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).

References

1. 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.
2. Joseph Bernstein, Israel Gelfand, and Sergei Gelfand. Algebraic bundles over ${\displaystyle P^{n}}$ and problems of linear algebra. Funkts. Anal. Prilozh. 12 (1978); English translation in Functional Analysis and its Applications 12 (1978), 212-214
3. Martínez Villa, Roberto; Saorín, Manuel (2004). "Koszul Equivalence and Dualities" (PDF). Pacific Journal of Mathematics. 214 (2): 359–378. Retrieved 14 May 2025.

Further reading

• Peter Jørgensen, A noncommutative BGG correspondence, February 2005, Pacific Journal of Mathematics 218(2):357-377, DOI:10.2140/pjm.2005.218.357
• http://pantodon.jp/index.rb?body=Koszul_duality in Japanese
• https://www.mathsoc.jp/section/algebra/algsymp_past/algsymp13_files/yamaura.pdf in Japanese