The Stacks project

Proposition 108.73.3. Let $A$ be a Noetherian integral domain. Let $K$ denote its field of fractions. Let $\text{Mod}_ A^{fg}$ denote the category of finitely generated $A$-modules and $\mathcal{T}^{fg}$ its Serre subcategory of finitely generated torsion modules. Then $\text{Mod}_ A^{fg}/\mathcal{T}^{fg}$ is canonically equivalent to the category of finite dimensional $K$-vector spaces.

Proof. The equivalence given in Proposition 108.73.2 restricts along the embedding $\text{Mod}_ A^{fg}/\mathcal{T}^{fg} \to \text{Mod}_ A/\mathcal{T}$ to an equivalence $\text{Mod}_ A^{fg}/\mathcal{T}^{fg} \to \text{Vect}_ K^{fd}$. The Noetherian assumption guarantees that $\text{Mod}_ A^{fg}$ is an abelian category (see More on Algebra, Section 15.53) and that the canonical functor $\text{Mod}_ A^{fg}/\mathcal{T}^{fg} \to \text{Mod}_ A/\mathcal{T}$ is full (else torsion submodules of finitely generated modules might not be objects of $\mathcal{T}^{fg}$). $\square$


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