Homological Algebra of Twisted Quiver Bundles |
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Authors: | Gothen Peter B; King Alastair D |
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Institution: | Departamento de Matemática Pura, Faculdade de Ciências, Universidade do Porto Rua do Campo Alegre 687, 4169-007 Porto, Portugal, pbgothen{at}fc.up.pt
Mathematical Sciences, University of Bath Claverton Down, Bath BA2 7AY, United Kingdom, a.d.king{at}maths.bath.ac.uk |
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Abstract: | Several important cases of vector bundles with extra structure(such as Higgs bundles and triples) may be regarded as examplesof twisted representations of a finite quiver in the categoryof sheaves of modules on a variety/manifold/ringed space. Itis shown that the category of such representations is an abeliancategory with enough injectives by the construction of an explicitinjective resolution. Using this explicit resolution, a longexact sequence is found that computes the Ext groups in thisnew category in terms of the Ext groups in the old category.The quiver formulation is directly reflected in the form ofthe long exact sequence. It is also shown that under suitablecircumstances, the Ext groups are isomorphic to certain hypercohomologygroups. |
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