The Stacks project

Exercise 111.3.2. Let $R$ be a Noetherian domain. Let $\mathcal{C}$ be the category of finitely generated torsion free $R$-modules.

  1. Show that this is an additive category.

  2. Let $f : N \to M$ be a morphism of $\mathcal{C}$. Show that $f$ has a kernel and cokernel (make sure you define precisely what the kernel and cokernel of $f$ are).

  3. Give an example of a Noetherian domain $R$ and a map of $\mathcal{C}$ such that the canonical map $\mathop{\mathrm{Coim}}(f) \to \mathop{\mathrm{Im}}(f)$ is not an isomorphism.

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View Exercise 111.3.2 as pdf