Grothendieck-Lefschetz for vector bundles

2018/02/22 by Kęstutis Česnavičius, Cesnavicius, Kestutis
Mathematics · #13D45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary 14B15 #Secondary 13D10

paper · doi:10.48550/arxiv.1802.08203

openalex publication_date 2018/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

According to the Grothendieck-Lefschetz theorem from SGA 2, there are no nontrivial line bundles on the punctured spectrum U_R of a local ring R that is a complete intersection of dimension ≥ 4. Dao conjectured a generalization for vector bundles \mathscrV of arbitrary rank on U_R: such a \mathscrV is free if and only if depth_R(End_R(Γ(U_R, \mathscrV))) ≥ 4. We use deformation theoretic techniques to settle Dao's conjecture. We also present examples showing that its assumptions are sharp and draw consequences for splitting of vector bundles on complete intersections in projective space.

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