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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]].\nLet $A = \\bigoplus_{n \\ge 0} A_n$ be a [[graded|8KB4SQGC0C]] [[associative algebra|3CTW1P4HBP]] over $k$ with $A_0 = k$ and each $A_n$ finite-dimensional.\nLet $k$ also denote the graded $A$-module $A/A_{\u003e0}$, where $A_{\u003e0} = \\bigoplus_{n \u003e 0} A_n$.\nConsider a graded [[projective resolution|HK2AA1YSQ2]] of $k$ by graded free $A$-modules:\n$$\\cdots \\to P_2 \\to P_1 \\to P_0 \\to k \\to 0.$$\nThe algebra $A$ is *Koszul* if $k$ has such a resolution in which each $P_i$, $i \\ge 0$, is generated by homogeneous elements of degree $i$.\n\n## Remarks\n\nExample: $A = k[x]$ with $\\deg x = 1$.\nThe resolution $0 \\to A(-1) \\xrightarrow{x} A \\to k \\to 0$ has $P_0 = A$ generated in degree $0$ and $P_1 = A(-1)$ generated in degree $1$.\nMore generally, the [[Koszul complex|YZY11W08SD]] is a linear resolution of $k$ over $k[x_1, \\dots, x_n]$.\nNon-example: $A = k[x]/(x^3)$ with $\\deg x = 1$.\nThe minimal resolution starts with $P_0 = A$ and $P_1 = A(-1)$, mapping the generator to $x$.\nThe kernel of multiplication by $x$ on $A$ is spanned by $x^2$, so $P_2$ is generated in degree $1 + 2 = 3$, not $2$.\nEquivalent form: $A$ is Koszul if and only if $\\operatorname{Ext}^i_A(k, k)$ is concentrated in internal degree $i$ for every $i$.\nEvery Koszul algebra is quadratic: it is generated by $A_1$ with relations in degree $2$.\n",
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