Orientations and transfers in cohomology of algebraic varieties
Orientations and transfers in cohomology of algebraic varieties
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代数簇上同调的定向和传递
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发表时间：
2007
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通讯作者：
A. Smirnov
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作者：
A. Smirnov
代数几何上同调理论以公理化方式描述，并对其方向进行系统处理。对于每个有向理论，传递映射是为在支持上正确的光滑簇的映射而构造的。在一些基本情况下，会计算转移。该演示通过动机上同调、K 理论、代数共边主义和其他示例进行说明。本文涉及代数簇的上同调理论。其中我们可以提到早已为人所知的étale上同调和代数K理论，以及最近的理论，例如动机上同调和代数共边。在拓扑学中，上同调理论可以通过稳定同伦范畴中的表示谱或公理化方式来描述。根据布朗的可表示性定理，这些方法是等效的。 Voevodsky 和 Morel 在 [1] 和 [2] 中发现了稳定同伦范畴的代数对应项。在本文中，我们提出了 Steenrod-Eilenberg 公理的代数模拟。我们本文的主要目标是研究导向理论。在拓扑中，理论的方向是一组数据，可以选择与复向量丛相关的球丛纤维的兼容基本类别（方向）。本论文使用了相同的方法，尽管通过在共边理论上引入模块结构可以获得更概念化的方向定义。拓扑定向理论的一个重要优点是上同调和相关运算的可计算性，例如特征类和传递。在本文中，我们提出了这些运算的代数版本并开发了适当的计算工具。要应用这些工具，需要知道如何定位具体理论。为此，我们提出了代数共边的自然方向，并描述了任意理论的所有方向（特别是从这个描述中可以得出，具有陈类的理论是可定向的）。此外，我们引入了传递的概念（上同调类积分的类比，作为空间上的函数）并证明了主要结果，即可定向理论的传递的存在性。现在，我们更详细地描述本文的内容。在1.2小节中，我们定义了上同调理论并给出了基本例子；在第 1.3 小节中，我们介绍了上同调理论的一般性质并描述了一些构造（这里相当重要的是法向圆锥的变形，请参见第 1.3.6 小节）；在1.4小节中，我们介绍产品；在第 1.5 和 1.6 小节中，我们讨论了暂停和组结构。 2000年数学学科分类。小学 14F99。
Algebro-geometric cohomology theories are described axiomatically, with a systematic treatment of their orientations. For every oriented theory, transfer mappings are constructed for mappings of smooth varieties that are proper on supports. In some basic cases, transfers are calculated. The presentation is illustrated by motivic cohomology, K-theory, algebraic cobordism, and other examples. The present paper concerns cohomology theories for algebraic varieties. Among these we can name étale cohomology and algebraic K-theory, known for a long time, as well as more recent theories such as motivic cohomology and algebraic cobordism. In topology, a cohomology theory is described either by a representing spectrum in the stable homotopy category or axiomatically. By Brown’s representability theorem, these approaches are equivalent. An algebraic counterpart of the stable homotopy category was found by Voevodsky and Morel in [1] and [2]. In the present paper, we present an algebraic analog of the Steenrod–Eilenberg axioms. Our main goal in this paper is to study oriented theories. In topology, an orientation of a theory is a set of data that makes it possible to choose compatible fundamental classes (orientations) of the fibers of spherical bundles related to complex vector bundles. The same approach is used in the present paper, though a more conceptual definition of orientation would have been obtained by introducing a module structure over cobordism theory. An essential advantage of oriented theories in topology is the computability of cohomology and related operations, e.g., the characteristic classes and transfers. In this paper, we present an algebraic version of these operations and develop appropriate computational tools. To apply these tools, it is necessary to know how to orient specific theories. For this, we present a natural orientation of algebraic cobordism and describe all orientations of an arbitrary theory (from this description it follows, in particular, that a theory with Chern classes is orientable). Furthermore, we introduce the notion of a transfer (an analog of the integral for cohomology classes as functions on spaces) and prove the main result, namely, the existence of a transfer for an orientable theory. Now, we describe the content of the present paper in more detail. In Subsection 1.2, we define a cohomology theory and give basic examples; in Subsection 1.3, we present general properties of cohomology theories and describe some constructions (of considerable importance here is the deformation to the normal cone, see Subsection 1.3.6); in Subsection 1.4, we introduce products; in Subsections 1.5 and 1.6, we discuss suspension and group structures. 2000 Mathematics Subject Classification. Primary 14F99.