Koszul duality

Koszul duality, as treated by Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel[4] can be formulated using the notion of Koszul algebra. An example of such a Koszul algebra A is the symmetric algebra ${\displaystyle S(V)}$ on a finite-dimensional vector space. More generally, any Koszul algebra can be shown to be a quadratic algebra, i.e., of the form

${\displaystyle A=T(V)/R,}$

where ${\displaystyle T(V)}$ is the tensor algebra on a finite-dimensional vector space, and ${\displaystyle R}$ is a submodule of ${\displaystyle T^{2}(V)=V\otimes V}$. The Koszul dual then coincides with the quadratic dual

${\displaystyle A^{!}:=T(V^{*})/R'}$

where ${\displaystyle V^{*}}$ is the (k-linear) dual and ${\displaystyle R'\subset V^{*}\otimes V^{*}}$ consists of those elements on which the elements of R (i.e., the relations in A) vanish. The Koszul dual of ${\displaystyle A=S(V)}$ is given by ${\displaystyle A^{!}=\Lambda (V^{*})}$, the exterior algebra on the dual of V. In general, the dual of a Koszul algebra is again a Koszul algebra. Its opposite ring is given by the graded ring of self-extensions of the underlying field k, thought of as an A-module:

${\displaystyle (A^{!})^{\text{opp}}=\operatorname {Ext} _{A}^{*}(k,k).}$

If an algebra ${\displaystyle A}$ is Koszul, there is an equivalence between certain subcategories of the derived categories of graded ${\displaystyle A}$- and ${\displaystyle A^{!}}$-modules. These subcategories are defined by certain boundedness conditions on the grading vs. the cohomological degree of a complex.

As an alternative to passing to certain subcategories of the derived categories of ${\displaystyle A}$ and ${\displaystyle A^{!}}$ to obtain equivalences, it is possible instead to obtain equivalences between certain quotients of the homotopy categories.[5] Usually these quotients are larger than the derived category, as they are obtained by factoring out some subcategory of the category of acyclic complexes, but they have the advantage that every complex of modules determines some element of the category, without needing to impose boundedness conditions. A different reformulation gives an equivalence between the derived category of ${\displaystyle A}$ and the 'coderived' category of the coalgebra ${\displaystyle (A^{!})^{*}}$.

An extension of Koszul duality to D-modules states a similar equivalence of derived categories between dg-modules over the dg-algebra ${\displaystyle \Omega _{X}}$ of Kähler differentials on a smooth algebraic variety X and the ${\displaystyle D_{X}}$-modules. [6][7][8]