I'm trying to understand a particular point in the proof of Koszul duality between $D^b(\Lambda(V))$ and $D^b(S(V^*))$ as seen in "Algebraic Bundles over $\mathbb{P}^n$ and Problems of Linear Algebra" by Bernstein, Gelfand and Gelfand. In particular, I am trying to work through Prop. 3, where we introduce functors

$F: C^b(\Lambda(V)) \rightarrow C^b(S(V^*)) $ and

$G: C(S(V^*)) \rightarrow C(\Lambda(V))$.

The goal of the proof is to show these functors define an equivalence of derived categories.

I have two particular questions:

1) In the proof, it is first demonstrated that $F$ and $G$ are adjoint. While I can prove this, I do not understand how it helps the argument; I cannot find a source which connects adjoint functors with equivalences on the derived category level.

2) In both BGG and in Gelfand-Manin "Methods of Homological Algebra", the next step is to show that $F$ and $G$ lift to derived functors, and that they form an equivalence of categories; both sources simply say use the Koszul complex for this step. As for the lifting, I know I need to demonstrate that $F$ and $G$ map acyclic complexes to acyclic complexes, but I don't see how to use the acyclicity of the Koszul resolution here. Moreover, once this is done, I'm stuck again on showing that $F_D$ and $G_D$ actually form an equivalence of categories. How does all of this follow from the properties of the Koszul complex?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.