Schemes over symmetric monoidal categories and torsion theories

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• 作者：Abhishek Banerjee ^{abhishekbanerjee1313@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：13D30 ; 18D10 ; 19D23
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：220
• 期：9
• 页码：3017-3047
• 全文大小：659 K

文摘

Let (C,⊗,1)$(\mathcal{C},\otimes ,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let X   be a scheme over (C,⊗,1)$(\mathcal{C},\otimes ,1)$ in the sense of Toën and Vaquié. In this paper, we construct torsion theories on the categories O_X-Mod${\mathcal{O}}_X\text{-}\mathit{Mod}$ and 65fd" title="Click to view the MathML source">QCoh(X)$\mathit{QCoh}(X)$ respectively of O_X${\mathcal{O}}_X$-modules and quasi-coherent sheaves on X, when X   is Noetherian and integral over (C,⊗,1)$(\mathcal{C},\otimes ,1)$. Thereafter, we study these torsion theories with respect to the quasi-coherator 65ff75c2f887b70" title="Click to view the MathML source">Q_X:O_X-Mod⟶QCoh(X)$Q_X:{\mathcal{O}}_X\text{-}\mathit{Mod}⟶\mathit{QCoh}(X)$ that is right adjoint to the inclusion f994c0b1a92e2610b428ef37e5fd" title="Click to view the MathML source">i_X:QCoh(X)⟶O_X-Mod$i_X:\mathit{QCoh}(X)⟶{\mathcal{O}}_X\text{-}\mathit{Mod}$. Finally, we obtain an alternative description of the quasi-coherator Q_X(F)$Q_X(\mathcal{F})$ as a subsheaf of F$\mathcal{F}$, when F∈O_X-Mod$\mathcal{F}\in {\mathcal{O}}_X\text{-}\mathit{Mod}$ satisfies certain conditions. Along the way, we present further results on the notions of “Noetherian” and “integral” for schemes over (C,⊗,1)$(\mathcal{C},\otimes ,1)$ that we believe to be of independent interest.