Holmstrom Weight filtration

http://mathoverflow.net/questions/117432/splitting-of-the-weight-filtration (there are also other MO questions on the weight filtration)

In the letter to Beilinson, Voevodsky formulates axioms for a homology theory. He considers $\partial \Delta^n$ as an (n-1)-dim sphere, write also S for the 1-dim sphere in this sense. Let Sch/k be the cat of separated schemes of finite type over a base k. Then a homological theory is a functor from Sch/k together with a family of natural isos $H(X \times S) \to H(X) \oplus H(X)[1]$. This functor should satisfy some conditions: Morally, homotopy invariance, MV exact triangle, an exact triangle for blowups, and transfer for flat finite morphisms. Get a 2-cat of homological theories over $k$. Examples: Algebraic K-th with rational coeffs, l-adic homology, Hodge homology ass to a complex embedding. Thm: There is an initial object $DM_k^{ft}$ in this cat, which we call the triang cat of eff mixed motives over k. Notion of reduced homological theory, and reduced motive of a scheme. Any motive in the above sense is of the form $\tilde{M}(X)[n]$, where we may assume $X$ affine and $n \leq 0$. Tate object and comparison with K-theory. Bigger cat $DM_k$ which contains the previous as a full triang subcat, but admits a more explicit description rather than just the universal property. Can also be viewed as the closure of the previous, wrt direct sums and inductive limits. Need the h-topology, in particular coverings including surjective blowups, finite surjetive maps, etale coverings. Various filtrations on $DM_k$ (homotopy canonical, geometrical, motivic canonical, weight). The weight filtr should be related to pure numerical motives.

nLab page on Weight filtration