Abstract.
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 \(\mathcal{V} \to g^{ * } \mathcal{V}\) 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 \(\mathbb{P}^{n}_{k} .\)
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 [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 \({ \ifmmode\expandafter\tilde\else\expandafter\~\fi{H}}\) is either H, or a bigger subgroup in the Cremona group (generated by H and a certain pair of involutions), then any semilinear \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{H}{\text{ - representation}}\) of degree one is an integral L-tensor power of \(\hbox{det}_{L} \Omega ^{1}_{{L/k}} .\) It is also shown that this bigger subgroup has no non-trivial representations of finite degree if n > 1.
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Rovinsky, M. Semilinear representations of PGL. Sel. math., New ser. 11, 491 (2006). https://doi.org/10.1007/s00029-005-0015-8
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DOI: https://doi.org/10.1007/s00029-005-0015-8