---
tag: WXRH0CYKYF
kind: statement
role: theorem
name: Yoneda Ext and extensions
msc: [18G15, 13D07, "16E30"]
uses: [0B1261B5ED, 0NZ8JA25WK, 4P7GBWJ4RR, KJKNJMGKZZ, Y9CZ3MTNF1]
about: [4P7GBWJ4RR]
sources:
  - url: https://en.wikipedia.org/wiki/Ext_functor
---

## Statement

Let $R$ be a [[ring|0B1261B5ED]] and let $M$ and $N$ be [[$R$-modules|0NZ8JA25WK]].
An *extension* of $M$ by $N$ is a [[short exact sequence|KJKNJMGKZZ]] $0 \to N \to E \to M \to 0$ of $R$-modules.
Two extensions with middle terms $E$ and $E'$ are *equivalent* if some homomorphism $E \to E'$ commutes with the maps from $N$ and to $M$.

1. There is a natural bijection between $\operatorname{Ext}^1_R(M, N)$ and the set of equivalence classes of extensions of $M$ by $N$. Under it, $0$ corresponds to the class of the split extension $0 \to N \to N \oplus M \to M \to 0$.
2. Let $n \ge 1$, and consider exact sequences $0 \to N \to E_n \to \cdots \to E_1 \to M \to 0$. Use the equivalence relation generated by maps of such sequences that are the identity on $N$ and on $M$. Then there is a natural bijection between $\operatorname{Ext}^n_R(M, N)$ and the set of classes.

## Remarks

Given an extension, apply $\Hom_R(-, N)$ and take the image of $\id_N$ under the connecting map $\Hom_R(N, N) \to \operatorname{Ext}^1_R(M, N)$; this gives the class of the extension.
Conversely, a class in $\operatorname{Ext}^1_R(M, N)$ is represented by a map $P_1 \to N$ from the first term of a projective resolution of $M$, and a pushout along this map gives the extension.
The group law on $\operatorname{Ext}^1$ becomes the Baer sum of extensions.
Example: for a prime $p$, $\operatorname{Ext}^1_\Z(\Z/p\Z, \Z/p\Z) \cong \Z/p\Z$.
Its $p$ elements are the split extension and the $p - 1$ classes of $0 \to \Z/p\Z \xrightarrow{\cdot p} \Z/p^2\Z \to \Z/p\Z \to 0$, whose quotient maps differ by the automorphisms of $\Z/p\Z$.
The description in part 2 makes sense in any [[abelian category|Y9CZ3MTNF1]], even without enough projectives or injectives; this is the *Yoneda Ext*.

## Explanations

### L1 human

The group $\operatorname{Ext}^1(M, N)$ has a concrete meaning: it lists the ways to glue $N$ and $M$ into a bigger module $E$ with $N$ inside and $M$ as quotient.
The trivial way, the direct sum, is the zero element.
For example, $\Z/p\Z$ and $\Z/p\Z$ can be glued into $\Z/p\Z \oplus \Z/p\Z$ or into $\Z/p^2\Z$.

### L2 human

This is how Ext got its name: it classifies extensions.
The higher groups $\operatorname{Ext}^n$ classify longer exact sequences, and the Yoneda product, which splices sequences end to end, corresponds to the composition of morphisms $M \to N[n]$ in the [[derived category|5ST7NVJFSM]].
In group theory, the same idea shows that $H^2$ of a group with coefficients in a module classifies extensions with abelian kernel, see [[group cohomology|YXJXA0Q52R]].