Draft:The BBG correspondence

Bernstein-Gelfand-Gelfand correspondence (BGG correspondence for short), established by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand, is an explicit triangulated equivalence that relates the bounded derived category of coherent sheaves on the projective space $$\mathbb{P}^{n}$$ and the stable category of graded modules $$\underline{gr \wedge V}$$ over the Exterior algebra $$\wedge V$$. In the noncommutative setting, Martinez-Villa and Saorin R generalized the BGG correspondence to finite-dimensional self-injective Koszul algebras $$A $$ with coherent Koszul duals $$A^{!}$$. Roughly speaking, they proved that the stable category of finite-dimentional graded modules over a finite-dimensional self-injective Koszul algebra $$A$$ is triangulated equivalent to the bounded derived category of the category of coherent modules over its Koszul dual $$A^{!}$$ (when $$A^{!}$$ is coherent).