The theory of pure motives over a field $k$ is designed to caputre the properties of smooth projective varieties over $k$, in particular, those which can be detected in all geometric (or Weil) cohomology theories in a parallel way. For example, the $i$th cohomology of such a variety typically gives rise to a semi-simple structure of weight $i$, e.g., if $k={\bf C}$ then the singular cohomology $H^(X({\bf C}),{\bf Q})$ carries a pure polarized Hodge structure of weight $i$ (cf. [2]). Correspondingly, the category $M_k$ of pure motives over $k$ is a semi-simple category, whose objects can be thought of as direct factors of "the" cohomology of smooth projective varieties. By definition, the morphisms in $M_k$ are given by algebraic cycles modulo numerical equivalence.
By work of Deligne ([2], [3]), the cohomology of arbitrary varieties (not necessarily smooth or proper) typically gives rise to mixed structures, i.e., successive extensions of pure structures. These extensions are in general non-trivial, and they show quite parallel features in very different theories, like Hodge theory and $l$-adic cohomology. For example, if an extension splits in one theory, one expects that it also splits in the others.
According to Beilinson and Deligne ([1], [4]), these phenomena should be explained in the setting of a suitable category $MM_k$ of mixed motives over $k$ (which are successive extensions of pure motives). Moreover, the Yoneda Ext-groups of this category should be closely related to algebraic $K$-theory and algebraic cycles. For example, algebraic cycles which are numerically equivalent to zero should give rise to 1-extensions, 2-extensions and so on, simiarly for elements in $K$-theory. Although a completely satisfactory definition of such a category does not yet exist, there are various approximations in which some of the wanted features can be observed and studied.
Finally, the category $MM_k$ should capture the properties of non-geometric (or absolute) cohomology theories, like Beilinson-Deligne cohomology $H_{\scr D}(X,{\bf Q}(\cdot))$ or $l$-adic cohomology $H(X,{\bf Q}_t(\cdot))$ over a non-algebraically closed field $k$. In fact, one hopes for a universal such absolute theory, the so-called motivic cohomology $H_M(X,{\bf Q}(\cdot))$, which is related to the Yoneda Ext-groups by a spectral sequence. Again, there are various proposals for such a motivic cohomology, one of them being Be\u\i linson's proposal $H^i_M(X,{\bf Q}(j))=K_{2j-i}(X)^{(j)}_{\bf Q}=\{x\in K_{2j-i}(X)\otimes {\bf Q}|\Psi^kx=k^jx$ for all $k\in{\bf N\}$, where $\Psi^k$ is the $k$th Adams operator acting on algebraic $K$-theory.
The aim of the lecture is to discuss and illustrate some of these ideas by way of some simple examples and to give an overview about known results. It will be shown that a rather complete and instructive picture can be obtained for so-called 1-motives [3].
Bibliography: