2002/02/28 by Peter B. Gothen, PETER B. GOTHEN, Alastair D. King +1 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.AG #msc:14F05 #msc:18F20 #msc:32L10
paper · pdf · doi:10.1112/s0024610704005952
published as J. London Math. Soc. 71 (2005), 85-99 · 20 pages; v2: substantially revised version; v3: minor clarifications and corrections
arxiv created 2004/05/03 · openalex publication_date 2005/02/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. It is shown that the category of such representations is an abelian category with enough injectives by the construction of an explicit injective resolution. Using this explicit resolution, a long exact sequence is found that computes the Ext groups in this new category in terms of the Ext groups in the old category. The quiver formulation is directly reflected in the form of the long exact sequence. It is also shown that under suitable circumstances, the Ext groups are isomorphic to certain hypercohomology groups.