Koszul duality

Summary

Koszul duality pairs a Koszul algebra $A$ with its quadratic dual $A^!$, which is again Koszul and computes the Ext algebra of the trivial $A$-module. The basic pair is the symmetric algebra of a vector space and the exterior algebra of its dual. Under suitable finiteness conditions it gives an equivalence between derived categories of graded modules over $A$ and $A^!$. The same pattern appears in representation theory, algebraic topology and the theory of operads.