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The Deligne product of finite vector spaces is finite vector spaces

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and let $v e c t$ be the category of finite-dimensional $k$ -vector spaces (Vector space over a field, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis), a finite $k$ -linear abelian category whose algebra model is $k - m o d$ for the algebra $k$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms, k-linear categories and k-linear functors, Abelian category). Then Finite Deligne products exist via tensor-product algebras with $R = S = k$ identifies $v e c t ⊠ v e c t$ with $k \otimes_{k} k - m o d = v e c t$ : the tensor-product algebra is $k \otimes_{k} k ≅ k$ (The tensor product of $R$ -algebras has multiplication $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ ), the universal bifunctor is the ordinary tensor product $\otimes_{k} : v e c t \times v e c t \to v e c t$ , and the universal property is that of The Deligne product of finite linear categories. The equivalence respects the universal bifunctors up to canonical natural isomorphism (Equivalence, quasi-inverse, and adjoint equivalence of categories).

Example

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice), a field $k$ , $v e c t$ the category of finite-dimensional $k$ -vector spaces, the algebra $k$ , and the algebra $k \otimes_{k} k$ .

[F1]

A left $k$ -module is exactly a $k$ -vector space, and the finite-dimensional $k$ -vector spaces form the category $v e c t$ , so the algebra model of $v e c t$ is $k - m o d$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms, Vector space over a field, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis).

[F2]

The $k$ -algebra $k \otimes_{k} k$ has multiplication $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ and unit $1 \otimes 1$ (The tensor product of $R$ -algebras has multiplication $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ ).

[F3]

For finite-dimensional $k$ -algebras $R, S$ the category $(R \otimes_{k} S) - m o d$ with the tensor bifunctor is a Deligne product of $R - m o d$ and $S - m o d$ , and a Deligne product is unique up to an equivalence respecting the universal bifunctors, its universal property being an equivalence between $k$ -linear right exact functors out of it and $k$ -linear bifunctors right exact in each variable (Finite Deligne products exist via tensor-product algebras, The Deligne product of finite linear categories).

Verification

1.1givenF1F2

The algebra $k$ is finite-dimensional and unital over the field $k$ [F1], and the multiplication of [F2] on $k \otimes_{k} k$ satisfies $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ ; the $k$ -linear map $k \to k \otimes_{k} k$ , $a \mapsto a \otimes 1$ , has the multiplication map $k \otimes_{k} k \to k$ , $\sum_{i} a_{i} \otimes b_{i} \mapsto \sum_{i} a_{i} b_{i}$ , as a two-sided inverse, so $k \otimes_{k} k ≅ k$ as $k$ -algebras and $(k \otimes_{k} k) - m o d = v e c t$ [F1].

2.1step 1.1F3

By [F3] with $R = S = k$ , the category $(k \otimes_{k} k) - m o d$ together with the tensor bifunctor $(X, Y) \mapsto X \otimes_{k} Y$ is a Deligne product of $k - m o d$ and $k - m o d$ ; under the algebra isomorphism of step 1.1, a module over $k \otimes_{k} k$ is a $k$ -vector space, so $(k \otimes_{k} k) - m o d = v e c t = k - m o d$ , and the universal bifunctor is the ordinary tensor product $\otimes_{k}$ on $v e c t$ (k-linear categories and k-linear functors).

3.1step 2.1F3∎

Hence $v e c t ⊠ v e c t$ is identified with $v e c t$ , the universal bifunctor being $\otimes_{k} : v e c t \times v e c t \to v e c t$ , and its universal property is exactly the defining property of a Deligne product of finite $k$ -linear categories [F3]; since Deligne products are unique up to an equivalence respecting the universal bifunctors, the identification respects the universal bifunctors up to canonical natural isomorphism (Equivalence, quasi-inverse, and adjoint equivalence of categories, The Deligne product of finite linear categories).

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