristic zero, and $H$ be the group of
projective transformations. An $H$-sheaf ${\mathcal V}$ on ${\mathbb P}^n_k$ is
a collection of isomorphisms ${\mathcal V} \longrightarrow g^{\ast}{\mathcal
V}$ for each $g\in H$ satisfying the chain rule.
  We construct, for any $n>1$, a fully faithful functor from the category of
finite-dimensional $L$-semi-linear representations of $H$ extendable to the
semi-group ${\rm End}(L/k)$ to the category of coherent $H$-sheaves on
${\mathbb P}^n_k$.
  The paper is motivated by a study of admissible representations of the
automorphism group $G$ of an algebraically closed extension of $k$ of countable
transcendence degree undertaken in \cite{rep}. The semi-group ${\rm End}(L/k)$
is considered as a subquotient of $G$, hence the condition on extendability.
  In the appendix it is shown that, if $\tilde{H}$ is either $H$, or a bigger
subgroup in the Cremona group (generated by $H$ and a standard involution),
then any semi-linear $\tilde{H}$-representation of degree one is an integral
$L$-tensor power of $\det_L\Omega^1_{L/k}$. It is shown also that this bigger
subgroup has no non-trivial representations of finite degree if $n>1$.
