Here are some random thoughts when going through Lecture notes on motivic cohomology.

# \(\mu_n\) and \(\mathbb{Z}/n(q)\)

Recall that \[ \mathbb{Z}(1)=\mathbb{Z}_{tr}(\mathbb{G}_m)\cong\mathcal{O}^*[-1] \] as presheaves with transfers. Étale sheafication gives that \[ \mathbb{Z}/l(1)_{ét}\cong\mathcal{O}_{ét}^*[-1]\otimes\mathbb{Z}/l\cong H_0(\mathcal{O}_{ét}^*[-1]\otimes^L\mathbb{Z}/l)=\mu_l. \] Note that \(\mathcal{O}_{ét}^*[-1]\otimes\mathbb{Z}/l\) is indeed quasi-isomorphic to \(\mathcal{O}_{ét}^*[-1]\otimes^L\mathbb{Z}/l\) when \(l\) is prime to the characteristic of our base field \(k\).

Moreover, in proposition 10.6 we see that \(\mu_n^{\otimes q}\rightarrow\mathbb{Z}/n(q)\) is an \(\mathbb{A}^1\)-weak equivalance in \(D^-(Sh_{ét}(Cor_k,\mathbb{Z}/n))\) when \(n\in k^\times\). Here we need some lemmas from the lecture.

#### [Lemma 8.13] If \(F\) and \(F'\) are étale sheaves sheaves of \(R\)-modules with transfers, and \(F\) is locally constant, then the map\[ F\otimes_{ét}F'\rightarrow F\otimes_{ét}^{tr}F' \] is an isomorphism in \(D^-(Sh_{ét}(Cor_k, R))\).

#### [Lemma 8.18] Let \(F\) be a locally constant étale sheaf with \(R\)-flat sections. Then the object \(E\otimes_{L,ét}^{tr}F\) is acyclic at all non-zero degrees.

These two lemmas suggest that locally constant étale sheaves with \(R\)-flat sections form a model to compute \(\otimes_{ét}^{tr}\) in \(D^-(Sh_{ét}(Cor_k, R))\). We may formalute this strictly later on.

Now we easily see that

#### [Proposition 10.5] The morphism \[ \mathbb{Z}/n(1)^{\otimes_L^{tr}q}\rightarrow\mathbb{Z}/n(q) \] is an \(\mathbb{A}^1\)-weak equivalance in \(D^-(Sh_{ét}(Cor_k,\mathbb{Z}/n))\).

Combining the facts above we see that \(\mu_n^{\otimes q}\rightarrow\mathbb{Z}/n(q)\) is indeed an \(\mathbb{A}^1\)-weak equivalance.

Another important fact is that when \(n\in k^\times\), we can invert the Tate twist operation to identify the localized category \(DM_{ét}^{eff,-}(k,\mathbb{Z}/n)\) with \(DM_{ét}^-(k,\mathbb{Z}/n)\).

#### [Corollary 8.19] Tate twist \(M\mapsto M(1)=M\otimes_{L,ét}^{tr}\mathbb{Z}/n(1)\) is invertible in \(D^-(Sh_{ét}(Cor_k,\mathbb{Z}/n))\) when \(n\in k^\times\).

**Proof** Let \(\mu_n^*\) be the Cartier dual of \(\mu_n\) then we have \[
\begin{aligned}
\mu_n^*\otimes_{L,ét}^{tr}\mathbb{Z}/n(1)&\cong
\mu_n^*\otimes_{ét}^{tr}\mathbb{Z}/n(1)&(8.18)\\
&\cong\mu_n^*\otimes_{ét}\mathbb{Z}/n(1)&(8.13)\\
&\cong\mu_n^*\otimes_{ét}\mu_n&\\
&\cong\mu_n^*\otimes_{\mathbb{Z},ét}\mu_n\otimes_{\mathbb{Z},ét}\mathbb{Z}/n&\\
&\cong\mathbb{Z}/n&(duality)
\end{aligned}
\]

# \(\mathbb{Z}(1)\) and \(\mathcal{O}^*\)

We now have a closer look at the motivic complex \(\mathbb{Z}(1)\). Following lecture 7, for a
closed immersion \(i:Y\hookrightarrow\bar{X}\), define the
presheaf (a sheaf indeed) of relative units as \[
\mathbb{G}_{\bar{X},Y}=Ker(\mathbb{G}_\bar{X}\rightarrow
i_*\mathbb{G}_Y).
\] The bar over \(X\) suggests a
good model for this should be a pole \(Y\) of \(1\)-dimensional projective geometric
variety \(\bar{X}\), which involves the
perspection of a *relative curve* over a noetherian base. We can
go quite deep into this, enabling us to easily extend our theory of
motives and singular homology to schemes with singulars. This is
interesting since Grothendieck’s theory of pure motives is based on the
intersection theory of Chow groups, which only behaves well at smooth
case, when moving lemma holds. We will discuss this later on.

