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  "body": "\n## Statement\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $M$ and $N$ be [[$R$-modules|0NZ8JA25WK]].\nAn *extension* of $M$ by $N$ is a [[short exact sequence|KJKNJMGKZZ]] $0 \\to N \\to E \\to M \\to 0$ of $R$-modules.\nTwo extensions with middle terms $E$ and $E'$ are *equivalent* if some homomorphism $E \\to E'$ commutes with the maps from $N$ and to $M$.\n\n1. 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 $0 \\to N \\to N \\oplus M \\to M \\to 0$.\n2. 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.\n\n## Remarks\n\nGiven 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.\nConversely, 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.\nThe group law on $\\operatorname{Ext}^1$ becomes the Baer sum of extensions.\nExample: for a prime $p$, $\\operatorname{Ext}^1_\\Z(\\Z/p\\Z, \\Z/p\\Z) \\cong \\Z/p\\Z$.\nIts $p$ 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 $\\Z/p\\Z$.\nThe description in part 2 makes sense in any [[abelian category|Y9CZ3MTNF1]], even without enough projectives or injectives; this is the *Yoneda Ext*.\n\n## Explanations\n\n### L1 human\n\nThe 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.\nThe trivial way, the direct sum, is the zero element.\nFor example, $\\Z/p\\Z$ and $\\Z/p\\Z$ can be glued into $\\Z/p\\Z \\oplus \\Z/p\\Z$ or into $\\Z/p^2\\Z$.\n\n### L2 human\n\nThis is how Ext got its name: it classifies extensions.\nThe 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]].\nIn 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]].\n",
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