Boij-S\"oderberg theory: Introduction and survey

TL;DR

This paper introduces and surveys Boij-S"oderberg theory, which characterizes Betti diagrams of graded modules, and related Eisenbud-Schreyer theory for cohomology tables, highlighting their foundational concepts and developments.

Contribution

It provides an accessible introduction and comprehensive survey of the recent advances in Boij-S"oderberg and Eisenbud-Schreyer theories.

Findings

Betti diagrams are characterized up to rational multiples.

Cohomology tables of vector bundles are similarly described.

The theories unify and extend previous understanding of algebraic and geometric invariants.

Abstract

Boij-S\"oderberg theory describes the Betti diagrams of graded modules over the polynomial ring up to multiplication by a rational number. Analog Eisenbud-Schreyer theory also describes the cohomology tables of vector bundles on projective spaces up to rational multiple. We give an introduction and survey of these newly developed areas.