nLab mixed motive

Idea

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

2. Let $G_S$ denote the set of representable sheaves and $H_S$ as the set of complexes in $Cpx(N_S^{tr})$ which are the cones of morphisms $L_S[Y] \to L_S[X]$ induced by hypercovers $Y\to X$. The pair $(G_S, H_S)$ 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 $Cpx(N_S^{tr})$ where the weak equivalences are the quasi-isomorphisms.

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

4. Consider the left Bousfield localization of the above model structure at the class of morphisms $0 \to T[n]$ for $T \in \mathcal{T}_S$ and $n \in \mathbf{Z}$. The fibrant objects are $G_S$-local and $A^1$-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 $S$, denoted $DM^{eff}(S)$. It is canonically equivalent to the full subcategory of the derived category $D(N_S^{tr})$ spanned by Nisnevich fibrant and $A^1$-local complexes.

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

Related concepts

• motive

  • mixed motive,

    • mixed Tate motive

  • pure motive,

    • Chow motive, numerical motive

  • Voevodsky motive

  • noncommutative motive

• motivic cohomology

References

• Marc Levine, Mixed Motives, Handbook of K-theory (pdf)

• Denis-Charles Cisinski, Frédéric Déglise, Triangulated categories of mixed motives, Springer Monographs in Mathematics, Springer (2019) [arXiv:0912.2110, doi:10.1007/978-3-030-33242-6]

• Denis-Charles Cisinski, Frédéric Déglise, Local and stable homological algebra in Grothendieck abelian categories, Homology, Homotopy and Applications 11 1 (2009) 219–260 [arXiv:0712.3296, hha:1251832567]

Section 8.3 of

• Alain Connes, Matilde Marcolli, Noncommutative Geometry, Quantum Fields and Motives (pdf)