Resumen de Semilinear representations of PGL

M. Rovinsky

• Let L be the function field of a projective space Pnk over an algebraically closed field k of characteristic zero, and H be the group of projective transformations. An H-sheaf V on Pnk 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-semilinear representations of H extendable to the semigroup End(L/k) to the category of coherent H-sheaves on Pnk.

  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 [4]. The semigroup End(L/k) is considered as a subquotient of G, hence the condition on extendability.

  In the appendix it is shown that, if \ifmmode\expandafter\else~\expandafter\~\fiH is either H, or a bigger subgroup in the Cremona group (generated by H and a certain pair of involutions), then any semilinear \ifmmode\expandafter\else~\expandafter\~\fiH - representation of degree one is an integral L-tensor power of detLΩ1L/k. It is also shown that this bigger subgroup has no non-trivial representations of finite degree if n > 1.