Invariant vectors span a semilinear representation #
For a field L over a commutative ring k with finite automorphism group, every semilinear
representation is spanned over L by its invariant vectors. No finite-dimensionality of the
vector space is required. This is the surjectivity step in Galois descent, applied in particular
to the coordinate algebra of a split torus with a Galois action on its character lattice.
The result has no characteristic restriction and does not require the order of the
automorphism group to be invertible in L.
References #
- J. S. Milne, Algebraic Groups (2017), Appendix A.64 (Galois descent).
theorem
TauCeti.GaloisDescent.span_invariants_eq_top
{k : Type u_1}
{L : Type u_2}
{V : Type u_3}
[CommRing k]
[Field L]
[Algebra k L]
[AddCommGroup V]
[Module k V]
[Module L V]
[Finite (L ≃ₐ[k] L)]
{ρ : Representation k (L ≃ₐ[k] L) V}
(hsemi : ∀ (σ : L ≃ₐ[k] L) (a : L) (v : V), (ρ σ) (a • v) = σ a • (ρ σ) v)
:
Invariant vectors of a semilinear action span the vector space over the coefficient field. Only finiteness of the automorphism group is needed; the extension need not be Galois.