Domenico Fiorenza somethingsomething

tt-structures are normal torsion theories

Idea

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

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:

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)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]=tt[1]=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 tt 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,,nt_{1,\dots, n} to denote the tuple t 1t 2t nt_1\preceq t_2\preceq\cdots\preceq t_n, each of the t it_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 ωt_\omega. Then

Towers

The HN-filtration induced by a tt-structure and the factorization induced by a semiorthogonal decomposition? on CC both stem from the same construction:

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References

Revised on November 1, 2014 at 08:22:34 by Anonymous