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 ${\mathbf{Z}}(n)$?



Two motivic complexes, compared
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](http://www.claymath.org/library/monographs/cmim02.pdf).


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.


