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t-structure

Contents

Definition

Definition

Let CC be a triangulated category. A t-structure on CC is a pair 𝔱=(C 0,C 0)\mathfrak{t}=(C_{\ge 0}, C_{\le 0}) of full subcategories

C 0,C 0C C_{\geq 0}, C_{\leq 0} \hookrightarrow C

such that

  1. for all XC 0X \in C_{\geq 0} and YC 0Y \in C_{\leq 0} the hom object is the zero object: Hom C(X,Y[1])=0Hom_{C}(X, Y[-1]) = 0;

  2. the subcategories are closed under suspension/desuspension: C 0[1]C 0C_{\geq 0}[1] \subset C_{\geq 0} and C 0[1]C 0C_{\leq 0}[-1] \subset C_{\leq 0}.

  3. For all objects XCX \in C there is a fiber sequence YXZY \to X \to Z with YC 0Y \in C_{\geq 0} and ZC 0[1]Z \in C_{\leq 0}[-1].

Definition

Given a t-structure, its heart is the intersection

C 0C 0C. C_{\geq 0} \cap C_{\leq 0} \hookrightarrow C \,.

In stable \infty-categories

In the infinity-categorical setting tt-structures arise as torsion/torsionfree classes associated to suitable factorization systems on a stable infinity-category CC.

  • In a stable setting, the subcategories are closed under de/suspension simply because they are co/reflective and reflective and these operations are co/limits. Co/reflective subcategories of CC arise from co/reflective factorization systems on CC;

  • A bireflective factorization system on a \infty-category CC consists of a factorization system 𝔽=(E,M)\mathbb{F}=(E,M) where both classes satisfy the two-out-of-three property.

  • A bireflective factorization system (E,M)(E,M) on a stable \infty-category CC is called normal if the diagram SxxRxS x\to x\to R x obtained from the reflection R:CM/0R\colon C\to M/0 and the coreflection S:C*/ES\colon C\to *\!/E (where the category M/*={A(0A)M}M/\!* =\{A\mid (0\to A)\in M\} is obtained as Ψ(E,M)\Psi(E,M) under the adjunction ΦΨ\Phi\dashv \Psi described at reflective factorization system and in CHK; see also FL0, §1.1) is exact, meaning that the square in

    0 SX X 0 RX 0 \begin{array}{cccccc} 0 &\to& S X &\to& X\\ && \downarrow&&\downarrow\\ && 0 &\to& R X\\ && && \downarrow\\ && && 0 \end{array}

    is a fiber sequence for any object XX; see FL0, Def 3.5 and Prop. 3.10 for equivalent conditions for normality.

Remark. CHK established a hierarchy between the three notions of simple, semi-exact and normal factorization system: in the setting of stable \infty-category the three notions turn out to be equivalent: see FL0, Thm 3.11.

Theorem. There is a bijective correspondence between the class TS(C)TS( C ) of tt-structures and the class of normal torsion theories on a stable \infty-category CC, induced by the following correspondence:

  • On the one side, given a normal, bireflective factorization system (E,M)(E,M) on CC we define the two classes (C 0(𝔽),C <0(𝔽))(C_{\ge0}(\mathbb{F}), C_{\lt 0}(\mathbb{F})) of a tt-structure 𝔱(𝔽)\mathfrak{t}(\mathbb{F}) to be the torsion and torsionfree classes (*/E,M/*)(*\!/E, M/\!*) associated to the factorization (E,M)(E,M).
  • On the other side, given a tt-structure on CC we set
    E(t)={fC Δ[1]τ <0(f) is an equivalence};E(t)=\{f\in C^{\Delta[1]} \mid \tau_{\lt 0}(f) \;\text{ is an equivalence}\};
    M(t)={fC Δ[1]τ 0(f) is an equivalence}.M(t)=\{f\in C^{\Delta[1]} \mid \tau_{\geq0}(f) \;\text{ is an equivalence}\}.

Proof. This is FL0, Theorem 3.13

Theorem. There is a natural monotone action of the group \mathbb{Z} of integers on the class TS(C)TS( C ) (now confused with the class FS ν(C)FS_\nu( C ) of normal torsion theories on CC) given by the suspension functor: 𝔽=(E,M)\mathbb{F}=(E,M) goes to 𝔽[1]=(E[1],M[1])\mathbb{F}[1] = (E[1], M[1]).

This correspondence leads to study families of tt-structures {𝔽 i} iI\{\mathbb{F}_i\}_{i\in I}; more precisely, we are led to study \mathbb{Z}-equivariant multiple factorization systems JTS(C)J\to TS( C ).

Theorem. Let 𝔱TS(C)\mathfrak{t} \in TS(C) and 𝔽=(E,M)\mathbb{F}=(E,M) correspond each other under the above bijection; then the following conditions are equivalent:

  1. 𝔱[1]=𝔱\mathfrak{t}[1]=\mathfrak{t}, i.e. C 1=C 0C_{\geq 1}= C_{\geq 0};
  2. C 0=*/EC_{\geq 0}=*\!/E is a stable \infty-category;
  3. the class EE is closed under pullback.

In each of these cases, we say that 𝔱\mathfrak{t} or (E,M)(E,M) is stable.

Proof. This is FL1, Theorem 2.16

This results allows us to recognize tt-structures with stable classes precisely as those which are fixed in the natural \mathbb{Z}-action on TS(C)TS( C ).

Two “extremal” choices of \mathbb{Z}-chains of tt-structures draw a connection between two apparently separated constructions in the theory of derived categories: Harder-Narashiman filtrations and semiorthogonal decompositions on triangulated categories: we adopt the shorthand 𝔱 1,,n\mathfrak{t}_{1,\dots, n} to denote the tuple 𝔱 1𝔱 2𝔱 n\mathfrak{t}_1\preceq \mathfrak{t}_2\preceq\cdots\preceq \mathfrak{t}_n, each of the 𝔱 i\mathfrak{t}_i being a tt-structure ((C i) 0,(C i) <0)((C_i)_{\ge 0}, (C_i)_{\lt 0}) on CC, and we denote similarly 𝔱 ω\mathfrak{t}_\omega. Then

  • In the stable case the tuple t 1,,nt_{1,\dots, n} is endowed with a (monotone) \mathbb{Z}-action, and the map {0<1<n}TS(C)\{0\lt 1\cdots\lt n\}\to TS( C ) is equivariant with respect to this action; the absence of nontrivial \mathbb{Z}-actions on {0<1<n}\{0\lt 1\cdots\lt n\} forces each t it_i to be stable.
  • In the orbit case we consider an infinite family t ωt_\omega of tt-structures on CC, obtained as the orbit of a fixed (E 0,M 0)TS(C)(E_0, M_0)\in TS( C ) with respect to the natural \mathbb{Z}-action.

Towers

The HN-filtration induced by a tt-structure and the factorization induced by a semiorthogonal decomposition on CC both are the byproduct of the tower associated to a tuple 𝔱 1,,n\mathfrak{t}_{1,\dots, n}:

(…)

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, FL0, Ex. 4.1 and FL1, §3.1)

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

For reflective factorization systems and normal torsion theories in stable \infty-categories

  • Cassidy and Hébert and Kelly, “Reflective subcategories, localizations, and factorization systems”. J. Austral. Math Soc. (Series A) 38 (1985), 287–329 (pdf)

Revised on November 1, 2014 12:36:49 by Fosco Loregian (87.15.179.49)