In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototypical example of Koszul duality was introduced by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand. It establishes a duality between the derived category of a symmetric algebra and that of an exterior algebra, as well as the BGG correspondence, which links the stable category of finite-dimensional graded modules over an exterior algebra to the bounded derived category of coherent sheaves on projective space. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature.
Koszul duality for graded modules over Koszul algebras The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space V over a field k, with dual vector space V ∗ {\displaystyle V^{*}} , the exterior algebra of V has two non-trivial components, namely
⋀ 1 V = V , ⋀ 0 V = k . {\displaystyle \bigwedge ^{1}V=V,\quad \bigwedge ^{0}V=k.}
This exterior algebra and the symmetric algebra of V ∗ {\displaystyle V^{*}} , Sym ( V ∗ ) {\displaystyle \operatorname {Sym} (V^{*})} , serve to build a two-step chain complex
V ⊗ k Sym ( V ∗ ) → k ⊗ k Sym ( V ∗ ) {\displaystyle V\otimes _{k}\operatorname {Sym} (V^{*})\to k\otimes _{k}\operatorname {Sym} (V^{*})}
whose differential is induced by natural evaluation map
V ⊗ k V ∗ → k , v ⊗ k φ ↦ φ ( v ) . {\displaystyle V\otimes _{k}V^{*}\to k,\quad v\otimes _{k}\varphi \mapsto \varphi (v).}
Choosing a basis of V, Sym ( V ∗ ) {\displaystyle \operatorname {Sym} (V^{*})} can be identified with the polynomial ring in one variable, k [ t ] {\displaystyle k[t]} , and the previous chain complex becomes isomorphic to the complex
k [ t ] ⟶ t k [ t ] {\displaystyle k[t]{\stackrel {t}{\longrightarrow }}k[t]}
whose differential is multiplication by t. This computation shows that the cohomology of the above complex is 0 at the left hand term, and is k at the right hand term. In other words, k (regarded as a chain complex concentrated in a single degree) is quasi-isomorphic to the above complex, which provides a close link between the exterior algebra of V and the symmetric algebra of its dual.
Koszul dual of a Koszul algebra Koszul duality, as treated by Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel can be formulated using the notion of Koszul algebra. An example of such a Koszul algebra A is the symmetric algebra S ( V ) {\displaystyle S(V)} on a finite-dimensional vector space. More generally, any Koszul algebra can be shown to be a quadratic algebra, i.e., of the form
A = T ( V ) / R , {\displaystyle A=T(V)/R,}
where T ( V ) {\displaystyle T(V)} is the tensor algebra on a finite-dimensional vector space, and R {\displaystyle R} is a submodule of T 2 ( V ) = V ⊗ V {\displaystyle T^{2}(V)=V\otimes V} . The Koszul dual then coincides with the quadratic dual
A ! := T ( V ∗ ) / R ′ {\displaystyle A^{!}:=T(V^{*})/R'}
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