Gerbe patching and a Mayer-Vietoris sequence over arithmetic curves

2017/06/11 by Bastian Haase, Haase, Bastian
Mathematics · #18G50 Secondary: 14M17 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary: 14H25

paper · pdf · doi:10.48550/arxiv.1706.03387

openalex publication_date 2017/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We discuss patching techniques and local-global principles for gerbes over arithmetic curves. Our patching setup is that introduced by Harbater, Hartmann and Krashen. Our results for gerbes can be viewed as a 2-categorical analogue on their results for torsors. Along the way, we also discuss bitorsor patching and local-global principles for bitorsors. As an application of these results, we obtain a Mayer-Vietoris sequence with respect to patches for non-abelian hypercohomology sets with values in the crossed module G->Aut(G) for G a linear algebraic group. Using local-global principles for gerbes, we also prove local-global principles for points on homogeneous spaces under linear algebraic groups H that are special (e.g. SL_n and Sp₂n) for certain kind of stabilizers.

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