Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)
By M. Bondarko. 2007
299 citations recorded by the source corpus.
In this paper we introduce a new notion of a weight structure (w) for a triangulated category C; this notion is an important natural counterpart to the notion of t-structures. This allows to generalize certain results of the previous paper [11] to a large class of triangulated categories and functors. The heart of w is an additive Hw C; there are no non-trivial C-distinguished triangles in Hw. We prove that a weight structure defines Postnikov towers for any X2 ObjC (whose "factors" are X i 2 ObjHw); these towers are canonical and functorial "up to zero cohomology". For any (co)homological functor H : C! A (A is abelian) we construct a weight spectral sequence T : H(X i [j]) =) H(X[i+j]); it is canonical and functorial starting from E2. This spectral sequences specializes to the usual weight spectral sequences for "classical" realizations of (Voevodsky’s) motives and to the Atiyah-Hirzebruch sequence in topology. We prove that K0(C) = K0(Hw) in the bounded case if Hw is idempotent complete. Under certain restrictions, we prove a similar equality for K0(EndC). We define a canonical conservative weakly exact functor t from C to a certain weak category of complexes Kw(Hw).
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/123e7c76b4799be5a20d30392707bded40e46fe4
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Silting theory in triangulated categories with coproducts (2015, background): “For instance, with co-t-structures (equivalently, weight structures), as defined in [13] and [47], with t-structures as defined in [11] and, in the context of Representation theory, also with the so-called simple minded collections (see [30]).”
- Torsion pairs in silting theory (2016, unknown): “We consider the notion of a torsion pair in a triangulated category (see, for example, [Iyama and Yoshino 2008]), which gives rise to the notions of a t-structure [Be˘ılinson et al. 1982] and a co-t-structure [Bondarko 2010; Pauksztello 2008].”
- Weight structures and simple dg modules for positive dg algebras (2010, unknown): “Section 6 of [7].”