Algebra Seminar 

15/12/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Mike Paluch, Instituto Superior Técnico

A simplicial guide to Voevodsky's motives

Let $V$ be a smooth complex projective variety. Since $V$, when endowed with the classical topology carries the homotopy type of a finite CW complex, it follows that the normalized chain complex $NZ\mathrm{Sin}(V)$ of the free simplicial abelian group of the singular complex of V can be viewed as an object in the derived category $D(\mathrm{Ab})$ of bounded complexes of finitely generated abelian groups. As stated by Beilinson and Vologodsky a basic objective of the theory of motives is to find a triangulated category $\mathrm{DM}$, whose formulation does not depend on the metric topology of the field of complex numbers, and functors from the category algebraic varieties to $\mathrm{DM}$ and from $\mathrm{DM}$ to $D(\mathrm{Ab})$ such that the composite functor carries $V$ to $NZ\mathrm{Sin}(V)$. In this talk I shall present a simplicial perspective of Voevodsky's theory of motives in terms of additive and simplicially additive Grothendieck topologies and a simplicial candidate for $\mathrm{DM}$.