Section 2. (0MZQ) · ScalingStacks

2. Weight structures and applications

We recall the definition and examples of weight structures due to Bondarko [Bon10]. The main goal is the definition of the weight complex functor and applications to Koszul and Ringel duality.

2.1. Weight structures

We start with the definition of a weight structure, see [Bon10, Definition 1.1.1].

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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.

Following [Bon10, Theorem 4.3.2] we now show how to define weight structures by specifying their heart.

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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.

2.2. 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.

2.3. Locally unital algebras

It is sometimes convenient to pass from additive categories, such as the heart of a weight structure $\mathcal{C}^{w=0},$ to categories of modules over some algebra with idempotents. We use this perspective to prove a small variation on Corollary 2.12.

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Definition 2.13. A (non-unital) algebra $A$ with a distinguished set of idempotents $\{e_{i}|i\in I\}$ is called locally unital if the canonical map

\bigoplus_{i,j\in I}e_{i}Ae_{j}\to A

is an isomorphism. A right module over a locally unital algebra $A$ is called finitely generated (projective) if it is isomorphic to a quotient (direct summand) of a finite direct sum of the modules $e_{i}A$ for $i\in I.$ The perfect derived category of $A$ is the bounded homotopy category of the finitely generated projective right modules

\operatorname{D_{perf}}(A)=\operatorname{K}^{b}(\operatorname{mod_{fgp}-}A).

If $A$ is moreover $\mathbb{Z}$ -graded we consider the category $\operatorname{mod}^{\mathbb{Z}}\operatorname{-}A$ of graded right modules over $A$ with morphisms of degree $0$ and the graded perfect derived category

\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(A)=\operatorname{K}^{b}(\operatorname{mod}^{\mathbb{Z}}_{\operatorname{fgp}}\operatorname{-}A).

For a family of objects $\mathcal{T}$ in an idempotent-closed additive category $\mathcal{C}$ we consider the locally unital algebra with the distinguished idempotents $e_{M}=\operatorname{id}_{M}$ for $M\in\mathcal{T}$

(2.2)

\displaystyle\operatorname{End}_{\mathcal{C}}(\mathcal{T})=\bigoplus_{M,N\in\mathcal{T}}\operatorname{Hom}_{\mathcal{C}}(M,N).

By general nonsense we obtain:

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Proposition 2.14. The functor

\bigoplus_{M\in\mathcal{T}}\operatorname{Hom}_{\mathcal{C}}(M,-):\langle\mathcal{T}\rangle_{\cong,\inplus,\oplus}\to\operatorname{mod_{fgp}-}\operatorname{End}_{\mathcal{C}}(\mathcal{T})

is an equivalence of categories.

For a graded version, assume that $\mathcal{C}$ is equipped with an autoequivalence $\langle 1\rangle.$ Then one can define the $\mathbb{Z}$ -graded locally unital algebra

(2.3)

\displaystyle\operatorname{End}_{\mathcal{C}}^{\bullet}(\mathcal{T})=\bigoplus_{M,N\in\mathcal{T}}\operatorname{Hom}^{\bullet}_{\mathcal{C}}(M,N)\text{ where }\operatorname{Hom}^{n}_{\mathcal{C}}(M,N)=\operatorname{Hom}_{\mathcal{C}}(M,N\langle n\rangle).

Again, general nonsense yields

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Proposition 2.15. The functor

\bigoplus_{M\in\mathcal{T}}\operatorname{Hom}^{\bullet}_{\mathcal{C}}(M,-):\langle\mathcal{T}\rangle_{\cong,\inplus,\oplus,\langle\pm 1\rangle}\to\operatorname{mod}^{\mathbb{Z}}_{\operatorname{fgp}}\operatorname{-}\operatorname{End}_{\mathcal{C}}^{\bullet}(\mathcal{T})

is an equivalence of categories.

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Corollary 2.16. In the situation of Corollary 2.12 the weight complex functor induces an equivalence

t:\langle\mathcal{T}\rangle_{\cong,\inplus,\Delta}\to\operatorname{D_{perf}}(\operatorname{End}_{\mathcal{C}}(\mathcal{T})).

If $\mathcal{C}$ admits and autoequivalence $\langle 1\rangle$ such that $\widehat{\mathcal{T}}=\bigcup_{n}\mathcal{T}\langle n\rangle$ is also tilting then the weight complex functor induces an equivalence

t:\langle\widehat{\mathcal{T}}\rangle_{\cong,\inplus,\Delta}\to\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(\operatorname{End}^{\bullet}_{\mathcal{C}}(\mathcal{T})).

