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quasicoherent sheaf
Definition
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A sheaf \({\mathcal{F}}\in {\mathsf{Sh}}(X; {}_{{\mathcal{O}}_X}{\mathsf{Mod}} )\) is quasicoherent if \(X\) is covered by open affine subsets \(U_{i}=\operatorname{Spec} A_{i}\) such that for each \(i\) there is an \(A_{i}\)-module \(M_{i}\) with \(\left.\mathcal{F}\right|_{U_{i}} \cong \widetilde{M}_{i}\), the sheaf associated to a module for \(M_i\).
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\(\mathcal{F}\) is coherent if it is quasicoherent and each \(M_{i}\) is finitely generated.
Notes
Reference: https://arxiv.org/pdf/1410.1716.pdf #resources/papers
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A Noetherian scheme \(X\) can be reconstructed from \(X\) can be reconstructed from \({\mathsf{QCoh}}(X)\), see Gabriel 1962.
- Idea: associate to an ringed space \(\operatorname{Spec}\mathsf{A}\) and show \begin{align*}X \underset{\mathsf{RingSp}}{\xrightarrow{\cong}} \operatorname{Spec}\mathsf{{\mathsf{QCoh}}(A)}\end{align*}
- A smooth variety \(X\) can not generally be reconstructed from its ample bundle.
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In 2002 Balmer reconstructs a Noetherian scheme from its perfect complexes.
- Lurie 2006ish: some more general result?
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