Moduli problems in abelian categories and the reconstruction theorem
Moduli problems in abelian categories and the reconstruction theorem
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阿贝尔范畴中的模问题和重构定理
DOI：
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发表时间：
2013
期刊：
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通讯作者：
M. Groechenig
中科院分区：
文献类型：
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作者：
John Calabrese;M. Groechenig
给出了经典Gabriel定理的模理论证明，证明了在拟相干束的阿贝尔范畴上可以重构一个格式。所采用的方法是基本的，并允许我们将定理扩展到（拟紧的和分离的）代数空间。我们还利用更先进的技术（并假设）证明了拟相干束范畴的自等价群是由底层空间的自同构和线束的扭曲组成的。我们应用我们的策略来证明由Gm-gerbe扭曲的麦束类别的类似陈述。我们的方法允许我们处理甚至不是来自Brauer类的gerbes。作为一个令人愉快的结果，我们推导出了一束阿贝尔范畴的森田理论。
We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the theorem to (quasi-compact and separated) algebraic spaces. Using more advanced technology (and assuming atness) we also give a proof of the folklore result that the group of auto-equivalences of the category of quasi-coherent sheaves consists of automorphisms of the underlying space and twists by line bundles. We apply our strategy to prove analogous statements for categories of sheaves twisted by a Gm-gerbe. Our methods allow us to treat even gerbes not coming from a Brauer class. As a pleasant consequence, we deduce a Morita theory for sheaves of abelian categories.