Knowlpedia

Theorem

Galois tensor-product identity

For a finite Galois extension, the self-tensor product splits into one copy for each automorphism.

Statement

Base-changing Spec⁡K→Spec⁡F\operatorname{Spec}K\to\operatorname{Spec}F along itself should reveal one sheet for every FF-automorphism of KK. On coordinate rings, that geometric splitting is exactly the following identity.

Let K/FK/F be a finite and G=Gal⁡(K/F)G=\operatorname{Gal}(K/F). The KK-algebra homomorphism

Φ:K⊗FK⟶∏σ∈GK,Φ(a⊗b)=(a σ(b))σ∈G\Phi:K\otimes_F K\longrightarrow\prod_{\sigma\in G}K, \qquad \Phi(a\otimes b)=\bigl(a\,\sigma(b)\bigr)_{\sigma\in G}

is an isomorphism. Taking spectra reverses products and arrows, giving

Spec⁡K×Spec⁡FSpec⁡K≅∐σ∈GSpec⁡K≅Spec⁡K×G.\operatorname{Spec}K\times_{\operatorname{Spec}F}\operatorname{Spec}K \cong \coprod_{\sigma\in G}\operatorname{Spec}K \cong \operatorname{Spec}K\times G.

This is the affine form of the and proves that a .