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 \(\mathcal{A}\).
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Barakat, M., Lange-Hegermann, M. Characterizing Serre Quotients with no Section Functor and Applications to Coherent Sheaves. Appl Categor Struct 22, 457–466 (2014). https://doi.org/10.1007/s10485-013-9314-y
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DOI: https://doi.org/10.1007/s10485-013-9314-y
Keywords
- Serre quotient
- Fundamental homomorphism theorem
- Exact functors
- Abelian categories
- Gabriel localization
- Coherent sheaves