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 $\mathbb {P} ^{n}$ and the stable category of graded modules $\operatorname {gr} \wedge V$ over the exterior algebra $\wedge V$ ; i.e.,

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 $A$ with coherent Koszul duals $A^{!}$ .^[3] Roughly speaking, they proved that the stable category of finite-dimensional graded modules over a finite-dimensional self-injective Koszul algebra $A$ is triangulated equivalent to the bounded derived category of the tails category of the Koszul dual $A^{!}$ (when $A^{!}$ is coherent).

References

^ J.-W. He and Q.-S. Wu. “Koszul differential graded algebras and BGG correspondence”. In: J. Algebra 320.7 (2008), pp. 2934–2962. arXiv: 0712.1324. url: https://doi.org/10.1016/j.jalgebra.2008.06.021.
^ Joseph Bernstein, Israel Gelfand, and Sergei Gelfand. Algebraic bundles over $P^{n}$ and problems of linear algebra. Funkts. Anal. Prilozh. 12 (1978); English translation in Functional Analysis and its Applications 12 (1978), 212-214
^ Martínez Villa, Roberto; Saorín, Manuel (2004). "Koszul Equivalence and Dualities" (PDF). Pacific Journal of Mathematics. 214 (2): 359–378. Retrieved 14 May 2025.

References

Further reading