When \(\bar{X}\) is geometrically connected, denote \(\mathcal{M}^*(\bar{X},Y)\) be the presheaf which associates to a smooth connected scheme \(U/k\) the abelian group of rational functions of \(\bar{X}\times_kU\) which vanishes at \(Y\times_kU\) and remains regular at a (Zariski) neighborhood of \(Y\times_kU\) under multiplcation.

#### [Remark 7.11] For any connected \(S\), the “stalk” \((i^{-1}\mathbb{G}_{\bar{X}\times S,Y\times S})(Y\times_k S)\) is equal to \(\mathcal{M}^*(\bar{X},Y)(S)\).

**Proof** Since \(Y\)
is a closed subscheme of \(\bar{X}\),
we see that the presheaf pullback \(i_{pre}^{-1}\mathcal{F}\) of any zariski
sheaf on \(\bar{X}\) is already a
sheaf. Taking the global section of the pullbacked short exact
sequence

we see that \(H^0(Y,i^*\mathbb{G}_{\bar{X},Y})\) is just the set \(\{(U\hookrightarrow X,f\in\mathcal{O}^*(U))|f\equiv 1 (\mathcal{O}^*(Y))\}\).

#### [Statement] \(\mathcal{M}^*(\bar{X},Y)\) is a sheaf for the Zariski topology.

Here we only need the case where \(\bar{X}=\mathbb{P}_k^1\) and \(Y=\{0,\infty\}\).

#### [Lemma 2.18] Let \(F\) be a presheaf. Then the maps \[ i_0^*,i_1^*:C_*F(X\times\mathbb{A}^1)\rightarrow C_*F(X) \] are chain homotopic.

#### [Statement] Let \(u:P_*\rightarrow Q_*\) be a morphism of zero homotopy between complex \(P\) and \(Q\). Then the homotopy class \([u(f)]\) defined by any cycle \(f\in P_i\) represents a zero homology class in \(H_i(Q)\).

**Proof** Since \(u\cong
0\) in the homotopy category \(K\), the triangle \[
P\rightarrow Q\rightarrow P\oplus Q\rightarrow P[-1]
\] is distinguished. Hence we get a long exact sequence of
homology groups\[
\cdots\rightarrow H_i(P)\rightarrow H_i(Q)\rightarrow H_i(P)\oplus
H_i(Q)\rightarrow\cdots
\] the class \([f]\in H_i(P)\)
maps to 0 in \(H_i(P)\oplus H_i(Q)\),
hence maps to \([u(f)]=0\) in \(H_i(Q)\).

#### [Lemma 4.6] For \(S/k\) smooth, \(C_*(\mathcal{M}^*(\mathbb{P}^1,\{0,\infty\}))(S)\) is acyclic, hence \(C_*(\mathcal{M}^*(\mathbb{P}^1,\{0,\infty\}))\) is acyclic as complex of sheaves.

**Proof** Here we work over the normalized TSP 017V \(\mathcal{M}^*(\mathbb{P}^1,\{0,\infty\})\)
chain complex \[
C_*^{DK}(S)=C_*^{DK}(\mathcal{M}^*(\mathbb{P}^1,\{0,\infty\}))(S).
\]

For a cycle \(f\in C_i^{DK}(S)\), that is, an element vanishing in \(C_{i-1}^{DK}(S)\), we construct a path \(h_S(f,t)\in C_i^{DK}(S\times\mathbb{A}^1)\subset C_i(S\times\mathbb{A}^1)\) such that the restriction along \(i_0\) and \(i_1\) retracts \(f\) to \(0\). Since \(f\) is a cycle, it represents a regular function on some neightborhood \(U\) of \(Z=S\times\Delta^i\times\{0,\infty\}\) which vanishes on every face \(S\times\Delta^{i-1}\times\{0,\infty\}\) and \(Z\). The path we construct would be \(h_S(f)=1-t(1-f)\), which is obviously regular on the neighborhood \(\mathbb{A}^1\times U\) of \(\mathbb{A}^1\times Z\). Moreover, \(h_S(f)\) is a cycle in \(C_i(S\times\mathbb{A}^1)\) since it equals to \(1\) when \(f\) equals to \(1\). Hence we get the result.