A DG Guide to Voevodsky's Motives, by A. Beilinson and V. Vologodsky

We offer a concise exposition of results from Voevodsky's theory of motives, with the notable exception of comparison results (relating the motivic cohomology with Bloch's higher Chow groups and Milnor's K-groups, and the étale localized motives with finite coefficients in Galois modules). The more advanced subjects of A1-homotopy theory, the proof of the Milnor-Bloch-Kato conjecture, and the array of motivic hopes, are not touched.

We take time to spell out the basic constructions on the DG-category level. For the present material, this has only the limited advantage of making certain formulas possible to write down, but it seems to be necessary for some future developments (such as understanding of motivic descent).

A. Beilinson <sasha@math.uchicago.edu>
V. Vologodsky <vologods@math.uga.edu>