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.
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机译：定向同调理论的概念在拓扑结构中是众所周知的。这些类型的理论的例子有复杂的cobordism，复杂的$ K $理论，通常的奇异同调性和Morava $ K $理论。这些同调理论的一个特定特征是存在用于紧凑型复杂流形的态射的跟踪算子（或Thom-Gysin算子或前推）。本文的主要目的是开发该概念的代数形式。在给定的环同调理论上，构造方向，Chern结构，Thom结构和迹线结构之间的双射对应关系。奇异同调，动机同调，代数$ K $-理论，Voevodsky的代数同余论和其他示例说明了该理论。
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