## 111.3 Additive and abelian categories

Exercise 111.3.1. Let $k$ be a field. Let $\mathcal{C}$ be the category of filtered vector spaces over $k$, see Homology, Definition 12.19.1 for the definition of a filtered object of any category.

Show that this is an additive category (explain carefully what the direct sum of two objects is).

Let $f : (V, F) \to (W, F)$ be a morphism of $\mathcal{C}$. Show that $f$ has a kernel and cokernel (explain precisely what the kernel and cokernel of $f$ are).

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

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

Show that this is an additive category.

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).

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.

Exercise 111.3.3. Give an example of a category which is additive and has kernels and cokernels but which is not as in Exercises 111.3.1 and 111.3.2.

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