t-structure

In mathematics, more specifically in homological algebra, a t-structure is an additional piece of structure that can be put on a triangulated category or a stable infinity category that axiomatizes the properties of complexes whose positive or negative cohomology vanishes. The notion was introduced by Beilinson, Bernstein and Deligne.^[1] It allows to construct an abelian category, namely the heart of the t-structure, from a triangulated category.

Definition

The derived category D of an abelian category A contains, for each n, the full subcategories $D^{\geq n}$ and $D^{\leq n}$ consisting of complexes whose cohomology is "bounded below" or "bounded above" n, respectively, i.e., $H^m(X) = 0$ for $m < n$ and $m > n$, respectively. The subcategories have the following properties:

• $D^{\le 0}\subset D^{\le 1}$, $D^{\ge 1}\subset D^{\ge 0};$
• $\operatorname{Hom}(D^{\le 0},D^{\ge 1})=0;$
• Every object X can be embedded in a distinguished triangle $X^{\le 0} \to X \to X^{\ge 1} \to \$ with $X^{\le 0} \in D^{\leq 0}$, $X^{\ge 1} \in D^{\geq 1}.$

This prototypical basic example gives rise to the following definition: a t-structure on a triangulated category consists of full subcategories $D^{\le 0}$ and $D^{\ge 1}$ satisfying the conditions above. In Faisceaux pervers a triangulated category equipped with a t-structure is called a t-category. The above example is referred to as the standard t-structure or canonical t-structure.

The notion of a t-structure can also be defined on a stable model category or a stable infinity category by requiring that there is a t-structure in the above sense on the homotopy category (which is a triangulated category).^[2]

Constructing t-structures

Many t-structures arise by means of the following fact: in a triangulated category with arbitrary direct sums, and a set $S_0$ of compact objects in D, the subcategories

$D^{\ge 1} := \{X \in D, Hom(S_0[-n], X) = 0, n \ge 0\}$

$D^{\le 0} := \{Y \in D, Hom(Y, D^{\ge 1}) = 0\}$

can be shown to be a t-structure.^[3] It is called the t-structure generated by $S_0$.

Consequences of the definition

The core or heart (the original French word is "cœur") of a t-structure is the category $D^{\le 0}\cap D^{\ge 0}$. It is an abelian category, as can be shown (whereas a triangulated category is additive but almost never abelian).

The two subcategories $D^{\le 0}$ and $D^{\ge 1}$ actually determine each other: an object X is in $D^{\le 0}$ if and only if $Hom(X,D^{\ge 1}) = 0$ and vice versa.

The objects $X^{\le 0}$ and $X^{\ge 0}$ are not a priori required to be unique or functorial. However, as a consequence of the other two axioms, they can be shown to be actually functorial and unique up to unique isomorphism. In fact, the assignment $X \mapsto \tau^{\le 0} := X^{\le 0}$ can be shown to be a left adjoint to the inclusion functor $D^{\le 0} \to D$ and likewise for $\ge 1$. In other words, the above distinguished triangle is

$\tau^{\leq 0} X \to X \to \tau^{\geq 1} X \to.\$

The n-th cohomology functor $H^n$ is defined as

$H^n := \tau^{\le n} \tau^{\ge n}$

It is in fact a cohomological functor: for any triangle $X \to Y \to Z \to$ we obtain a long exact sequence

$\cdots \to H^i(X) \to H^i(Y) \to H^i(Z) \to H^{i + 1}(X) \to \cdots.\$

Examples

Perverse sheaves

The category of perverse sheaves is, by definition, the core of the so-called t-structure on the derived category of the category of sheaves on a complex analytic space X or (working with l-adic sheaves) an algebraic variety over a finite field. As was explained above, the heart of the standard t-structure simply contains ordinary sheaves, regarded as complexes concentrated in degree 0. For example, the category of perverse sheaves on a (possibly singular) algebraic curve X (or analogously a possibly singular surface) is designed so that it contains, in particular, objects of the form

$i_* F_Z, j_* F_U[1]$

where $i: Z \to X$ is the inclusion of a point, $F_Z$ is an ordinary sheaf, $U$ is a smooth open subscheme and $F_U$ is a locally constant sheaf on U. Note the presence of the shift according to the dimension of Z and U respectively. This shift causes the category of perverse sheaves to be well-behaved on singular spaces. This t-structure was introduced by Beilinson, Bernstein and Deligne.^[4] It was shown by Beilinson that the derived category of the heart $D^b(Perv(X))$ is in fact equivalent to the original derived category of sheaves. This is an example of the general fact that a triangulated category may be endowed with several distinct t-structures.^[5]

Graded modules

A non-standard example of a t-structure on the derived category of (graded) modules over a graded ring has the property that its heart consists of complexes

$\dots \to P^n \to P^{n+1} \to \dots$

where $P^n$ is a module generated by its (graded) degree n. This t-structure called geometric t-structure plays a prominent role in Koszul duality.^[6]

Spectra

The category of spectra is endowed with a t-structure generated, in the sense above, by a single object, namely the sphere spectrum. The category $Sp^{\le 0}$ is the category of connective spectra, i.e., those whose negative homotopy groups vanish. (In areas related to homotopy theory, it is common to use homological conventions, as opposed to cohomological ones, so in this case it is common to replace "$\le$" by "$\ge$". Using this convention, the category of connective spectra the notation is denoted $Sp^{\ge 0}$.)

Motives

A conjectural example in the theory of motives is the so-called motivic t-structure. Its (conjectural) existence is closely related to certain standard conjectures on algebraic cycles and vanishing conjectures, such as the Beilinson-Soulé conjecture.^[7]

Related concepts

If the requirement $D^{\le 0}\subset D^{\le 1}$, $D^{\ge 1}\subset D^{\ge 0};$ is replaced by the opposite inclusion

$D^{\le 0}\supset D^{\le 1}$, $D^{\ge 1}\supset D^{\ge 0};$

(and the other two axioms kept the same), the resulting notion is called a co-t-structure or weight structure.^[8]

References

1. Beĭlinson, A. A.; Bernstein, J.; Deligne, P. Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100, Soc. Math. France, Paris, 1982.
2. Jacob Lurie. Higher Algebra.
3. Beligiannis, Apostolos; Reiten, Idun. Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207 pp. Theorem III.2.3
4. Beĭlinson, A. A.; Bernstein, J.; Deligne, P. Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100, Soc. Math. France, Paris, 1982.
5. Beĭlinson, A. A. On the derived category of perverse sheaves. K-theory, arithmetic and geometry (Moscow, 1984–1986), 27–41, Lecture Notes in Math., 1289, Springer, Berlin, 1987.
6. Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang. Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.
7. Hanamura, Masaki. Mixed motives and algebraic cycles. III. Math. Res. Lett. 6 (1999), no. 1, 61–82.
8. Bondarko, M. V. Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general). J. K-Theory 6 (2010), no. 3, 387–504.