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Algebraic topology can be done in algebraic geometry, September 1, 2002
This review is from: Cycles, Transfers, and Motivic Homology Theories. Annals of Mathematics Studies, No. 143 (Paperback)
Beginning with the work of Grothendiek, the theory of motives is, very loosely speaking, an attempt at a "unified theory" of number theory and algebraic geometry. This years Fields Medals reflect the interest in motives, as one of the authors in this book (Voevodsky) was awarded for his research in this area. In a gist, this book tries to see how much of (standard) algebraic topology can be carried over to the study of algebraic varieties and schemes. By defining a new topology on algebraic cycles, the "qfh topology", Voevodsky showed that the techniques of sheaf theory can be used to study them from the standpoint of algebraic topology. This topology is finer than the etale topology and allows one to use sheaf cohomology to study algebraic cycles. The reader will be expected to have a substantial background in the theory of schemes, higher K-theory, algebraic topology, and sheaf theory. Reading this book will give one a deep appreciation of how difficult it is to do algebraic topology in algebraic geometry, requiring formidable technical machinery.
The use of K-theory in topology and algebra goes back half a century, beginning with the K-theory of CW-complexes and the construction of Atiyah and Hirzebruch of spectral sequences relating singular cohomology to topological K-theory. The K-theory of algebraic varieties is a little more subtle, and involves looking at the isomorphism classes of algebraic vector bundles on the variety. These form an abelian group with the group operation being defined via the existence of an exact sequence between the isomorphism classes.
As a warm-up to the scheme-theoretic setting, the K-theory of an arbitrary ring proceeds by analogy with the simplicial setting, the latter of which involves the classifying space of homotopy maps of the complex and the notion of stable equivalence. But for a general ring, the unit interval used in the definition of homotopy is replaced by the affine line. The work of Karoubi and Villamayor, and Quillen defined precisely higher algebraic K-theory for rings, the former using this simplicial motivation, the latter using what is called a "Q-construction". The definitions coincide for regular schemes but not for singular ones.
Motivic cohomology, which is an algebraic analog of singular cohomology, arose in the setting of the Chow ring of algebraic cycles modulo rational equivalence. A homology theory of the free abelian group of algebraic cycles of a variety, with the replacement of the unit interval with the affine line, was developed. The products existing in cohomology arise from the consideration of the intersection of subvarieties, leading to the familiar Chow ring. The Chow ring is functorial under pull-backs, and can be related to the zeroth K-group via the use of the Chern class and the Riemann-Roch theorem. The higher K-groups of Quillen give the desired long exact sequence of K-groups.
Bloch then defined motivic cohomology via the construction of higher Chow groups, again by analogy to the simplicial theory, and with a careful definition of intersection product, so as to insure the algebraic cycles intersect the faces in the correct codimension. It was then shown that the higher Chow groups are related to the the higher K-groups for a variety which is smooth over a field.
One of the authors (Frielander) and Dwyer, using the etale cohomology of Grothendieck, gave a mod-n topological K-theory, called etale K-theory, which led to the work of Suslin and Voevodsky on the motivic homology of algebraic cycles, which is the main focus of this book.
After a brief introduction to motivic cohomology in chapter 1 and an historical introduction, the second chapter deals with relative cycles on schemes and Chow sheaves. Relative cycles are defined for schemes of finite type over a Noetherian (base) scheme and are well-behaved for morphisms of of the base scheme. The authors concentrate most of their attention not to general schemes but to varieties over a field. The cdh-topology is introduced here as one which allows the construction of long exact sequences for sheaves of relative cycles.
Chapter 3 overviews the cohomological theory of presheaves and defines the notion of a transfer map. For smooth schemes over a field, these maps are used to define a "pretheory" over the field, and homotopy invariance of pretheories can then be defined. Examples of pretheories include etale cohomology, algebraic K-theory, and algebraic de Rham cohomology. The Mayer-Vietoris exact sequence for the Suslin homology is proven, giving another analogue of ordinary algebraic topology.
In chapter 4 the authors consider the generalization of the duality property of homology and cohomology in algebraic topology using bivariant cycle cohomology. The bivariant cycle cohomology groups are defined for schemes of finite type over a field in terms of the higher Chow groups. They have the origin in the generalization of the simplicial theory to the algebraic geometry setting. Homotopy invariance, suspension maps, and the Gysin sequence find their place here also. The authors detail to what extent the higher Chow groups can be considered to be a motivic cohomology theory. Motivic homology, motivic cohomology, and Borel-Moore motivic cohomology are shown to be related to the bivariant cycle cohomology and their algebraic topological properties discussed briefly.
Chapter 5 studies algebraic cycle cohomology theories categorically via the construction of triangulated categories of motives. This is the key step in allowing the techniques of (ordinary) sheaf cohomology to be applied to the category of motives. The discussion is done in the context of smooth schemes, but it would be interesting if the authors would have given some concrete examples, possibly with elliptic curves, showing how these constructions come into play for elementary algebraic varieties.
The book ends with a discussion of the higher Chow groups and how they relate to etale cohomology. A relatively concrete presentation, the author proves the equality between the higher Chow groups and etale cohomology with compact supports for quasiprojective schemes over algebraically closed fields of characteristic zero.
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