Semilinear representations of PGL

Abstract

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-semilinear representations of H extendable to the semigroup 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 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, ifH is either H, or a bigger subgroup in the Cremona group (generated by H and a certain pair of involutions), then any semilinearH-representation of degree one is an integral L-tensor power of det L Ω 1 L/k . It is also shown that this bigger subgroup has no non-trivial representations of finite degree if n > 1.

On maximal proper subgroups of field automorphism groups

“…Suppose n = ∞. Then (W 1 R2,Lemma 7.5]) implies the first inclusion, the second follows from the first.…”

Stable Birational Invariants with Galois Descent and Differential Forms

“…If the transcendence degree of L|k is minimal then, by [10], L is U -invariant, so A(F ) (U ) = A(L). ) 8 Schur's lemma= [ Proof. Let p ρ be the central projector in the group algebra of Q = K/ ker ρ onto the ρ-isotypical part.…”

“…Let p ρ be the central projector in the group algebra of Q = K/ ker ρ onto the ρ-isotypical part. As explained in [8,Prop.7.6], W contains a non-zero element ω fixed by the group G F |k(P M ) for an appropriate M ≥ 1 and an embedding k(P M ) ֒→ F . The finite field extension F ker ρ |F K can be considered as a purely transcendental extension of a function field extension k(Y )|k(Y ) Q of smooth projective k-varieties of dimension ≥ M .…”

“…[8]). Any simple A 1 -invariant sheaf can be embedded into the sheaf Ω• |k : Y → Ω • k(Y )|k .This conjecture is rather strong: it implies the Bloch's conjecture:"Corollary" 1.6 ([8]).…”

“…Proof. By[8, Lemma 7.7], the semilinear representation Ω j F |k is irreducible for any j ≥ 0. In particular, F -linear envelope of the G-orbit of ω is the direct sum of Ω j F |k over all j ≥ i such that the homogeneous component of ω of degree j is non-zero.…”