# Mixed motives and motivic cohomology

In Scholl's paper "Remarks on special values of $L$-functions", he defines that an object $M$ of $\textbf{MM}_{\mathbb{Q}}$ (the conjectured abelian category of mixed motives with coefficients $\mathbb{Q}$), is defined over $\mathbb{Z}$ if for every $p$, $\ell$, $p\neq \ell$, the weight filtration of the $\ell$-adic realisation $M_\ell$ splits as a representation of the inertia group $I_p$. Mixed motives defined over $\mathbb{Z}$ form a full subcategory $\textbf{MM}_{\mathbb{Z}}$. Suppose $X$ is smooth and proper over $\mathbb{Q}$, and let the pure motive $M$ and $N$ be $$M=h^i(X)(m),~N=M^\vee(1) \simeq h^i(X)(n), ~n=i+1-m$$ Then we will expect that $$\text{Ext}^0_{\textbf{MM}_{\mathbb{Z}}}(M,\mathbb{Q}(1))=\text{Ext}^0_{\textbf{MM}_{\mathbb{Z}}}(\mathbb{Q}(0),N)=\begin{cases} \text{CH}^n(X)/\text{CH}^n(X)^0 \otimes \mathbb{Q},~\text{if}~i=2n\\ 0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \text{if}~ i\neq 2n \end{cases}$$

$$\text{Ext}^1_{\textbf{MM}_{\mathbb{Z}}}(M,\mathbb{Q}(1))=\text{Ext}^1_{\textbf{MM}_{\mathbb{Z}}}(\mathbb{Q}(0),N)= \begin{cases} H^{i+1}_{\mathcal{M}}(X,\mathbb{Q}(n))_\mathbb{Z},~\text{if}~i+1 \neq 2n \\ \text{CH}^n(X)^0 \otimes \mathbb{Q}, ~~~~~\text{if}~i+1=2n \end{cases}$$ where $\text{CH}^n(X)$ is the Chow group of codimension-$n$ cycles on $X$ modulo rational equivalence and $\text{CH}^n(X)^0$ is the subgroup of classes that is homologically equivalent to zero. While $H_{\mathcal{M}}$ denotes the motivic cohomology: $$H^i_{\mathcal{M}}(X,\mathbb{Q}(j))=(K_{2j-i}(X) \otimes \mathbb{Q})^{(j)}$$ and $H^*_{\mathcal{M}}(X,\cdot)_{\mathbb{Z}}$ is the image in $H^*_{\mathcal{M}}(X,\cdot)$ of the $K$-theory of a regular model for $X$, proper and flat over $\mathbb{Z}$.

For a number field $F$ with rings of integers $\mathcal{O}_F$, there is the conjectured abelian category $\textbf{MM}_F$ (with coefficients $\mathbb{Q}$). A motive $M$ of $\textbf{MM}_F$ is defined over $\mathcal{O}_{F}$ is the weight filtration of the $\ell$-adic realisation $M_\ell$ splits as a representation of the inertia group $I_v$ for a prime $v$ of $\mathcal{O}_F$, $v \nmid \ell$. The motives defined over $\mathcal{O}_F$ form a full subcategory $\textbf{MM}_{\mathcal{O}_F}$. Now for a smooth proper variety $X$ defined over $F$, do we still expect the same properties are valid? i.e. if we still define $$M=h^i(X)(m),~N=M^\vee(1) \simeq h^i(X)(n), ~n=i+1-m$$ do we still expect the following to be valid? $$\text{Ext}^0_{\textbf{MM}_{\mathcal{O}_F}}(M,\mathbb{Q}(1))=\text{Ext}^0_{\textbf{MM}_{\mathcal{O}_F}}(\mathbb{Q}(0),N)=\begin{cases} \text{CH}^n(X)/\text{CH}^n(X)^0 \otimes \mathbb{Q},~\text{if}~i=2n\\ 0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \text{if}~ i\neq 2n \end{cases} \\ \text{Ext}^1_{\textbf{MM}_{\mathcal{O}_F}}(M,\mathbb{Q}(1))=\text{Ext}^1_{\textbf{MM}_{\mathcal{O}_F}}(\mathbb{Q}(0),N)= \begin{cases} H^{i+1}_{\mathcal{M}}(X,\mathbb{Q}(n))_{\mathcal{O}_F},~\text{if}~i+1 \neq 2n \\ \text{CH}^n(X)^0 \otimes \mathbb{Q}, ~~~~~\text{if}~i+1=2n \end{cases}$$

If this is expected, could anyone give a precise reference?

• There is an operation of restriction of scalars from mixed motives over F to mixed motives over Q, more precisely in a suitable category of "mixed realizations", see Jannsen's book "Mixed motives and algebraic K-theory". This might help you for this kind of question. – François Brunault Apr 5 '18 at 15:51
• @FrançoisBrunault Thank you, I will check this book. – Wenzhe Apr 5 '18 at 20:46