Category theory; homological algebra Derived functors Theorem

Yoneda Ext and extensions

Builds on Ring Module over a ring Ext functor Short exact sequence Abelian category

Statement

Let $R$ be a ring and let $M$ and $N$ be $R$ -modules. An extension of $M$ by $N$ is a short exact sequence $0 \to N \to E \to M \to 0$ 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 $M$ .

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 $0 \to N \to N \oplus M \to M \to 0$ .
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, even without enough projectives or injectives; this is the Yoneda Ext.

Going deeper

For a student

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. 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.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Tag	`WXRH0CYKYF`
URI	`m2://WXRH0CYKYF`
Kind	statement, theorem
Writing	W0 stub
MSC	18G15, 13D07, 16E30
Status	This copy of the site has no build data. Run `m2 status WXRH0CYKYF`.
About	Ext functor
Uses	Ring, Module over a ring, Ext functor, Short exact sequence, Abelian category