---
created: 2022-04-05T23:42
updated: 2024-04-19T16:19
---

---
date: 2021-10-21 18:42
modification date: Saturday 23rd October 2021 22:40:55
title: mapping cone
aliases: [mapping cone]

---

Tags: 
#todo #todo/learning/definitions 
Refs:
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# mapping cone

## Of chain complexes

For $f\in \Ch_{\rmod}(A,  B)$ a morphism of chain complex of $R\dash$modules, the **mapping cone** complex is 
\[
\cone(f) := A[1] \oplus B, 
\quad d =
\begin{bmatrix}
d_A & 0 
\\
f & d_B
\end{bmatrix}
\]

![](attachments/Pasted%20image%2020220429191554.png)

### Results

- If $\cone(f) \homotopic 0$, i.e $\cone(f)$ is an [quasi-isomorphism)](quasi-isomorphism.md)).

\[\begin{tikzcd}
	A && B \\
	\\
	& {\cone(f)}
	\arrow["f", from=1-1, to=1-3]
	\arrow[from=1-3, to=3-2]
	\arrow["{[1]}", from=3-2, to=1-1]
\end{tikzcd}\]

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJBIl0sWzIsMCwiQiJdLFsxLDIsIlxcY29uZShmKSJdLFswLDEsImYiXSxbMSwyXSxbMiwwLCJbMV0iXV0=)

![](attachments/Pasted%20image%2020220326014050.png)
![](attachments/Pasted%20image%2020220326014127.png)