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  "body": "\n## Definition\n\nLet $K$ be a [[commutative ring|0GYGHKX834]].\nLet $A = \\bigoplus_{n \\in \\Z} A^n$ be a [[graded|CAPQBQDRA7]] [[associative algebra|3CTW1P4HBP]] over $K$, so that $A^m A^n \\subseteq A^{m+n}$; for $a \\in A^n$ write $\\abs{a} = n$.\nA *differential graded algebra* is such an $A$ with a $K$-linear map $d \\colon A \\to A$ such that:\n\n1. $d(A^n) \\subseteq A^{n+1}$ for all $n$.\n2. $d \\circ d = 0$.\n3. $d(ab) = d(a)\\, b + (-1)^{\\abs{a}} a\\, d(b)$ for all homogeneous $a \\in A$ and all $b \\in A$.\n\n## Remarks\n\nThis is the cohomological convention; in the homological convention $d$ lowers the degree by one.\nThe cohomology $H(A) = \\ker d / \\operatorname{im} d$ is a graded algebra, because clause 3 makes $\\ker d$ a subalgebra and $\\operatorname{im} d$ a two-sided ideal of it.\nExample: let $K$ be a field, $A^0 = K[x]$, $A^1 = K[x]\\, dx$, and $A^n = 0$ otherwise, with $(dx)^2 = 0$.\nPut $d(f) = f'\\, dx$ on $A^0$ and $d = 0$ on $A^1$.\nClauses 1 and 2 are clear, and clause 3 reduces to the product rule $(fg)' = f'g + fg'$; for $a = f$, $b = g\\,dx$ both sides are $0$ since $(dx)^2 = 0$.\nNon-example: in the same algebra put $d(f) = f\\, dx$ instead.\nClauses 1 and 2 hold, but clause 3 fails: $d(1 \\cdot 1) = dx$ while $d(1)\\cdot 1 + 1 \\cdot d(1) = 2\\, dx$.\nEvery graded algebra with $d = 0$ is a differential graded algebra.\n",
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