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  "body": "\n## Summary\n\nKoszul duality pairs a [[Koszul algebra|FTKFCCZ8YA]] $A$ with its quadratic dual $A^!$, which is again Koszul and computes the Ext algebra of the trivial $A$-module.\nThe basic pair is the symmetric algebra of a vector space and the exterior algebra of its dual.\nUnder suitable finiteness conditions it gives an equivalence between [[derived categories|5ST7NVJFSM]] of graded modules over $A$ and $A^!$.\nThe same pattern appears in representation theory, algebraic topology and the theory of operads.\n\n## Explanations\n\n### L1 human\n\nKoszul duality is a way to trade one algebra for another, simpler or more familiar one, without losing homological information.\nThe polynomial ring $k[x_1, \\dots, x_n]$, where the variables commute, is paired with the exterior algebra, where the variables anticommute.\nThe relations of one algebra are, roughly, the orthogonal complement of the relations of the other.\nProblems about modules over one side can be translated into problems about modules over the other side.\nFor instance, the Bernstein–Gelfand–Gelfand correspondence relates sheaves on projective space with modules over an exterior algebra.\n",
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