Orientations and Transfers in Cohomology of Algebraic Varieties

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.