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  "body": "\n## Summary\n\nA Koszul algebra is a graded [[associative algebra|3CTW1P4HBP]] over a field whose trivial module has a linear projective resolution.\nThis means that the $i$-th term of the resolution is generated in degree $i$.\nPolynomial algebras and [[exterior algebras|0MDMZ965VE]] are the basic examples, resolved by the [[Koszul complex|YZY11W08SD]].\nKoszul algebras are quadratic, and they are the setting of [[Koszul duality|6AW49BZ8TM]].\n\n## Explanations\n\n### L1 human\n\nTo understand a graded algebra one often resolves its simplest module, the field $k$, by free modules.\nFor most algebras the generators of these free modules sit in irregular degrees.\nA Koszul algebra is one where the $i$-th step is generated exactly in degree $i$, as tidy as possible.\nFor the polynomial ring $k[x]$ the resolution is $0 \\to k[x] \\xrightarrow{x} k[x] \\to k \\to 0$, with generators in degrees $0$ and $1$.\nThis tidiness makes many invariants of a Koszul algebra computable from its quadratic relations.\n",
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