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  "body": "\n## Statement\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], such as the category of modules over a ring.\nConsider a [[commutative diagram|WV898P4X16]]\n$$\\begin{array}{ccccccccc} \u0026 \u0026 0 \u0026 \u0026 0 \u0026 \u0026 0 \u0026 \u0026 \\\\ \u0026 \u0026 \\downarrow \u0026 \u0026 \\downarrow \u0026 \u0026 \\downarrow \u0026 \u0026 \\\\ 0 \u0026 \\to \u0026 A_1 \u0026 \\to \u0026 B_1 \u0026 \\to \u0026 C_1 \u0026 \\to \u0026 0 \\\\ \u0026 \u0026 \\downarrow \u0026 \u0026 \\downarrow \u0026 \u0026 \\downarrow \u0026 \u0026 \\\\ 0 \u0026 \\to \u0026 A_2 \u0026 \\to \u0026 B_2 \u0026 \\to \u0026 C_2 \u0026 \\to \u0026 0 \\\\ \u0026 \u0026 \\downarrow \u0026 \u0026 \\downarrow \u0026 \u0026 \\downarrow \u0026 \u0026 \\\\ 0 \u0026 \\to \u0026 A_3 \u0026 \\to \u0026 B_3 \u0026 \\to \u0026 C_3 \u0026 \\to \u0026 0 \\\\ \u0026 \u0026 \\downarrow \u0026 \u0026 \\downarrow \u0026 \u0026 \\downarrow \u0026 \u0026 \\\\ \u0026 \u0026 0 \u0026 \u0026 0 \u0026 \u0026 0 \u0026 \u0026 \\end{array}$$\nin $\\mathcal{A}$ in which the three columns are [[short exact sequences|KJKNJMGKZZ]].\n\n1. If the second and third rows are short exact, then the first row is short exact.\n2. If the first and second rows are short exact, then the third row is short exact.\n\n## Remarks\n\nFor part 1, apply the [[snake lemma|Q8AYGD93CW]] to the map from the second row to the third row given by the vertical morphisms.\nSince the columns are exact, the kernels of these vertical morphisms are $A_1, B_1, C_1$ and their cokernels are $0$.\nThe exact sequence of the snake lemma becomes $0 \\to A_1 \\to B_1 \\to C_1 \\to 0$, which is the first row.\nPart 2 is the dual statement and follows in the same way from the map from the first row to the second row, whose cokernels are $A_3, B_3, C_3$.\nExample: let $A \\subseteq B \\subseteq C$ be modules.\nTake the rows $0 \\to A \\to A \\to 0 \\to 0$, then $0 \\to B \\to C \\to C/B \\to 0$, then $0 \\to B/A \\to C/A \\to C/B \\to 0$, with the evident maps.\nThe columns $0 \\to A \\to B \\to B/A \\to 0$, $0 \\to A \\to C \\to C/A \\to 0$ and $0 \\to 0 \\to C/B \\to C/B \\to 0$ are short exact, and so are the first two rows.\nPart 2 gives that the third row is short exact, which is the third isomorphism theorem $(C/A)/(B/A) \\cong C/B$.\n\n## Explanations\n\n### L1 human\n\nThe nine lemma is about a three-by-three grid of objects in which every column is a short exact sequence.\nIf two of the rows are short exact as well, either the bottom two or the top two, then the remaining row is short exact too.\nIt is a bookkeeping tool: exactness in two directions propagates through the grid.\n\n### L2 human\n\nThe lemma is also called the $3 \\times 3$ lemma.\nIt is a quick consequence of the snake lemma, and it is used to compare filtrations and to build short exact sequences of quotients.\nA typical use is the third isomorphism theorem in an abelian category.\nIt also appears when one shows that constructions such as the mapping cone or the total complex respect short exact sequences.\n",
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