Sheaf Cohomology and Free Resolutions over Exterior Algebras

Abstract

In this paper we study the Bernstein-Gel'fand-Gel'fand (BGG) correspondence linking sheaves on a projective space to graded modules over an exterior algebra. We give an explicit construction of a Beilinson monad for a sheaf on projective space. The explicitness allows us to to prove two conjectures about the morphisms in the monad. We also construct all the monads for a sheaf that can be built from sums of line bundles. A large subclass, containing many monads introduced by others, are uniquely characterized by simple numerical data. Our methods also yield an efficient method for machine computation of the cohomology of sheaves. Along the way we study minimal free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their "linear parts" in the sense that erasing all terms of degree $>1$ in the complex yields a new complex which is eventually exact. This paper is a joint, extended version of papers previously written by Eisenbud-Schreyer and by Fløystad seperately.

36 pages, new improved version of last Section 8 on free monads

References in corpus (4)

• Sheaf Cohomology and Free Resolutions over Exterior Algebras
• Describing coherent sheaves on projective spaces via Koszul duality
• Exterior algebra methods for the Minimal Resolution Conjecture
• A Fitting Lemma for Z/2-graded modules

Cited by in corpus (59)

• Sheaf Cohomology and Free Resolutions over Exterior Algebras
• GL-equivariant modules over polynomial rings in infinitely many variables
• Edge ideals: algebraic and combinatorial properties
• Boij-Söderberg theory: Introduction and survey
• Linearity defects of modules over commutative rings
• Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence
• Categorified duality in Boij-Söderberg Theory and invariants of free complexes
• Tate Resolutions for Products of Projective Spaces
• Globally Generated Vector Bundles on P^n with c_1=3
• Globally Generated Vector Bundles on P^n with c_1=4
• Linearity defect of edge ideals and Fröberg's theorem
• Real Fibered Morphisms and Ulrich Sheaves
• Minimal resolutions of monomial ideals
• (Bi-)Cohen-Macaulay simplicial complexes and their associated coherent sheaves
• Homological properties of Orlik-Solomon algebras
• On the Ext-computability of Serre quotient categories
• On monads of exact reflective localizations of Abelian categories
• A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2
• Enriched homology and cohomology modules of simplicial complexes
• Inequalities for the Hodge numbers of irregular compact Kaehler manifolds
• Betti Numbers of Graded Modules and Cohomology of Vector Bundles
• Zipping Tate resolutions and exterior coalgebras
• Positivity and nonstandard graded Betti numbers
• The Banks of the Cohomology River
• Computing representation matrices for the action of Frobenius to cohomology groups
• Supernatural analogues of Beilinson monads
• A constructive approach to the module of twisted global sections on relative projective spaces
• The exterior algebra and central notions in mathematics
• Current Challenges in Developing Open Source Computer Algebra Systems
• Globally generated vector bundles with $c_1=5$ on ${\mathbb{P}}^3$
• Homology of subgroups of right-angled Artin groups
• Poset structures in Boij-Söderberg theory
• Questions about Boij-Söderberg theory
• Betti numbers of Z^n-graded modules
• Dynamical systems on chain complexes and canonical minimal resolutions
• Linear maps in minimal free resolutions of Stanley-Reisner rings
• Determinantal representations of the cubic discriminant
• Tate Resolutions on Products of Projective Spaces: Cohomology and Direct Image Complexes
• Multigraded Hilbert Schemes
• Splitting criterion for reflexive sheaves
• Small projective spaces and Stillman uniformity for sheaves
• Minimal free resolution of a finitely generated module over a regular local ring
• Cohomology of Coherent Sheaves and Series of Supernatural Bundles
• Castelnuovo-Mumford regularity for complexes and weakly Koszul modules
• Linearity Defects of Face Rings
• Bezoutians and Tate Resolutions
• Lyubeznik numbers of monomial ideals
• Subcomplexes of Certain Free Resolutions
• Triangulated categories of periodic complexes and orbit categories
• Stable rank 3 vector bundles on ${\mathbb{P}}^3$ with $c_1=0$, $c_2=3$
• A Remark on Lazarsfeld's Approach to Castelnuovo-Mumford Regularity
• Constructing equivariant vector bundles via the BGG correspondence
• Tate Resolutions and Weyman Complexes
• Linearity Defect and Regularity over a Koszul Algebra
• Tate Resolutions for Segre Embeddings
• Regularity and Segre-Veronese embeddings
• Hierarchies of simplicial complexes via the BGG-correspondence
• BGG correspondence and Roemer's theorem on an exterior algebra
• Construction of rational surfaces of degree 12 in projective fourspace (with an appendix by Kristian Ranestad)