BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

When thinking about this book, three questions come to mind: What are motives? What is motivic cohomology? How does this book fit into these frameworks? Let me begin with a very brief answer to these questions. The theory of motives is a branch of algebraic geometry dealing with algebraic varieties over a fixed field k. The basic idea is simple in its audacity: enlarge the category of varieties into one which is abelian, meaning that it resembles the category of abelian groups: we should be able to add morphisms, take kernels and cokernels of maps, etc. The objects in this enlarged category are to be called motives, whence the notion of the motive associated to an algebraic variety. Any reasonable cohomology theory on varieties should factor through this category of motives. Even better, the abelian nature of motives allows us to do homological algebra. In particular, we can use Ext groups to form a universal cohomology theory for varieties. Not only does this motivate our interest in motives, but the universal property also gives rise to the play on words “motivic cohomology”. Different ways of thinking about varieties leads to different classes of motives and different aspects of the theory. And each aspect of the theory quickly leads us into conjectural territory. The best understood part of the theory is the abelian category of pure motives. This is what we get by restricting our attention to smooth projective varieties, identifying numerically equivalent maps and taking coefficients in the rational numbers Q. The pure motives of smooth projective varieties are the analogues of semisimple modules over a finite-dimensional Q-algebra. The theory of pure motives is related to many deep unsolved problems in algebraic geometry, via what are known as the standard conjectures. At the other extreme, we have the most general part of the theory of motives, obtained by considering all varieties and taking coefficients in the integers Z. This is the theory of mixed motives, and it is where homological algebra comes into play. It is also related to deep problems, such as the behavior of Hasse-Weil ζ-functions over a number field. Sadly, I must report bad news: an actual category of mixed motives is not yet fully known to exist, even with coefficients in Q. Here’s the good news. It turns out that in order to construct the motivic cohomology of a variety we do not need to know whether or not (mixed) motives exist. Using homological algebra, all we need is a candidate for the derived category of motives. Indeed, most reasonable cohomology groups of a variety may be viewed as morphisms in some derived category.