Mathematical Sciences: Motives and Motivic Cohomology

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Lichtenbaum The investigator continues his researcn in the closely related areas of motives and motivic cohomology. Having given a definition of mixed m otives over an algebraically closed field, he extends this definition to motives over an arbitrary base scheme. He then studies the relationship between the motive of X relative to Y, the motive of Y relative to Z, and the motive of X relative to Z. Using his definition of motivic cohomology in terms of motives, the investigator searches for a proof that this definition satisfies most, if not all, of the "axioms" proposed by Beilinson and himself. In particular, the Kummer axiom and the "finite dimensionality" axiom seem to be accessible. Finally, having already defined for each variety a homology motive in the category of mixed motives,the investigator studies possible definitions of motives corresponding to cohomology, Borel-Moore homology, and cohomology with compact supports, and explicitly computes these motives in the case of curves. The investigator also studies Poincare and Alexander duality for motives. This proposal constitutes fundamental research into the basic structures of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated curves in the plane that could be defined by polynomial equations, generalizing the classic conic sections known to the Greeks. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as physics, theoretical computer science, and robotics.