5 Semi - Linear Representations of Pgl

Let L be the function field of a projective space P n k over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf V on P n k is a collection of isomorphisms V −→ g * V for each g ∈ 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 End(L/k) to the category of coherent H-sheaves on 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 un-dertaken in [R]. The semi-group End(L/k) is considered as a subquotient of G, hence the condition on extendability. In the appendix it is shown that, if˜H is either H, or a bigger subgroup in the Cre-mona group (generated by H and a certain pair of involutions), then any semi-linear˜H-representation of degree one is an integral L-tensor power of detL Ω 1 L/k. It is shown also that this bigger subgroup has no non-trivial representations of finite degree if n > 1.