# Two motivic complexes, compared

Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986).

Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) := \mathbf{Z}(1)^{\otimes n} = \mathbf{G}_m[-1]^{\otimes n}$$ instead of $\mathbf{Z}(n)$.

What is the relation between $\check{\mathbf{Z}}(n)$ and ${\mathbf{Z}}(n)$?

• It has nothing to do with Cech cohomology. It’s just notation for the tensor powers of the complex $\mathbf{Z}(1)$.
– user92332
Feb 24, 2018 at 16:45
• What tensor product are you using? I think they are equivalent if you consider the tensor product in the category of Voevodsky motives, but that of course is not the tensor product of sheaves of complexes. Feb 24, 2018 at 17:08
• Surely the etale cohomology functor you are considering factors through the category of Galois representations - do you think one of them fails to correspond to the standard cyclotomic character? Feb 24, 2018 at 17:56

They are not quasi-isomorphic: your $\check{\mathbf Z}(n)$ is concentrated in a single degree. As Denis points out in the comments, your definition of $\check{\mathbf Z}(n)$ is wrong because you should use the tensor product of sheaves with transfers. In addition, you need to apply Suslin's $\mathbf A^1$-invariantification construction $C_*$ to the result.

Voevodsky's motivic complex $\mathbf Z(n)_V$ is $$\mathbf Z(n)_V = C_*(\mathbf G_m^{\otimes_{\mathrm{tr}}n})[-n],$$ where:

• $\mathbf G_m$ is regarded as a presheaf with transfers, i.e., a presheaf on Voevodsky's category $\mathrm{Cor}_k$ of smooth separated $k$-schemes and finite correspondences. The transfers are given by norms of invertible functions.
• $\otimes_{\mathrm{tr}}$ is the tensor product of sheaves with transfers, which is the Day convolution of the tensor product $X\otimes Y=X\times Y$ on $\mathrm{Cor}_k$.
• $C_*(F)(X)$ is a chain complex concentrated in nonnegative degrees with $C_n(F)(X)=F(X\times \mathbf A^n)$.

$\mathbf Z(n)_V$ is quasi-isomorphic to Bloch's $\mathbf Z(n)$ as a complex of Zariski sheaves on smooth $k$-schemes. This combines several deep results of Voevodsky, Suslin, and Friedlander. A more or less self-contained proof is in Mazza–Voevodsky-Weibel's Lecture notes on motivic cohomology.

On the other hand, $\mathbf Z(n)_V$ is also a complex of étale sheaves, and if $m$ is prime to the characteristic, then $\mathbf Z/m(n)_V$ is quasi-isomorphic to $\mu_m^{\otimes n}$ as a complex of étale sheaves. This is Theorem 10.3 in the above book. Perhaps this answers your last question.

Edit: Actually the tensor product in the above formula for $\mathbf Z(n)_V$ must be derived, so it's not very explicit. The "official" definition is $$\mathbf Z(n)_V = C_*(\mathbf Z_{\mathrm{tr}}(\mathbf G_m^{\wedge n}))[-n],$$ where $\mathbf Z_{\mathrm{tr}}(\mathbf G_m^{\wedge n})$ means the quotient of the sheaf $\mathbf Z_{\mathrm{tr}}(\mathbf G_m^{\times n})$ sending $U$ to $\mathrm{Cor}_k(U,\mathbf G_m^{\times n})$ by the subsheaf generated by $\mathbf Z_{\mathrm{tr}}(\mathbf G_m^{\times n-i-1}\times\{1\}\times\mathbf G_m^{\times i})$. The relation with the other formula comes from the fact that $C_*(\mathbf Z_{\mathrm{tr}}(\mathbf G_m^{\wedge 1}))\to\mathbf G_m$ is a free resolution of $\mathbf G_m$ as a sheaf with transfers.

• Great. I'm happy with this!
– user120812
Feb 25, 2018 at 0:51