nLab
t-structure

Contents

Definition

Definition

Let 𝒟\mathcal{D} be a triangulated category. A t-structure on 𝒟\mathcal{D} is a pair of full subcategories

𝒟 ≥0,𝒟 ≤0↪𝒟 \mathcal{D}_{\geq 0}, \mathcal{D}_{\leq 0} \hookrightarrow \mathcal{D}

such that

  1. for all X∈𝒟 ≥0X \in \mathcal{D}_{\geq 0} and Y∈𝒟 ≤0Y \in \mathcal{D}_{\leq 0} the hom object is the zero object: Hom 𝒟(X,Y[−1])=0Hom_{\mathcal{D}}(X, Y[-1]) = 0;

  2. the subcategories are closed under suspension/desuspension: 𝒟 ≥0[1]⊂𝒟 ≥0\mathcal{D}_{\geq 0}[1] \subset \mathcal{D}_{\geq 0} and 𝒟 ≤0[−1]⊂𝒟 ≤0\mathcal{D}_{\leq 0}[-1] \subset \mathcal{D}_{\leq 0}.

  3. For all objects X∈𝒟X \in \mathcal{D} there is a fiber sequence Y→X→ZY \to X \to Z with Y∈𝒟 ≥0Y \in \mathcal{D}_{\geq 0} and Z∈𝒟 ≤0[−1]Z \in \mathcal{D}_{\leq 0}[-1].

Definition

Given a t-structure, its heart is the intersection

𝒟 ≥0∩𝒟 ≤0↪𝒟. \mathcal{D}_{\geq 0} \cap \mathcal{D}_{\leq 0} \hookrightarrow \mathcal{D} \,.
Definition

A t-structure on a stable (∞,1)-category is a t-structure on its homotopy category of an (∞,1)-category, according to def. 1.

(Higher Algebra, def. 1.2.1.4).

Properties

General

Proposition

The heart of a stable (∞,1)(\infty,1)-category is an abelian category.

(BBD 82, Higher Algebra, remark 1.2.1.12)

Application to spectral sequence

If a the heart of a t-structure on a stable (∞,1)-category with sequential limits is an abelian category, then the spectral sequence of a filtered stable homotopy type converges (see there).

References

Related nnLab entries include Bridgeland stability?.

For triangulated categories

  • D. Abramovich, A. Polishchuk, Sheaves of t-structures and valuative criteria for stable complexes, J. reine angew. Math. 590 (2006), 89–130
  • A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, t-stabilities and t-structures on triangulated categories, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 4, 117–150
  • A. Polishchuk, Constant families of t-structures on derived categories of coherent sheaves, Moscow Math. J. 7 (2007), 109–134
  • John Collins, Alexander Polishchuk, Gluing stability conditions, arxiv/0902.0323

For stable (∞,1)-categories

Revised on November 19, 2013 03:38:32 by Zoran Å koda (161.53.130.104)