Representation of relative sheaf cohomology
Representation of relative sheaf cohomology
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DOI：
10.1515/forum-2022-0258
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发表时间：
2018-10
影响因子：
0.8
通讯作者：
T. Suwa
中科院分区：
文献类型：
--
作者：
T. Suwa
本文研究拓扑空间开嵌入层复形的上同调理论及相关问题。该理论位于一般切赫理论和派生范畴理论的交叉点。也就是说，一方面，上同调被描述为层复形的截面的相对上同调，这在层复形的切赫上同调理论中自然出现。另一方面，在导出范畴理论中，它被解释为复形对偶与复形的某个态射的映射锥的上同调。从这两个观点出发，我们证明了一个“相对de Rham型定理”。它说，在复形是某个层的软分解或细分解的情况下，上同调与该层的相对上同调正则同构。因此，前者提供了一种表示后者的方便方法。沿着我们发展了各种理论，并建立了其中出现的上同调之间的规范同构。第二种观点导致理论的推广到层态射的上同调的情况。文中还指出了一些特殊情况及其应用。特别值得注意的是应用Dolbeault复杂的情况下，佐藤超函数理论和其他问题的代数分析。
Abstract We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Čech theory and the theory of derived categories. That is to say, on the one hand the cohomology is described as the relative cohomology of the sections of the sheaf complex, which appears naturally in the theory of Čech cohomology of sheaf complexes. On the other hand it is interpreted as the cohomology of a complex dual to the mapping cone of a certain morphism of complexes in the theory of derived categories. We prove a “relative de Rham-type theorem” from the above two viewpoints. It says that, in the case the complex is a soft or fine resolution of a certain sheaf, the cohomology is canonically isomorphic with the relative cohomology of the sheaf. Thus the former provides a handy way of representing the latter. Along the way we develop various theories and establishes canonical isomorphisms among the cohomologies that appear therein. The second viewpoint leads to a generalization of the theory to the case of cohomology of sheaf morphisms. Some special cases together with applications are also indicated. Particularly notable is the application of the Dolbeault complex case to the Sato hyperfunction theory and other problems in algebraic analysis.