# 像就是余核的核　`prop.image-kernel-cokernel`
命题：$\operatorname{im} f = \ker(\operatorname{coker} f)$，每个态射都严格
layer 17 · 命题 · 加法与阿贝尔范畴 · 同调代数
[在星图里打开](../#n=prop.image-kernel-cokernel)　·　[节点索引](../index.md)

设 $\mathcal{C}$ 是阿贝尔范畴，$f : M \to N$。则

$$\operatorname{im} f \;=\; \ker\bigl(N \longrightarrow \operatorname{coker} f\bigr), \qquad \operatorname{coim} f \;=\; \operatorname{coker}\bigl(\ker f \longrightarrow M\bigr).$$
> 陈述续见 [nodes/prop.image-kernel-cokernel.2.md](./prop.image-kernel-cokernel.2.md)


## 为什么成立（入边，证明在 proofs/）
- …另有入边，续页见 [nodes/prop.image-kernel-cokernel.3.md](./prop.image-kernel-cokernel.3.md)

## 它能推出什么 / 谁在用它
- …另有出边，续页见 [nodes/prop.image-kernel-cokernel.4.md](./prop.image-kernel-cokernel.4.md)

refs: Le Stum, An introduction to condensed mathematics, Exercise 5.9

> 说明见 `notes/prop.image-kernel-cokernel.md`

引用：`prop.image-kernel-cokernel`（内容 v0.9.0）· 条款见 [LICENSE.txt](../LICENSE.txt)