additive_of_preserves_binary_products

English

Let X be simple and finite-dimensional over an algebraically closed field 𝕜. For any endomorphism f: X → X, there exists c ∈ 𝕜 such that c · id_X = f.

Русский

Пусть X прост и конечномерно над алгебраически замкнутым полем 𝕜. Для любого эндоморфизма f: X → X существует c ∈ 𝕜 такое, что c · id_X = f.

LaTeX

$$$\exists c \in \mathbb{k},\ c \cdot \mathrm{id}_X = f$$$

Lean4

theorem additive_of_preserves_binary_products [HasBinaryProducts C] [PreservesLimitsOfShape (Discrete WalkingPair) F]
    [F.PreservesZeroMorphisms] : F.Additive :=
  by
  have : HasBinaryBiproducts C := HasBinaryBiproducts.of_hasBinaryProducts
  have := preservesBinaryBiproducts_of_preservesBinaryProducts F
  exact Functor.additive_of_preservesBinaryBiproducts F

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