BGG correspondence

In mathematics, the Bernstein-Gelfand-Gelfand correspondence or BGG correspondence for short is the first example of the Koszul duality.^[1]

Established by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand,^[2] the correspondence is an explicit triangulated equivalence that relates the bounded derived category of coherent sheaves on the projective space ${\displaystyle \mathbb {P} ^{n}}$ and the stable category of graded modules ${\displaystyle \operatorname {gr} \wedge V}$ over the exterior algebra ${\displaystyle \wedge V}$; i.e.,

${\displaystyle D^{b}(\mathbb {P^{n}} )\simeq {\overline {\operatorname {gr} \wedge V}}}$.

In the noncommutative setting, Martínez Villa and Saorín generalized the BGG correspondence to finite-dimensional self-injective Koszul algebras ${\displaystyle A}$ with coherent Koszul duals ${\displaystyle A^{!}}$.^[3] Roughly speaking, they proved that the stable category of finite-dimensional graded modules over a finite-dimensional self-injective Koszul algebra ${\displaystyle A}$ is triangulated equivalent to the bounded derived category of the tails category of the Koszul dual ${\displaystyle A^{!}}$ (when ${\displaystyle A^{!}}$ is coherent).