Theorem Galois Descent for Hopf Algebras
theorem admitted

Galois Descent for Hopf Algebras

Let \(E/k\) be a finite Galois extension with group \(\Gamma = \operatorname{Gal}(E/k)\). Let \(A\) be a commutative \(E\)-Hopf algebra equipped with a semilinear \(\Gamma\)-action: ring automorphisms \(\rho_\gamma : A \to A\) with \(\rho_1 = \mathrm{id}\), \(\rho_{\gamma\delta} = \rho_\gamma \circ \rho_\delta\), \(\rho_\gamma(c\,a) = \gamma(c)\,\rho_\gamma(a)\) for \(c \in E\), compatible with the Hopf structure: \[ \Delta \circ \rho_\gamma = (\rho_\gamma \otimes \rho_\gamma) \circ \Delta, \qquad \varepsilon(\rho_\gamma a) = \gamma\bigl(\varepsilon(a)\bigr), \qquad S \circ \rho_\gamma = \rho_\gamma \circ S, \] where the middle identity uses the semilinear action of \(\gamma\) on the codomain \(E\) of \(\varepsilon\), and the first uses the diagonal semilinear action on \(A \otimes_E A\) (well defined because \(\rho_\gamma \otimes \rho_\gamma\) is balanced over the \(\gamma\)-twist).

Then, writing \(B := A^\Gamma\) for the fixed points:

  1. \(B\) is a \(k\)-Hopf algebra under the restrictions of the structure maps of \(A\);
  2. the natural multiplication map \(E \otimes_k B \longrightarrow A\) is an isomorphism of \(E\)-Hopf algebras (where \(E \otimes_k B\) carries the base-change Hopf structure);
  3. the assignments \(A \mapsto A^\Gamma\) and \(B \mapsto E \otimes_k B\) are quasi-inverse equivalences between the category of commutative \(E\)-Hopf algebras with compatible semilinear \(\Gamma\)-action (morphisms: \(\Gamma\)-equivariant \(E\)-Hopf maps) and the category of commutative \(k\)-Hopf algebras; in particular \[ \operatorname{Hom}_{k\text{-Hopf}}(B, B') \;\cong\; \operatorname{Hom}_{\Gamma,\,E\text{-Hopf}}(A, A'). \]
Proof

Step 1: linear descent. By Galois Descent for Vector Spaces applied to the underlying \(E\)-vector space of \(A\), the fixed points \(B\) span \(A\) over \(E\) and \(k\)-linearly independent fixed vectors stay \(E\)-independent; hence \(E \otimes_k B \to A\) is an isomorphism of \(E\)-vector spaces. Applied to \(A \otimes_E A\) with the diagonal semilinear action, the same theorem gives \((A \otimes_E A)^\Gamma \cong B \otimes_k B\): the spanning half shows every fixed tensor is an \(E\)-combination of fixed decomposables, and independence identifies the \(k\)-structure.

Step 2: the structure maps restrict. Multiplication and unit are \(\Gamma\)-equivariant because the \(\rho_\gamma\) are ring maps. The compatibility hypotheses say exactly that \(\Delta\), \(\varepsilon\), \(S\) are \(\Gamma\)-equivariant; hence they carry fixed points to fixed points: \(\Delta(B) \subseteq (A \otimes_E A)^\Gamma = B \otimes_k B\), \(\varepsilon(B) \subseteq E^\Gamma = k\) (Galois), \(S(B) \subseteq B\). The Hopf axioms for the restricted maps hold because they hold in \(A\) and \(B \hookrightarrow A\), \(B \otimes_k B \hookrightarrow A \otimes_E A\) are injective (Step 1). This proves (1), and (2) upgrades the linear isomorphism of Step 1 to Hopf: it is a ring map by construction, and comultiplication is compatible because it is determined on the spanning set \(B\).

Step 3: homs. A \(\Gamma\)-equivariant \(E\)-Hopf map \(\phi : A \to A'\) carries fixed points to fixed points, giving \(\phi^\Gamma : B \to B'\), a \(k\)-Hopf map. Conversely a \(k\)-Hopf map \(\psi : B \to B'\) base-changes to \(E \otimes \psi : E \otimes_k B \to E \otimes_k B'\), which is \(\Gamma\)-equivariant for the actions \(\gamma \otimes 1\) — these correspond to the given actions under (2). The two constructions are mutually inverse: \(\phi\) is determined by its restriction to \(B\) because \(B\) spans \(A\) over \(E\) and \(\phi\) is \(E\)-linear. This proves (3). \(\square\)

Remarks

  • The compatibility of \(\Delta\) is stated through the diagonal semilinear action on \(A \otimes_E A\); making this precise is the only delicate point of the formalization (plan G0.C M3, where it is done for the concrete twisted group algebra rather than axiomatically — design decision D-d).
  • Specialized to \(A = E[M]\) with the action twisted by a lattice action of \(\Gamma\), this theorem produces all twisted forms of split tori; see The Twisted Form of a Galois Lattice.

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