Subsection 2.1. (0MZR) · ScalingStacks

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Definition 2.1. Let $\mathcal{C}$ be a triangulated category. A weight structure $w$ on $\mathcal{C}$ is a pair $w=(\mathcal{C}^{w\leq 0},\mathcal{C}^{w\geq 0})$ of idempotent-closed full subcategories of $\mathcal{C},$ such that with $\mathcal{C}^{w\leq n}:=\mathcal{C}^{w\leq 0}[-n]$ and $\mathcal{C}^{w\geq n}:=\mathcal{C}^{w\geq 0}[-n]$ the following conditions are satisfied:

(1)

$\mathcal{C}^{w\leq 0}\subseteq\mathcal{C}^{w\leq 1}$ and $\mathcal{C}^{w\geq 1}\subseteq\mathcal{C}^{w\geq 0};$
(2)

for all $X\in\mathcal{C}^{w\geq 0}$ and $Y\in\mathcal{C}^{w\leq-1}$ , we have $\operatorname{Hom}_{\mathcal{C}}(X,Y)=0;$
(3)

for any $X\in\mathcal{C}$ there is a distinguished triangle

with $A\in\mathcal{C}^{w\geq 1}$ and $B\in\mathcal{C}^{w\leq 0}.$

The full subcategory $\mathcal{C}^{w=0}=\mathcal{C}^{w\leq 0}\cap\mathcal{C}^{w\geq 0}$ is called the heart. A weight structure is called bounded if $\bigcup_{i}\mathcal{C}^{w\leq i}=\bigcup_{i}\mathcal{C}^{w\geq i}=\mathcal{C}.$ A triangulated functor $F:\mathcal{C}\to\mathcal{D}$ between two categories with weight structures is called weight exact if $F(\mathcal{C}^{w\leq 0})\subset\mathcal{D}^{w\leq 0}$ and $F(\mathcal{C}^{w\geq 0})\subset\mathcal{D}^{w\geq 0}.$

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Example 2.2. (1) The prototypical example of a weight structure arises from the stupid filtration of complexes. Let $\mathcal{A}$ be an idempotent-closed additive category. Then there is a bounded weight structure on $\operatorname{K}^{b}(\mathcal{A})$ given by

	$\displaystyle\operatorname{K}^{b}(\mathcal{A})^{w\geq 0}$	$\displaystyle=\langle X\mid X_{i}=0\text{ for all }i<0\rangle_{\cong}\,\,\,\,\text{ and}$
	$\displaystyle\operatorname{K}^{b}(\mathcal{A})^{w\leq 0}$	$\displaystyle=\langle X\mid X_{i}=0\text{ for all }i>0\rangle_{\cong},$

the subcategories generated by the complexes in non-negative and non-positive degrees under isomorphism. Most of the axioms of a weight structure are straigtforward to check. The idempotent-completeness $\operatorname{K}^{b}(\mathcal{A})^{w\geq 0}$ and $\operatorname{K}^{b}(\mathcal{A})^{w\leq 0}$ is discussed in [Sch11a]. The heart of the weight structure is $\operatorname{K}^{b}(\mathcal{A})^{w=0}=\mathcal{A}.$ In general, the heart of a weight structure is additive and idempotent closed but not necessarily abelian.
(2) Assume that $\mathcal{A}$ is abelian and that every object in $\mathcal{A}$ has a finite projective resolution. Then one can identify the bounded derived category of $\mathcal{A}$ with the bounded homotopy category of projectives

\operatorname{D}^{b}(\mathcal{A})=\operatorname{K}^{b}(\operatorname{Proj}(\mathcal{A})).

Hence $\operatorname{D}^{b}(\mathcal{A})$ is equipped with a weight structure with heart $\operatorname{Proj}(\mathcal{A}).$ This should be compared to the natural $t$ -structure on $\operatorname{D}^{b}(\mathcal{A})$ with heart $\mathcal{A}.$
(3) A particularly interesting example of weight structures arises in the world of motives, namely for Voevodsky’s triangulated category of geometric motives $\operatorname{DM}_{gm}(k,\mathbb{Q})$ over a perfect field $k$ , see [VSF00]. The existence of a $t$ -structure on $\operatorname{DM}_{gm}(k,\mathbb{Q})$ is a notoriously difficult problem that implies, see [Bei10], for example Grothendieck’s standard conjectures.

Instead of a $t$ -structure, Bondarko [Bon10] showed the existence of a weight structure $w$ called Chow weight structure on the category $\operatorname{DM}_{gm}(k,\mathbb{Q})$ whose heart

\operatorname{DM}_{gm}(k,\mathbb{Q})^{w=0}\cong\operatorname{Chow}(k,\mathbb{Q})

is equivalent to the category of Chow motives. The category of Chow motives was introduced by Grothendieck and has an elementary definition, see [Mil12]. Namely, one first considers the category of correspondences of smooth projective varieties up to rational equivalence. Here, objects are smooth projective varieties $X$ over $k$ and morphisms from $X$ to $Y$ are elements in the rational Chow group

\operatorname{CH}_{\dim(Y)}(X\times Y)_{\mathbb{Q}}.

Morphisms are composed via convolution. The category $\operatorname{Chow}(k,\mathbb{Q})$ is then obtained from the category of correspondences by idempotent completion and tensor-inverting the Lefschetz motive $\mathbb{L}=\ker(\mathbb{P}_{k}^{1}\to\operatorname{Spec}(k)).$ In other words, objects of weight zero in $\operatorname{DM}_{gm}(k,\mathbb{Q})$ arise from motives of smooth projective varieties.

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Definition 2.3. A collection of objects $\mathcal{T}$ in a triangulated category $\mathcal{C}$ is called positive, negative or tilting, respectively, if

\operatorname{Hom}_{\mathcal{C}}(M,N[n])=0

for all $M,N\in\mathcal{T}$ where $n<0,n>0$ or $n\neq 0,$ respectively.

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Proposition 2.4. Let $\mathcal{C}$ be an idempotent closed triangulated category and $\mathcal{T}$ be a collection of objects in $\mathcal{C}.$ Assume that $\mathcal{T}$ is negative and generates $\mathcal{C}$ with respect to isomorphisms, direct summands and triangles

\mathcal{C}=\langle\mathcal{T}\rangle_{\cong,\inplus,\Delta}.

Then there is a bounded weight structure on $\mathcal{C}$ whose heart

\mathcal{C}^{w=0}=\langle\mathcal{T}\rangle_{\cong,\inplus,\oplus}

is the full subcategory of $\mathcal{C}$ generated by $\mathcal{T}$ under isomorphisms, direct summands and finite direct sums.

2.1. Weight structures