On the Bernstein-Gel'fand-Gel'fand correspondence and a result of Eisenbud, Fl{\o}ystad, and Schreyer

TL;DR

This paper combines classical Bernstein-Gel'fand-Gel'fand (BGG) ideas with modern Tate resolution techniques to provide streamlined proofs of key results in algebraic geometry and homological algebra.

Contribution

It introduces a lemma connecting BGG and Tate resolutions, simplifying proofs of important theorems in the field.

Findings

Quick proofs of main results from BGG and Eisenbud et al.

A new lemma linking cohomology of sheaves with Tate resolutions.

Enhanced understanding of the BGG correspondence's applications.

Abstract

We show that a combination between a remark from the well known note of I.N. Bernstein, I.M. Gel'fand and S.I. Gel'fand and the idea, systematically investigated in a recent work of D. Eisenbud, G. Fl{\o}ystad and F.-O. Schreyer, of taking Tate resolution over exterior algebras leads to quick proofs of the main results of these two papers. This combination is expressed by a lemma which we prove directly using the cohomology of invertible sheaves on a projective space.