2.4. Koszul duality

Weight structures and weight complex functors can be used to provide a convenient language for the Koszul duality formalism from [BGS96], and slightly more generally [MOS09].

Let $A$ be a $\mathbb{Z}$ -graded algebra which is positively graded, that is, $A^{i}=0$ for $i<0$ and assume that $A_{0}$ is semisimple and finite dimensional. Denote by $\langle 1\rangle$ the shift of grading functor on the category $A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}}$ of finitely generated graded $A$ -modules and let $\mathcal{T}(A)$ be a set of representatives of simple objects concentrated in degree $0.$

The algebra $A$ is called Koszul if $\operatorname{Ext}^{i}(L,L^{\prime}\langle j\rangle)=0\text{ for all }i\neq j$ and $L,L^{\prime}\in\mathcal{T}(A).$ Equivalently, $A$ is Koszul if the family $\widehat{\mathcal{T}}(A)=\bigcup_{i}\mathcal{T}(A)\langle i\rangle[i]$ is tilting in $\operatorname{D}^{b}(A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}}).$ In this case, the Koszul dual $A^{!}$ of $A$ is the graded algebra

A^{!}=\operatorname{End}^{\bullet}_{\operatorname{D}^{b}(A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}})}(\mathcal{T}(\mathcal{A}))

where the right hand side is defined as in (2.2) using the autoequivalence $\langle 1\rangle[1].$

Corollary 2.16 implies that the weight complex functor induces an equivalence of categories

(2.4)

\displaystyle t:\langle\widehat{\mathcal{T}}(A)\rangle_{\cong,\Delta}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{D_{perf}}(A^{!}).

In the case that $A$ is finitely generated as an $A_{0}$ -left and right module and $A^{!}$ is left Noetherian, (2.4) specializes to the Koszul duality from [BGS96, Theorem 2.12.5]

(2.5)

\displaystyle\operatorname{D}^{b}(A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}})\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{D}^{b}(A^{!}\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}}).

The heart of the weight structure defined by $\mathcal{T}(A)$ on the left hand side maps to projective modules in cohomological degree $0$ on the right. Moreover, one can show that the heart of the standard $t$ -structure on the right hand side corresponds to the category of linear complexes on the left hand side, see [MOS09].

2.5. Ringel duality

Similarly to the last paragraph, weight structures and weight complex functors can be applied to the theory of tilting objects and Ringel duality for highest weight categories and their semi-infinite and stratified generalisation. For the general setup we refer to Brundan–Stroppel [BS21] and freely use the terminology from there. Let $\mathcal{R}$ be a lower finite or essentially finite $\epsilon$ -stratified category over an algebraically closed field (in particular it could be a highest weight category).

Then by [BS21, Theorems 4.2, 4.13, 4.18] tilting modules in the sense of highest weight categories exist in $\mathcal{R}$ and form an additive subcategory generated by the family $\mathcal{T}(\mathcal{R})$ of representatives of isomorphism classes of indecomposable tilting modules. For $E=\operatorname{End}_{\mathcal{R}}(\mathcal{T}(\mathcal{R})),$ see (2.2), the category $\mathcal{R}^{\prime}=\operatorname{mod_{lfd}-}E$ of locally finite dimensional right modules is called the Ringel dual of $\mathcal{R}.$

By [BS21, Theorems 3.11, 3.56] the family of tilting modules $\mathcal{T}(\mathcal{R})$ is tilting in $\operatorname{D}^{b}(\mathcal{R})$ in the sense of Definition 2.3. Therefore Corollary 2.16 implies that the weight complex functor induces an equivalence of categories

(2.6)

\displaystyle t:\langle\mathcal{T}(\mathcal{R})\rangle_{\cong,\Delta}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{D_{perf}}(E).

Now assume that $\mathcal{T}(\mathcal{R})$ generates $\operatorname{D}^{b}(\mathcal{R})$ as a triangulated category and $E$ has finite cohomological dimension. For example, this is the case if $\mathcal{R}$ is a highest weight category. Then (2.6) yields a derived equivalence, called Ringel duality, between $\mathcal{R}$ and its Ringel dual $\mathcal{R}^{\prime}$

\operatorname{D}^{b}(\mathcal{R})\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{D}^{b}(\mathcal{R}^{\prime}).

This interpretation of Ringel duality in terms of weight complex functors has interesting applications. For example, one can use Proposition 2.10 to show that Ringel duality commutes with functors preserving tilting modules (under the correct technical assumptions).

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2