[Paper Review] Betti Numbers of Syzygies and Cohomology of Coherent Sheaves

David Eisenbud, Frank–Olaf Schreyer|arXiv (Cornell University)|Feb 17, 2011

Commutative Algebra and Its Applications16 references18 citations

TL;DR

This paper provides a simplified proof of the Boij-Söderberg conjecture, characterizing the rational cone of Betti tables of finitely generated graded modules over a polynomial ring via pure resolutions. It establishes that Betti tables and cohomology tables of coherent sheaves on projective space are fully described by extremal rays corresponding to supernatural bundles, leading to a proof of the Multiplicity Conjecture and a deeper understanding of syzygy invariants.

ABSTRACT

The Betti numbers of a graded module over the polynomial ring form a table of numerical invariants that refines the Hilbert polynomial. A sequence of papers sparked by conjectures of Boij and Söderberg have led to the characterization of the possible Betti tables up to rational multiples---that is, to the rational cone generated by the Betti tables. We will summarize this work by describing the cone and the closely related cone of cohomology tables of vector bundles on projective space, and we will give new, simpler proofs of some of the main results. We also explain some of the applications of the theory, including the one that originally motivated the conjectures of Boij and Söderberg, a proof of the Multiplicity Conjecture of Herzog, Huneke and Srinivasan.

Motivation & Objective

• To provide new, simplified proofs of the main results on the Boij-Söderberg cone of Betti tables of graded modules.
• To clarify the duality between Betti tables of modules and cohomology tables of coherent sheaves on projective space.
• To establish the existence of pure resolutions and characterize the extremal rays of the Betti table cone.
• To apply the theory to prove the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan.
• To extend the framework to arbitrary coherent sheaves and explore its implications for Ulrich sheaves and varieties.

Proposed method

• The authors use the duality between Betti tables of modules and cohomology tables of coherent sheaves on projective space to relate the structure of the Boij-Söderberg cone to extremal rays of cohomology tables.
• They identify the extremal rays of the cohomology table cone as corresponding to supernatural vector bundles, which are shown to exist via explicit constructions.
• The proof relies on analyzing Hilbert series and comparing them to normalized Betti tables of pure resolutions to establish inequalities that characterize the cone.
• The authors employ convex geometry to describe the rational cone of Betti tables as the convex hull of pure resolution tables, up to rational scaling.
• They use the fact that cohomology tables of arbitrary coherent sheaves are infinite convergent sums of supernatural sheaf tables with non-negative coefficients.
• The theory is extended to arbitrary modules and coherent sheaves on projective varieties, with a characterization of when the cone coincides with that of projective space via the existence of Ulrich sheaves.

Experimental results

Research questions

• RQ1 What is the complete structure of the rational cone generated by Betti tables of finitely generated graded modules over a polynomial ring?
• RQ2 How are the extremal rays of the Betti table cone related to the cohomology tables of vector bundles on projective space?
• RQ3 Can the Multiplicity Conjecture be proven using the Boij-Söderberg decomposition of Betti tables?
• RQ4 Under what conditions does the Boij-Söderberg cone of coherent sheaves on a variety X coincide with that of projective space?
• RQ5 What is the role of Ulrich sheaves in determining the cohomology table cone of a projective variety?

Key findings

• The Boij-Söderberg cone of Betti tables is completely characterized as the rational convex cone spanned by the Betti tables of pure resolutions, up to rational scaling.
• The extremal rays of the Betti table cone correspond to Betti tables of modules that are Cohen-Macaulay and have all Betti numbers concentrated in a single degree per syzygy level.
• The Multiplicity Conjecture is proven: the multiplicity of a graded module is bounded above by β₀₀(M) · (b₁⋯bₛ)/s!, with equality if and only if the module is Cohen-Macaulay with a pure resolution.
• For any coherent sheaf on ℙⁿ, its cohomology table is an infinite convergent sum with non-negative coefficients of cohomology tables of supernatural sheaves.
• The cone of cohomology tables of coherent sheaves on a variety X is isomorphic to that of ℙᵈ if and only if X admits an Ulrich sheaf.
• The monoid of actual Betti tables is finitely generated for Cohen-Macaulay modules with bounded degree sequences, though it depends on the characteristic of the ground field.