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Applied Categorical Structures

, Volume 22, Issue 3, pp 457–466 | Cite as

Characterizing Serre Quotients with no Section Functor and Applications to Coherent Sheaves

  • Mohamed Barakat
  • Markus Lange-HegermannEmail author
Article

Abstract

We prove an analogon of the the fundamental homomorphism theorem for certain classes of exact and essentially surjective functors of Abelian categories \(\mathcal{Q}:\mathcal{A} \to \mathcal{B}\). It states that \(\mathcal{Q}\) is up to equivalence the Serre quotient \(\mathcal{A} \to \mathcal{A} / \ker \mathcal{Q}\), even in cases when the latter does not admit a section functor. For several classes of schemes X, including projective and toric varieties, this characterization applies to the sheafification functor from a certain category \(\mathcal{A}\) of finitely presented graded modules to the category \(\mathcal{B}=\mathfrak{Coh}\, X\) of coherent sheaves on X. This gives a direct proof that \(\mathfrak{Coh}\, X\) is a Serre quotient of \(\mathcal{A}\).

Keywords

Serre quotient Fundamental homomorphism theorem Exact functors Abelian categories Gabriel localization Coherent sheaves 

Mathematics Subject Classifications (2010)

18E35 18F20 18A40 18E40 

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Lehrstuhl B für MathematikRWTH Aachen UniversityAachenGermany

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