mixed motive



Where the category of pure motives has smooth projective varieties as its objects, the category of mixed motives is supposed to be constructed from all smooth varieties.

The category of mixed motives is supposed to be an abelian tensor category which contains the pure motives as the full subcategory of semisimple objects.

So far there is no realisation of such a category, but there are proposals by Vladimir Voevodsky and Marc Levine of triangulated categories that behave as its derived category is expected to.

Voevodsky’s mixed motives

Here we construct Voevodsky’s triangulated category of mixed motives following Cisinski-Deglise.

  1. Let SS be a regular and noetherian base scheme. Let Sm SSm_S be the category of schemes smooth and of finite type over SS. Let N S trN_S^{tr} denote the closed symmetric monoidal category of Nisnevich sheaves with transfer. We will write L S[X]L_S[X] for the sheaf represented by XSm SX \in Sm_S.

  2. Let G SG_S denote the set of representable sheaves and H SH_S as the set of complexes in Cpx(N S tr)Cpx(N_S^{tr}) which are the cones of morphisms L S[Y]L S[X]L_S[Y] \to L_S[X] induced by hypercovers YXY\to X. The pair (G S,H S)(G_S, H_S) then defines a weakly flat descent structure? (in the sense of Cisinski-Deglise, see discussion at model structure on chain complexes) and therefore induces a symmetric monoidal model structure on Cpx(N S tr)Cpx(N_S^{tr}) where the weak equivalences are the quasi-isomorphisms.

  3. Let 𝒯 S\mathcal{T}_S denote the set of complexes Cone(L S[A X 1]L S[X])Cone(L_S[A^1_X] \to L_S[X]) where A X 1=A S 1× SXA^1_X = A^1_S \times_S X is the affine line over XX. Call a complex KK A 1A^1-local if Hom D(A)(T,K[n])=0Hom_{D(A)}(T, K[n]) = 0 for all T𝒯T \in \mathcal{T} and integers nn. Equivalently, the Nisnevich? hypercohomology sheaves? are homotopy invariant. In particular, for FF fibrant with respect to the above model structure, FF is A 1A^1-local iff the morphism F(X)F(A X 1)F(X) \to F(A^1_X) induced by the projection is a quasi-isomorphism for all XSm SX \in Sm_S. This is again equivalent to the cohomology presheaves of FF being homotopy invariant.

  4. Consider the left Bousfield localization of the above model structure at the class of morphisms 0T[n]0 \to T[n] for T𝒯 ST \in \mathcal{T}_S and nZn \in \mathbf{Z}. The fibrant objects are G SG_S-local and A 1A^1-local complexes. One can prove that this model structure is still symmetric monoidal?.

  5. The homotopy category of this model category is called the triangulated category of effective motives over SS, denoted DM eff(S)DM^{eff}(S). It is canonically equivalent to the full subcategory of the derived category D(N S tr)D(N_S^{tr}) spanned by Nisnevich fibrant and A 1A^1-local complexes.

  6. (introduce symmetric Tate spectra and DM(S))


Section 8.3 of

Revised on March 2, 2015 14:21:56 by Adeel Khan (