Finite-flat affine group schemes and Galois descent via invariants of commutative Hopf algebras
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Abstract
Let L/K be a finite Galois extension with group Γ = Gal(L/K). Finite-flat affine group schemes over a field are equivalently encoded by finite-dimensional commutative Hopf algebras. This paper develops a clean Hopf-algebraic descent criterion for such group schemes: given a finite-dimensional commutative Hopf algebra HL over L equipped with a compatible semilinear Γ-action by Hopf algebra automorphisms, we show that the fixed subalgebra HK = HΓ is a commutative Hopf algebra over K and that base change recovers HL via an explicit canonical isomorphism HK ⊗K L =∼ HL. Consequently, the finite-flat affine group scheme GL = Spec(HL) descends to a finite-flat affine group scheme GK = Spec(HK) over K with GK ⊗K L =∼ GL. We also explain how nontrivial twists are measured by a Galois cohomology set, and we illustrate the criterion on standard families such as µn and constant finite group schemes.
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References
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