Subsection 3.2. (0MZY) · ScalingStacks

3.2. Mixed Tate motives

We will now recall the definition of the category of pure/mixed Tate motives, see [Lev05], which will play for us the role of a graded version of the derived category of sheaves on the point.

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Definition 3.2. The category of mixed Tate motives²² 2 In the literature, there are many notations for the category of mixed Tate motives. For example, $\operatorname{TDM}$ in [HK06], $\operatorname{MTDer}$ in [SW18, SVW18], $\operatorname{DMT}$ in [Spi16] or $\operatorname{DTM}$ in [Lev05]. is the subcategory

\operatorname{DTM}(k,\mathbb{Q})=\langle\mathbb{Q}\rangle_{\cong,\inplus,\Delta,(\pm 1)}\subset\operatorname{DM}(k,\mathbb{Q})

generated by $\mathbb{Q}$ under isomorphism, direct summands, Tate twists and triangles. The category of pure Tate motives is the category generated by objects $\mathbb{Q}(n)[2n]$

\operatorname{DTM}(k,\mathbb{Q})^{w=0}=\langle\mathbb{Q}(n)[2n]\mid n\in\mathbb{Z}\rangle_{\cong,\inplus,\oplus}\subset\operatorname{DTM}(k,\mathbb{Q})

with respect to isomorphism, direct summands and finite direct sums.

For example, the decompostion of a projective space into affine spaces

\mathbb{P}^{n}=\mathbb{A}^{0}\uplus\mathbb{A}^{1}\uplus\dots\uplus\mathbb{A}^{n}

induces a decomposition of its motive into a direct sum of Tate motives

\operatorname{M}(\mathbb{P}^{n})=\mathbb{Q}\oplus\mathbb{Q}(1)[2]\oplus\dots\oplus\mathbb{Q}(n)[2n]

which shows that $\operatorname{M}(\mathbb{P}^{n})$ is pure Tate.

We will use the following generalization.

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Definition 3.3. A partition of a variety $X$ into subvarieties $X_{1},\dots,X_{n}$ (called strata) is an affine paving, if $X_{\leq k}=\bigcup_{i=1,\dots,k}X_{i}$ is closed in $X$ for all $1\leq l\leq n$ and each stratum $X_{i}$ is isomorphic to an affine space $\mathbb{A}^{n}.$

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Proposition 3.4. Let $X$ be a variety that admits an affine paving. Then the motive with compact support $\operatorname{M}^{c}(X)\in\operatorname{DTM}(k,\mathbb{Q})^{w=0}$ is pure Tate.

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Proof. This follows from the localisation sequence and $\operatorname{M}^{c}(\mathbb{A}^{n})=\mathbb{Q}(n)[2n].$ A proof can be found in [Ebe21, Lemma 2.3(10)]. ∎

As the notation suggests, the category of pure Tate motives is the heart of a weight structure:

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Proposition 3.5. The category $\operatorname{DTM}(k,\mathbb{Q})^{w=0}$ is the heart of a weight structure $w$ on $\operatorname{DTM}(k,\mathbb{Q}).$

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Proof. By (3.2) the collection of objects in the $\operatorname{DTM}(k,\mathbb{Q})^{w=0}$ is negative. Moreover $\operatorname{DTM}(k,\mathbb{Q})$ is idempotent closed and generated by $\operatorname{DTM}(k,\mathbb{Q})^{w=0}$ . Now the statement follows from Proposition 2.4. ∎

Using that $k=\overline{\mathbb{F}}_{p}$ we obtain a simple decription of the category of Tate motives:

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Proposition 3.6. The weight complex functor yields an equivalence of categories

t:\operatorname{DTM}(k,\mathbb{Q})\to\operatorname{K}^{b}(\operatorname{DTM}(k,\mathbb{Q})^{w=0})\cong\operatorname{D}^{b}(\mathbb{Q}\operatorname{-mod}^{\mathbb{Z}})

where $\mathbb{Q}(1)[2]$ corresponds to the one-dimensional vector space in degree one $\mathbb{Q}\langle 1\rangle$ in the category $\mathbb{Q}\operatorname{-mod}^{\mathbb{Z}}$ of graded finite-dimensional vector spaces over $\mathbb{Q}.$

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Proof. Lemma 3.1 implies that $\operatorname{DTM}(k,\mathbb{Q})^{w=0}$ is tilting in $\operatorname{DTM}(k,\mathbb{Q}).$ Further, $\operatorname{DM}$ admits an enhancement as stable $\infty$ -category. The statement follows from Corollary 2.16. ∎

We note that the weight structure on mixed Tate motives considered here is just the shadow of the Chow weight structure on $\operatorname{DM}^{gm}(k,\mathbb{Q}),$ see Example 2.2.

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2