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-structure on consists of two subcategories 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.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.
Definition Fix a triangulated category with translation functor . A t-structure on is a pair of full subcategories, each of which is stable under isomorphism, which satisfy the following three axioms. If X is an object of and Y is an object of , then If X is an object of , then X is also an object of . Similarly, if Y is an object of , then Y[-1] is also an object of . If A is an object of , then there exists a distinguished triangle such that X is an object of and Y is an object of . It can be shown that the subcategories and are closed under extensions in . In particular, they are stable under finite direct sums.
Suppose that is a t-structure on . In this case, for any integer n, we define to be the full subcategory of whose objects have the form , where is an object of . Similarly, is the full subcategory of objects , where is an object of . More briefly, we define
With this notation, the axioms above may be rewritten as: If X is an object of and Y is an object of , then and . If A is an object of , then there exists a distinguished triangle such that X is an object of and Y is an object of .
The heart or core of the t-structure is the full subcategory consisting of objects contained in both and , 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 and specify and . Some authors define a t-structure to be the pair .
The two subcategories and determine each other. An object X is in if and only if for all objects Y in , and vice versa. That is, are left and right orthogonal complements of each other. Consequently, it is enough to specify only one of and . 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 on is a pair such that, if we define
then is a (cohomological) t-structure on . That is, the definition is the same except that upper indices are converted to lower indices and the roles of and are swapped. If we define
then the axioms for a homological t-structure may be written explicitly as If X is an object of and Y is an object of , then and . If A is an object of , then there exists a distinguished triangle such that X is an object of and Y is an object of .
Examples The natural t-structure The most fundamental example of a t-structure is the natural t-structure on a derived category. Let be an abelian category, and let 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-structure, the existence of a certain distinguished triangle, can be made explicit as follows. Suppose that is a cochain complex with values in . Define
It is clear that 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).Beilinson, Bernstein, and Deligne, 1.3.22. There are also similar t-structures for the bounded, bounded above, and bounded below derived categories. If is an abelian subcategory of , then the full subcategory of consisting of those complexes whose cohomology is in has a similar t-structure whose heart is .Beilinson, Bernstein, and Deligne, p. 13.
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
where is the inclusion of a point, is an ordinary sheaf, is a smooth open subscheme and 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.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. It was shown by Beilinson that the derived category of the heart 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.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.
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
where is a module generated by its (graded) degree n. This t-structure called geometric t-structure plays a prominent role in Koszul duality.Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang. Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9 (1996), no. 2, 473–527.
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 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 "" (superscript) by "" (subscript). Using this convention, the category of connective spectra is denoted as .)
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.Hanamura, Masaki. Mixed motives and algebraic cycles. III. Math. Res. Lett. 6 (1999), no. 1, 61–82.
Truncation functors In the above example of the natural t-structure on the derived category of an abelian category, the distinguished triangle guaranteed by the third axiom was constructed by truncation. As operations on the category of complexes, the truncations and are functorial, and the resulting short exact sequence of complexes is natural in . 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-structure do not assume the existence of truncation functors, such functors can always be constructed and are essentially unique. Suppose that is a triangulated category and that is a t-structure. The precise statement is that the inclusion functors
admit adjoints. These are functors
such that
Moreover, for any object of , there exists a unique
such that d and the counit and unit of the adjunctions together define a distinguished triangle
Up to unique isomorphism, this is the unique distinguished triangle of the form with and objects of and , respectively. It follows from the existence of this triangle that an object lies in (resp. ) if and only if (resp. ).
The existence of implies the existence of the other truncation functors by shifting and taking opposite categories. If is an object of , the third axiom for a t-structure asserts the existence of an in and a morphism fitting into a certain distinguished triangle. For each , fix one such triangle and define . The axioms for a t-structure imply that, for any object of , we have
with the isomorphism being induced by the morphism . This exhibits as a solution to a certain universal mapping problem. Standard results on adjoint functors now imply that is unique up to unique isomorphism and that there is a unique way to define on morphisms that makes it a right adjoint. This proves the existence of and hence the existence of all the truncation functors.
Repeated truncation for a t-structure behaves similarly to repeated truncation for complexes. If , then there are natural transformations
which yield natural equivalences
Cohomology functors The nth cohomology functor is defined as
As the name suggests, this is a cohomological functor in the usual sense for a triangulated category. That is, for any distinguished triangle , we obtain a long exact sequence
In applications to algebraic topology, the cohomology functors may be denoted instead of . The cohomology functors take values in the heart . By one of the repeated truncation identities above, up to natural equivalence it is equivalent to define
For the natural t-structure on a derived category , the cohomology functor is, up to quasi-isomorphism, the usual nth cohomology group of a complex. However, considered as functors on complexes, this is not true. Consider, for example, as defined in terms of the natural t-structure. By definition, this is
This complex is non-zero in degrees and , so it is clearly not the same as the zeroth cohomology group of the complex . However, the non-trivial differential is an injection, so the only non-trivial cohomology is in degree , where it is , the zeroth cohomology group of the complex . It follows that the two possible definitions of are quasi-isomorphic.
A t-structure is non-degenerate if the intersection of all , as well as the intersection of all , consists only of zero objects. For a non-degenerate t-structure, the collection of functors is conservative. Moreover, in this case, (resp. ) may be identified with the full subcategory of those objects for which for (resp. ).
Exact functors For , let be a triangulated category with a fixed t-structure . Suppose that is an exact functor (in the usual sense for triangulated categories, that is, up to a natural equivalence it commutes with translation and preserves distinguished triangles). Then is: Left t-exact if , Right t-exact if , and t-exact if it is both left and right t-exact.
It is elementary to see that if is fully faithful and t-exact, then an object of is in (resp. ) if and only if is in (resp. ). It is also elementary to see that if is another left (resp. right) t-exact functor, then the composite is also left (resp. right) t-exact.