Subsection 2.2. (0MZS) · ScalingStacks

2.2. Weight complex functor

An object in a triangulated category $\mathcal{C}$ with bounded weight structure can be built from objects which are pure, that is in $\mathcal{C}^{w=n}=\mathcal{C}^{w=0}[-n]$ for $n\in\mathbb{Z},$ using the distinguished triangles in Definition 2.1( 3), see [Bon10, Proposition 1.5.6]. This will imply that any geometric motive can be built from motives of smooth projective varieties. This should be seen as a reflection of Deligne’s yoga of weights in the context of mixed Hodge structures and $\ell$ -adic cohomology.

We will now show that, most remarkably, these pure pieces can be assembled into a complex called weight complex. If the category $\mathcal{C}$ admits some enhancement the weight complex gives a well-defined element in the homotopy category of $\mathcal{C}^{w=0}$ in a functorial way.

We will use the following description of weight exact functors due to Sosnilo, see [Sos17, Proposition 3.3(b)].

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Proposition 2.5. Let $\mathcal{C}_{\infty}$ and $\mathcal{D}_{\infty}$ be stable $\infty$ -categories. Assume that their homotopy categories $\mathcal{C}=\operatorname{h}\!\mathcal{C}_{\infty}$ and $\mathcal{D}=\operatorname{h}\!\mathcal{D}_{\infty}$ are equipped with bounded weight structures. The hearts $\mathcal{C}_{\infty}^{w=0}$ and $\mathcal{D}_{\infty}^{w=0}$ are the full subcategories of $\mathcal{C}_{\infty}$ and $\mathcal{D}_{\infty}$ consisting of all objects in $\mathcal{C}^{w=0}$ and $\mathcal{D}^{w=0},$ respectively. Then restriction gives an equivalence of categories

\operatorname{Res}:\operatorname{Fun}^{\operatorname{w-ex}}(\mathcal{C}_{\infty},\mathcal{D}_{\infty})\to\operatorname{Fun}^{\operatorname{add}}(\mathcal{C}_{\infty}^{w=0},\mathcal{D}_{\infty}^{w=0})

between the $\infty$ -categories of exact functors from $\mathcal{C}_{\infty}$ to $\mathcal{D}_{\infty}$ that induce weight exact functors from $\mathcal{C}$ to $\mathcal{D}$ and of additive functors between the hearts $\mathcal{C}_{\infty}^{w=0}$ and $\mathcal{D}_{\infty}^{w=0}.$

The result can be used to construct a weight complex functor, see [Sos17, Corollary 3.5]:

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Proposition 2.6. Let $\mathcal{C}$ be an idempotent complete triangulated category with bounded weight structure $w$ . Assume that $\mathcal{C}=\operatorname{h}\!\mathcal{C}_{\infty}$ arises as the homotopy category of a stable $\infty$ -category $\mathcal{C}_{\infty}$ . Then there is a functor of triangulated categories called weight complex functor

(2.1)

\displaystyle t:\mathcal{C}\to\operatorname{K}^{b}(\mathcal{C}^{w=0})

that restricts to the natural embedding $\mathcal{C}^{w=0}\to\operatorname{K}^{b}(\mathcal{C}^{w=0})$ into degree $0.$

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Proof. We apply Proposition 2.5 to $\mathcal{C}_{\infty}$ and $\mathcal{D}_{\infty}=\operatorname{N}\!\operatorname{Ch}\!^{b}(\mathcal{C}^{w=0}),$ the stable $\infty$ -category of bounded chain complexes with values in $\mathcal{C}^{w=0}$ from [Lur09, Example 4.4.5.1] and [Lur12, Section 1.3.1]. The homotopy category $\operatorname{h}\!\mathcal{D}_{\infty}=\operatorname{K}^{b}(\mathcal{C}^{w=0})$ of $\mathcal{D}_{\infty}$ is the bounded homotopy category of chain complexes in $\mathcal{C}^{w=0}$ and equipped with the canonical weight structure with heart $\mathcal{C}^{w=0},$ see Example 2.2(1). The heart of the $\infty$ -category $\mathcal{D}_{\infty}$ is $\mathcal{D}_{\infty}^{w=0}=\operatorname{N}\!(\mathcal{C}^{w=0}),$ the nerve of $\mathcal{C}^{w=0}.$

The unit of the adjunction between the nerve and homotopy category functors yields a functor $\epsilon:\mathcal{C}_{\infty}^{w=0}\to\mathcal{D}_{\infty}^{w=0}=\operatorname{N}\!(\operatorname{h}\!\mathcal{C}_{\infty}^{w=0}).$ The $\infty$ -categorical weight complex functor $t_{\infty}:\mathcal{C}_{\infty}\to\mathcal{D}_{\infty}=\operatorname{N}\!\operatorname{Ch}\!^{b}(\mathcal{C}^{w=0})$ is the unique (up to equivalence) weight exact functor such that $\operatorname{Res}(t_{\infty})=\epsilon,$ where $\operatorname{Res}$ is the functor defined in Proposition 2.5. On the homotopy categories, $t_{\infty}$ induces the weight complex functor $t$ . ∎

