In the branch of mathematics called homological algebra, a t-structure is a way to axiomatize the properties of an abelian subcategory of a derived category. A t of a triangulated category or stable infinity category which abstract the idea of complexes whose cohomology vanishes in positive, respectively negative, degrees. There can be many distinct t-structures on the same category, and the interplay between these structures has implications for algebra and geometry. The notion of a t-structure arose in the work of Beilinson, Bernstein, Deligne, and Gabber on perverse sheaves.

Definition

. A t-structure of full subcategories, each of which is stable under isomorphism, which satisfy the following three axioms.

1. If X and Y

2. If X, then X. Similarly, if Y, then Y.

3. If A such that X and Y.

. In particular, they are stable under finite direct sums.

is a t. In this case, for any integer n. More briefly, we define

With this notation, the axioms above may be rewritten as:

1. If X and Y

2. .

3. If A such that X and Y.

The heart or core of the t, that is,

The heart of a t-structure is an abelian category (whereas a triangulated category is additive but almost never abelian), and it is stable under extensions.

A triangulated category with a choice of t-structure is sometimes called a t-category.

Variations

It is clear that, to define a t-structure, it suffices to fix integers m and n. Some authors define a t.

determine each other. An object X for all objects Y. Moreover, because these subcategories are full by definition, it is enough to specify their objects.

The above notation is adapted to the study of cohomology. When the goal is to study homology, slightly different notation is used. A homological t-structure such that, if we define

is a (cohomological) t are swapped. If we define

then the axioms for a homological t-structure may be written explicitly as

1. If X and Y

2. .

3. If A such that X and Y.

Examples

The natural t-structure

The most fundamental example of a t-structure is the natural t-structure be its derived category. Then the natural t-structure is defined by the pair of subcategories

It follows immediately that

In this case, the third axiom for a t. Define

and that there is a short exact sequence of complexes

This exact sequence furnishes the required distinguished triangle.

This example can be generalized to exact categories (in the sense of Quillen). There are also similar t has a similar t.

Perverse sheaves

The category of perverse sheaves is, by definition, the core of the so-called perverse 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

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. The simple objects in this category are the intersection cohomology sheaves of subvarieties with coefficients in an irreducible local system. This t-structure was introduced by Beilinson, Bernstein and Deligne. 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.

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

is a module generated by its (graded) degree n. This t-structure called geometric t-structure plays a prominent role in Koszul duality.

Spectra

The category of spectra is endowed with a t-structure generated, in the sense above, by a single object, namely the sphere spectrum is the category of connective spectra, i.e., those whose negative homotopy groups.)

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.

Truncation functors

In the above example of the natural t. Using this, it can be shown that there are truncation functors on the derived category and that they induce a natural distinguished triangle.

In fact, this is an example of a general phenomenon. While the axioms for a t is a t-structure. The precise statement is that the inclusion functors

admit adjoints. These are functors

such that

, there exists a unique

such that d and the counit and unit of the adjunctions together define a distinguished triangle

).

, the third axiom for a t. The axioms for a t, we have

and hence the existence of all the truncation functors.

Repeated truncation for a t, then there are natural transformations

which yield natural equivalences

Cohomology functors

The nth cohomology functor is defined as

, we obtain a long exact sequence

. By one of the repeated truncation identities above, up to natural equivalence it is equivalent to define

For the natural t is, up to quasi-isomorphism, the usual nth cohomology group of a complex. However, considered as functors on complexes, this is not as defined in terms of the natural t-structure. By definition, this is

are quasi-isomorphic.

A t-structure is non-degenerate, consists only of zero objects. For a non-degenerate t).

Exact functors

be a triangulated category with a fixed t is:

• Left t-exact,

• Right t-exact, and

• t-exact if it is both left and right t-exact.

is fully faithful and t is another left (resp. right) t is also left (resp. right) t-exact.

The motivation for the study of one-sided t be the inclusion. Then there is a composite functor

.

is right (resp. left) t).

is right t is left t.

Constructions of t-structures

be a t. If n is an integer, then the translation by n t. The dual t-structure is the t-structure on the opposite category.

is a t, then

is a t. When this condition holds, the t is called the induced t-structure. The truncation and cohomology functors for the induced t is t.

To construct the category of perverse sheaves, it is important to be able to define a t-structure on a category of sheaves over a space by working locally in that space. The precise conditions necessary for this to be possible can be abstracted somewhat to the following setup. Suppose that there are three triangulated categories and two morphisms

satisfying the following properties.

• .

• .

• There are unique differentials making, for every K, exact triangles

In this case, given t, respectively, there is a t defined by

This t-structure is said to be the gluing of the t-structures on U and F are bounded below derived categories of sheaves on a space X, an open subset U, and the closed complement F of U on X and when the sheaves are ℓ-adic sheaves.

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

can be shown to be a t-structure. The resulting t-structure is said to be .

and a t.

On stable ∞-categories

The elementary theory of t-structures carries over to the case of be a stable -category. A t-structure is defined to be a t (which is a triangulated category). A t-structure on an -category can be notated either homologically or cohomologically, just as in the case of a triangulated category.

is an is a t. Then, for each integer n, respectively. Define

to be the inclusion functors. Just as in the case of a triangulated category, these admit a right and a left adjoint, respectively, the truncation functors

These functors satisfy the same repeated truncation identities as in the triangulated category case.

The heart of a t is defined to be the .

is, by definition, a localization functor. In fact, there is a bijection between t and certain kinds of localization functors called t-localizations. These are localization functors L is a fiber sequence with X and Z in the essential image of L, then Y is also in the essential image of L. Given such a localization functor L, the corresponding t-structure is defined by

t-localization functors can also be characterized in terms of the morphisms f for which Lf is an equivalence. A set of morphisms S in an is quasisaturated with two of its non-degenerate edges in S has its third non-degenerate edge in S is a localization functor, then the set S of all morphisms f for which Lf is an equivalence is quasisaturated. Then L is a t-localization functor if and only if S.

The derived category of an abelian category has several subcategories corresponding to different boundedness conditions. A t-structure on a stable -category can be used to construct similar subcategories. Specifically,

is left bounded (with respect to the given t, right bounded, and bounded.

It is also possible to form a left or right completion with respect to a t-structure. This is analogous to formally adjoining directed limits or directed colimits. The left completion is the homotopy limit of the diagram

The right completion is defined dually. The left and right completions are themselves stable -categories which inherit a canonical t to either of its completions, and this map is t is left complete or right complete if the canonical map to its left or right completion, respectively, is an equivalence.

Related concepts

is replaced by the opposite inclusion

and the other two axioms kept the same, the resulting notion is called a co-t-structure or weight structure.

References