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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]], for example the category of modules over a ring.\nA *short exact sequence* in $\\mathcal{A}$ is a diagram\n$$0 \\to A \\xrightarrow{f} B \\xrightarrow{g} C \\to 0$$\nof objects and morphisms of $\\mathcal{A}$ such that:\n\n1. $f$ is a [[monomorphism|Y2NP1JFR86]];\n2. $g$ is an [[epimorphism|RVXDSY04KG]];\n3. the image of $f$ equals the [[kernel|S417YNEJ6M]] of $g$.\n\n## Remarks\n\nEquivalently, the sequence is [[exact|JB7ZBFBA9C]] at $A$, at $B$ and at $C$.\nEquivalently again, $g \\circ f = 0$, $f$ is a kernel of $g$, and $g$ is a cokernel of $f$.\nThen $C$ is isomorphic to the quotient $B / f(A)$, and $B$ is called an *extension* of $C$ by $A$.\n\nExample: $0 \\to \\Z \\xrightarrow{4} \\Z \\xrightarrow{\\pi} \\Z/4\\Z \\to 0$ is short exact, with $\\pi$ the reduction modulo $4$.\nMultiplication by $4$ is injective, $\\pi$ is onto, and the image $4\\Z$ of the first map is the kernel of $\\pi$.\n\nNon-example: $0 \\to 2\\Z \\xrightarrow{\\iota} \\Z \\xrightarrow{\\pi} \\Z/4\\Z \\to 0$, with $\\iota$ the inclusion, fails clause 3.\nClauses 1 and 2 hold, but the image of $\\iota$ is $2\\Z$ while the kernel of $\\pi$ is $4\\Z$.\n",
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