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Remark 2.7. The weight complex functor also exists and admits an explicit construction if $\mathcal{C}$ admits an enhancement as an $f$ -category, see [Bon10] and [Sch11c], or as a differential graded category, see [Bon09]. Moreover it also exists if $\mathcal{C}$ admits an enhancement as a stable derivator using the fact that stable derivators yield $f$ -categories, see [Mod19].

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Example 2.8. As an application we obtain a weight complex functor for the Chow weight structure

t:\operatorname{DM}_{gm}(k,\mathbb{Q})\to\operatorname{K}^{b}(\operatorname{Chow}(k,\mathbb{Q})),

see Example 2.2(3). Here we use that $\operatorname{DM}_{gm}(k,\mathbb{Q})$ has an enhancement as a stable $\infty$ -category.

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Remark 2.9. The weight complex functor for $\operatorname{DM}_{gm}(k,\mathbb{Q})$ allows to decompose the motive $\operatorname{M}(X)$ of any (not necessarily smooth or projective) variety $X$ into a complex of motives of smooth projective varieties. Again, this reflects Deligne’s yoga of weights for mixed Hodge structures and $\ell$ -adic cohomology: the cohomology of a variety admits a weight filtration whose graded pieces behave like the cohomology of smooth projective varieties. Similarly to Deligne’s approach, the existence of the Chow weight structure and the weight complex functor relies on resolutions of singularities or de Jong’s and Gabber’s theory of alterations, see [Bon11].

Sosnilo [Sos17, Corollary 3.5] and Aoki [Aok20, Theorem 4.3] show that the weight complex functor is compatible with weight exact functors and also with symmetric monoidal structures as follows.

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Proposition 2.10. Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories with bounded weight structures and $F:\mathcal{C}\to\mathcal{D}$ be a weight exact functor. Assume that there is an enhancement to a functor $F:\mathcal{C}_{\infty}\to\mathcal{D}_{\infty}$ between stable $\infty$ -categories. Then

(1)

The following diagram of functors commutes up to natural isomorphism
(2)

If $\mathcal{C}_{\infty}$ is symmetric monoidal and $\mathcal{C}^{w\geq 0}$ and $\mathcal{C}^{w\leq 0}$ are closed with respect to the monoidal structures then the weight complex functor can be turned into a symmetric monoidal functor.

The weight complex functor reflects isomorphisms, but is not necessarily an equivalence. For example, the heart $\mathcal{C}^{w=0}$ is clearly tilting in $\operatorname{K}^{b}(\mathcal{C}^{w=0})$ while in general it is just negative in $\mathcal{C}.$ This is the main obstruction as the following result shows.

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Proposition 2.11. In the situation of Proposition 2.6, assume that $\mathcal{C}^{w=0}$ is tilting in $\mathcal{C}$ . Then the weight complex functor $t$ is an equivalence of categories.

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Proof. Clearly, $t$ is fully faithful when restricted to $\mathcal{C}^{w=0}.$ Since $\mathcal{C}^{w=0}$ is tilting, $t$ is also fully faithful when restricted to $\bigcup_{n}\mathcal{C}^{w=n}.$ Now $\mathcal{C}^{w=0}$ generates $\mathcal{C}$ as triangulated subcategory since $w$ is bounded, see [Bon10, Corollary 1.5.7]. Hence $t$ is fully faithful on $\mathcal{C}$ by induction (dévissage) using the long exact sequence of $\operatorname{Hom}$ -groups for distinguished triangles and the $5$ -lemma. Essential surjectivity follows from dévissage as well since $\mathcal{C}^{w=0}$ generates $\operatorname{K}^{b}(\mathcal{C}^{w=0})$ as triangulated subcategory. ∎

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Corollary 2.12. Let $\mathcal{C}$ be an idempotent-closed triangulated category that admits an enhancement as in Proposition 2.6 or Remark 2.7. Let $\mathcal{T}$ be a collection of objects in $\mathcal{C}$ which is tilting. Then there is an equivalence of categories

t:\langle\mathcal{T}\rangle_{\cong,\inplus,\Delta}\to\operatorname{K}^{b}(\langle\mathcal{T}\rangle_{\cong,\inplus,\oplus})

between the full subcategory of $\mathcal{C}$ generated by $\mathcal{T}$ under isomorphisms, direct summands and triangles and the bounded homotopy category of the category generated by $\mathcal{T}$ under isomorphisms, direct summands and finite direct sums.

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2