[Paper Review] Weight structures for triangulated categories: weight filtrations, weight spectral sequences and weight complexes; applications to motives and to the stable homotopy category

Mikhail V. Bondarko|arXiv (Cornell University)|Apr 30, 2007

Homotopy and Cohomology in Algebraic Topology3 citations

TL;DR

This paper introduces weight structures in triangulated categories, providing a dual framework to t-structures that axiomatizes stupid truncations in K(B). It constructs canonical weight spectral sequences, proves K-theory isomorphisms K₀(C) ≅ K₀(Hw), and applies the framework to motives and stable homotopy theory, recovering Deligne’s and Atiyah-Hirzebruch spectral sequences as special cases.

ABSTRACT

This paper is dedicated to triangulated categories endowed with weight structures (a new notion; D. Pauksztello has independently introduced them as co-t-structures). This axiomatizes the properties of stupid truncations of complexes in $K(B)$. We also construct weight structures for Voevodsky's categories of motives and for various categories of spectra. A weight structure $w$ defines Postnikov towers of objects; these towers are canonical and functorial 'up to morphisms that are zero on cohomology'. For $Hw$ being the heart of $w$ (in $DM_{gm}$ we have $Hw=Chow$) we define a canonical conservative 'weakly exact' functor $t$ from our $C$ to a certain weak category of complexes $K_w(Hw)$. For any (co)homological functor $H:C o A$ for an abelian $A$ we construct a weight spectral sequence $T:H(X^i[j])\implies H(X[i+j])$ where $(X^i)=t(X)$; it is canonical and functorial starting from $E_2$. This spectral sequences specializes to the 'usual' (Deligne's) weight spectral sequences for 'classical' realizations of motives and to Atiyah-Hirzebruch spectral sequences for spectra. Under certain restrictions, we prove that $K_0(C)\cong K_0(Hw)$ and $K_0(End C)\cong K_0(End Hw)$. The definition of a weight structure is almost dual to those of a t-structure; yet several properties differ. One can often construct a certain $t$-structure which is 'adjacent' to $w$ and vice versa. This is the case for the Voevodsky's $DM^{eff}_-$ (one obtains certain new Chow weight and t-structures for it; the heart of the latter is 'dual' to $Chow^{eff}$) and for the stable homotopy category. The Chow t-structure is closely related to unramified cohomology.

Motivation & Objective

• To introduce and axiomatize weight structures in triangulated categories as a dual to t-structures, inspired by stupid truncations in K(B).
• To construct weight structures for Voevodsky's categories of motives and for categories of spectra.
• To define a canonical conservative weakly exact functor t: C → K_w(Hw), where Hw is the heart of the weight structure.
• To establish a canonical, functorial weight spectral sequence T: H(X^i[j]) ⇒ H(X[i+j]) for any (co)homological functor H.
• To prove K₀(C) ≅ K₀(Hw) and K₀(End C) ≅ K₀(End Hw) under suitable conditions, linking K-theory of the category to its heart.

Proposed method

• Define a weight structure w on a triangulated category C via axioms dual to those of a t-structure, capturing properties of stupid truncations.
• Construct a Postnikov tower for each object in C that is canonical and functorial up to morphisms vanishing on cohomology.
• Define the heart Hw of w as the full subcategory of objects of weight zero, and construct a weak category of complexes K_w(Hw).
• Define a canonical conservative 'weakly exact' functor t: C → K_w(Hw) that sends objects to their weight complex.
• Derive a weight spectral sequence T: H(X^i[j]) ⇒ H(X[i+j]) from the image of X under t, with E₂-page determined by cohomology of the weight complex.
• Establish adjunctions between weight structures and t-structures, particularly in DM^{eff}_- and the stable homotopy category, yielding dual Chow structures.

Experimental results

Research questions

• RQ1 How can one axiomatize the properties of stupid truncations in triangulated categories using a dual framework to t-structures?
• RQ2 Can weight structures be systematically constructed in categories such as Voevodsky's motives and spectra?
• RQ3 To what extent do weight spectral sequences recover known spectral sequences like Deligne’s or Atiyah-Hirzebruch’s?
• RQ4 What is the relationship between the K-theory of a triangulated category and the K-theory of its weight heart?
• RQ5 How do weight structures interact with t-structures, and can they be used to define dual or adjacent structures in key categories?

Key findings

• A weight structure w on a triangulated category C induces a canonical, functorial Postnikov tower for each object, unique up to morphisms that vanish on cohomology.
• The construction yields a canonical, functorial weight spectral sequence T: H(X^i[j]) ⇒ H(X[i+j]) starting from the E₂-page, which specializes to Deligne’s weight spectral sequence for motives and the Atiyah-Hirzebruch spectral sequence for spectra.
• There exists a conservative, weakly exact functor t: C → K_w(Hw) that maps objects to their weight complexes in the weak category of complexes over the heart Hw.
• Under suitable conditions, the K-theory of the category C satisfies K₀(C) ≅ K₀(Hw), and similarly for endomorphism categories: K₀(End C) ≅ K₀(End Hw).
• In DM^{eff}_-, a weight structure w exists that is adjacent to a t-structure, leading to a dual Chow t-structure whose heart is dual to the effective Chow category.
• The Chow t-structure in DM^{eff}_- is closely related to unramified cohomology, suggesting deeper arithmetic connections via the weight structure framework.