ORIENTED COHOMOLOGY THEORIES OF ALGEBRAIC VARIETIES II (AFTER I. PANIN AND A. SMIRNOV)
ORIENTED COHOMOLOGY THEORIES OF ALGEBRAIC VARIETIES II (AFTER I. PANIN AND A. SMIRNOV)
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代数簇的有向上同调理论 II（继 I. Panin 和 A. Smirnov）
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发表时间：
2009
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通讯作者：
K. Zainoulline
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作者：
K. Zainoulline
有向上同调理论的概念在拓扑学中是众所周知的。这类理论的例子有复共论、复k理论、通常奇异上同论和Morava k理论。这些上同调理论的一个特殊特征是紧复流形的态射存在迹算子（或称tom - gysin算子，或前推算子）。本文的主要目的是发展这个概念的代数版本。在给定环上同调理论下，构造了取向、Chern结构、Thom结构和迹结构之间的双射对应关系。用奇异上同调、动机上同调、代数k理论、Voevodsky的代数上同调和其他例子说明了该理论。
The concept of oriented cohomology theory is well-known in topology. Examples of these kinds of theories are complex cobordism, complex K-theory, usual singular cohomology, and Morava K-theories. A specific feature of these cohomology theories is the existence of trace operators (or Thom-Gysin operators, or push-forwards) for morphisms of compact complex manifolds. The main aim of the present article is to develop an algebraic version of the concept. Bijective correspondences between orientations, Chern structures, Thom structures and trace structures on a given ring cohomology theory are constructed. The theory is illustrated by singular cohomology, motivic cohomology, algebraic K-theory, the algebraic cobordism of Voevodsky and by other examples.