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The Motivation Behind Motivic Cohomology

Some of the most significant developments in mathematics in the past year stem from a breakthrough achieved by Vladimir Voevodsky, and from work by Voevodsky and Andrei Suslin which builds on this breakthrough. Providing bridges across different areas of mathematics, this work constitutes a significant step toward resolving some questions that had eluded mathematicians for several decades. Voevodsky has been invited to present a plenary lecture about his work at the International Congress of Mathematicians, the most important meeting in the mathematical world which takes place every four years and will next be held in Berlin in August 1998.

On its most general level, the work of Voevodsky provides a new link between two circles of ideas in mathematics: the algebraic and the topological. The term algebra as used here refers to a much broader and deeper field than that studied by high school students. What mathematicians mean by algebra is, roughly speaking, a theory for studying the general structure of sets endowed with algebraic operations, like addition and multiplication of the integers {...-3, -2, -1, 0,1, 2, 3,...}.

There are many different kinds of algebraic objects, the most basic one being an abelian group. An abelian group is a set together with an operation on the elements of the set, where the operation has all the properties of addition of the integers. More complicated structures, such as commutative rings and fields, arise when one considers more than one operation and how the operations interact. The simplest example of a ring is the integers, together with the two operations of addition and multiplication. If one introduces a third operation, division, one obtains the rational numbers, i.e., fractions, and such a structure is formalized in the notion of a field. Other examples of rings are the sets of polynomials (in any number of variables) whose coefficients belong to a given field. For any finite set of polynomials one can look at the common zeros of those polynomials. This set of zeros is called the algebraic variety defined by the polynomials. Adding more polynomials will often cut down the size of the common zero set, yielding subvarieties of the algebraic variety. Subvarieties are sometimes referred to as algebraic cycles on the variety.

Two algebraic objects are said to be isomorphic if they have the same size and structure. This means there must be a way to match the elements of the two objects in a one-to-one correspondence so that the matching uses up all the elements, and so that the matching preserves the algebraic structure. This kind of matching is called an isomorphism. Two algebraic objects that are isomorphic are, from the algebraic viewpoint, exactly the same.

In contrast to algebraic structures, the objects of study in topology are a priori of a geometric nature. Some examples to keep in mind when thinking of topological spaces are a circle, the surface of a doughnut, and the interior of a sphere. In topological spaces one has a notion of when two points are "close together." One of the aims of topology is to study how topological spaces can be smoothly and continuously deformed in ways that preserve the "closeness" of points. In other words, such deformations can twist, stretch, and mold a topological space, but they cannot tear it.

Although the methods and viewpoints of algebra and topology are so different, mathematicians have often been able to use tools from one setting to illuminate the other. One example is the concept of homology. To handle the complexities of topological spaces, mathematicians have developed systematic ways of cutting the objects into simpler pieces. The way that these pieces fit together to form the topological space is encoded in the language of algebra through a set of homology or cohomology groups. There are several ways of making this precise, one being through what is called singular cohomology.

In the 1960s, mathematicians discovered a remarkable link between singular cohomology and another set of groups which are collectively referred to as K-theory. A cornerstone of topological K-theory is the Atiyah-Hirzebruch spectral sequence, developed by Michael Atiyah and Friedrich Hirzebruch, which provides a catalog of isomorphisms between certain cohomology groups of a topological space and its K-groups. In fact, more is true. The collection of all the cohomology groups of a space forms a ring, and the same is true of the K-groups. These two rings are isomorphic, meaning that both addition and multiplication in cohomology and K-theory are preserved by the isomorphism between them. This isomorphism is known as the Chern character, after S. S. Chern. These results were very surprising, because the definitions of cohomology and of K-theory are very different and they arise in widely separated contexts. Cohomology arises from cutting up a space into smaller pieces, whereas K-theory arises from consideration of larger objects called vector bundles over the space.

Ever since this connection was found between singular cohomology and topological K-theory, mathematicians have sought analogous structures in the algebraic setting. The problem is that algebraic varieties do not necessarily constitute topological spaces in any meaningful way. Mathematicians were able to construct a K-theory for algebraic objects, but the notion of homology in the algebraic setting remained elusive. It wasn't until the 1990s that mathematicians defined the appropriate groups to construct what is now known as motivic cohomology. The hope was that, in analogy to what was found in the topological setting, one could find isomorphisms between the groups of motivic cohomology and the groups of algebraic K-theory.

Crucial work on motivic cohomolgy was done by Spencer Bloch, who established in certain special cases an analog of the Chern character in the algebraic setting. Bloch and Stephen Lichtenbaum were also able to establish an analog of the Atiyah-Hirzebruch spectral sequence, but again only in certain special cases; the general question of whether such a sequence exists in the algebraic setting is still open. Because these results were only partial, it was unclear whether motivic cohomology would provide the strong link to algebraic K-theory mathematicians were aiming for. One of the major breakthroughs Voevodsky achieved was to utilize work of Suslin that had previously been poorly understood to uncover a very strong piece of evidence that motivic cohomolgy was indeed the right path to take. Subsequent work by Marc Levine using a different approach has provided further evidence.

Voevodsky has also made important progress on elucidating ways to compare the topological and algebraic settings. The problem, once again, is that an algebraic variety X does not constitute a topological space in any way that is useful for making such a comparison. However, given an algebraic variety X, there is a natural topological space X(C ) associated with it, consisting of the solutions in complex numbers of the equations defining X. The aim then is to compare the motivic cohomology of X with the singular cohomology of X(C ), and the algebraic K-theory of X with the topological K-theory of X(C ). The algebraic and the topological cases are wildly different, with the algebraic being far more complicated. A remarkable achievement of Suslin and Voevodsky is that, after making a standard modification of the motivic cohomology and singular cohomology, they were able to prove that the modified motivic cohomology of X and the modified singular cohomology of X(C ) are in fact isomorphic.

In establishing their results, Suslin and Voevodsky developed an entirely new perspective on these questions. One of their most fruitful inventions is a new kind of topology that applies to algebraic objects. This new perspective also revealed that certain powerful tools mathematicians developed for use in other areas can be applied to study algebraic cycles, which as subvarieties of an algebraic variety form a homology theory of their own by breaking up the variety into smaller pieces.

Although very new, these results have already spawned further work by mathematicians. There are many unsolved problems in this area of mathematics, and the new perspective developed by Suslin and Voevodsky is sure to provide further important advances.

--- Allyn Jackson

Note: The July 1997 issue of the Bulletin of the AMS carried an exposition, aimed at mathematicians, about this work: "Homology of algebraic varieties: An introduction to the works of Suslin and Voevodsky" by Marc Levine, Bulletin of the AMS, Volume 34, Number 3, pp. 293-312.

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