Localization functors and cosupport in derived categories of commutative Noetherian rings
Localization functors and cosupport in derived categories of commutative Noetherian rings
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DOI：
10.2140/pjm.2018.296.405
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发表时间：
2017-10
影响因子：
0.6
通讯作者：
Tsutomu Nakamura;Y. Yoshino
中科院分区：
文献类型：
--
作者：
Tsutomu Nakamura;Y. Yoshino
令 $R$ 为交换诺特环。我们引入了本地化函子 $\lambda^W$ 的概念，其在 $\text{Spec}\, R$ 的任意子集中 $W$ 中具有协支持；它是关于乘法闭子集和理想进补函子的左派生函子的局部化的常见推广。我们证明了关于定位函子 $\lambda^W$ 的几个结果，包括通过 Cech 复合体的概念计算 $\lambda^W$ 的显式方法。作为一个应用，我们可以给出 Gruson 和 Raynaud 的经典定理的更简单的证明，该定理指出平面 $R$ 模块的射影维数至多是 $R$ 的 Krull 维数。作为另一个应用，可以给出一种函数方法来用纯内射 $R$ 模块的复合体替换平面 $R$ 模块的复合体或有限生成的 $R$ 模块的复合体。
Let $R$ be a commutative Noetherian ring. We introduce the notion of localization functors $\lambda^W$ with cosupports in arbitrary subsets $W$ of $\text{Spec}\, R$; it is a common generalization of localizations with respect to multiplicatively closed subsets and left derived functors of ideal-adic completion functors. We prove several results about the localization functors $\lambda^W$, including an explicit way to calculate $\lambda^W$ by the notion of Cech complexes. As an application, we can give a simpler proof of a classical theorem by Gruson and Raynaud, which states that the projective dimension of a flat $R$-module is at most the Krull dimension of $R$. As another application, it is possible to give a functorial way to replace complexes of flat $R$-modules or complexes of finitely generated $R$-modules by complexes of pure-injective $R$-modules.