T-structure

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 formula_1 consists of two subcategories formula_2 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.
Fix a triangulated category formula_1 with translation functor formula_4. A "t"-structure on formula_1 is a pair formula_2 of full subcategories, each of which is stable under isomorphism, which satisfy the following three axioms.
It can be shown that the subcategories formula_7 and formula_8 are closed under extensions in formula_1. In particular, they are stable under finite direct sums.
Suppose that formula_2 is a "t"-structure on formula_1. In this case, for any integer "n", we define formula_23 to be the full subcategory of formula_1 whose objects have the form formula_25, where formula_26 is an object of formula_7. Similarly, formula_28 is the full subcategory of objects formula_29, where formula_30 is an object of formula_8. More briefly, we define
With this notation, the axioms above may be rewritten as:
The heart or core of the "t"-structure is the full subcategory formula_42 consisting of objects in both formula_7 and formula_8, 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 formula_46 and formula_28. Some authors define a "t"-structure to be the pair formula_48.
The two subcategories formula_7 and formula_34 determine each other. An object "X" is in formula_7 if and only if formula_52 for all objects "Y" in formula_34, and vice versa. That is, formula_48 are left and right orthogonal complements of each other. Consequently, it is enough to specify only one of formula_7 and formula_34. 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 formula_1 is a pair formula_58 such that, if we define
then formula_2 is a (cohomological) "t"-structure on formula_1. That is, the definition is the same except that upper indices are converted to lower indices and the roles of formula_62 and formula_63 are swapped. If we define
then the axioms for a homological "t"-structure may be written explicitly as
Examples.
The natural "t"-structure.
The most fundamental example of a "t"-structure is the natural "t"-structure on a derived category. Let formula_74 be an abelian category, and let formula_75 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 formula_78 is a cochain complex with values in formula_74. Define
It is clear that formula_81 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"-structures for the bounded, bounded above, and bounded below derived categories. If formula_83 is an abelian subcategory of formula_74, then the full subcategory formula_85 of formula_75 consisting of those complexes whose cohomology is in formula_83 has a similar "t"-structure whose heart is formula_83.
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 formula_90 is the inclusion of a point, formula_91 is an ordinary sheaf, formula_92 is a smooth open subscheme and formula_93 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. It was shown by Beilinson that the derived category of the heart formula_94 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
where formula_96 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. The category formula_97 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 "formula_63" by "formula_62". Using this convention, the category of connective spectra the notation is denoted formula_100.)
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"-structure on an abelian category, the distinguished triangle guaranteed by the third axiom was constructed by truncation. As operations on the category of complexes, the truncations formula_101 and formula_102 are functorial, and the resulting short exact sequence of complexes is natural in formula_78. 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 formula_1 is a triangulated category and that formula_2 is a "t"-structure. The precise statement is that the inclusion functors
admit adjoints. These are functors
such that
Moreover, for any object formula_109 of formula_1, 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 formula_39 with formula_26 and formula_30 objects of formula_7 and formula_34, respectively. It follows from the existence of this triangle that an object formula_109 lies in formula_23 (resp. formula_28) if and only if formula_121 (resp. formula_122).
The existence of formula_123 implies the existence of the other truncation functors by shifting and taking opposite categories. If formula_109 is an object of formula_1, the third axiom for a "t"-structure asserts the existence of an formula_26 in formula_7 and a morphism formula_128 fitting into a certain distinguished triangle. For each formula_109, fix one such triangle and define formula_130. The axioms for a "t"-structure imply that, for any object formula_131 of formula_7, we have
with the isomorphism being induced by the morphism formula_128. This exhibits formula_26 as a solution to a certain universal mapping problem. Standard results on adjoint functors now imply that formula_26 is unique up to unique isomorphism and that there is a unique way to define formula_123 on morphisms that makes it a right adjoint. This proves the existence of formula_123 and hence the existence of all the truncation functors.
Repeated truncation for a "t"-structure behaves similarly to repeated truncation for complexes. If formula_139, then there are natural transformations
which yield natural equivalences
Cohomology functors.
The "n"th "cohomology functor" formula_142 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 formula_144, we obtain a long exact sequence
In applications to algebraic topology, the cohomology functors may be denoted formula_146 instead of formula_142. The cohomology functors take values in the heart formula_42. 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 formula_75, the cohomology functor formula_142 is, up to quasi-isomorphism, the usual "n"th cohomology group of a complex. However, considered as functors on complexes, this is "not" true. Consider, for example, formula_152 as defined in terms of the natural "t"-structure. By definition, this is
This complex is non-zero in degrees formula_154 and formula_155, so it is clearly not the same as the zeroth cohomology group of the complex formula_78. However, the non-trivial differential is an injection, so the only non-trivial cohomology is in degree formula_155, where it is formula_158, the zeroth cohomology group of the complex formula_78. It follows that the two possible definitions of formula_160 are quasi-isomorphic.
A "t"-structure is non-degenerate if the intersection of all formula_23, as well as the intersection of all formula_28, consists only of zero objects. For a non-degenerate "t"-structure, the collection of functors formula_163 is conservative. Moreover, in this case, formula_23 (resp. formula_28) may be identified with the full subcategory of those objects formula_109 for which formula_167 for formula_168 (resp. formula_169).
Exact functors.
For formula_170, let formula_171 be a triangulated category with a fixed "t"-structure formula_172 Suppose that formula_173 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 formula_174 is:
It is elementary to see that if formula_174 is fully faithful and "t"-exact, then an object formula_109 of formula_179 is in formula_180 (resp. formula_181) if and only if formula_182 is in formula_183 (resp. formula_184). It is also elementary to see that if formula_185 is another left (resp. right) "t"-exact functor, then the composite formula_186 is also left (resp. right) "t"-exact.
The motivation for the study of one-sided "t"-exactness properties is that they lead to one-sided exactness properties on hearts. Let formula_187 be the inclusion. Then there is a composite functor
It can be shown that if formula_174 is left (resp. right) exact, then formula_190 is also left (resp. right) exact, and that if formula_191 is also left (resp. right) exact, then formula_192.
If formula_174 is left (resp. right) "t"-exact, and if formula_109 is in formula_7 (resp. formula_8), then there is a natural isomorphism formula_197 (resp. formula_198).
If formula_199 are exact functors with formula_200 left adjoint to formula_201, then formula_200 is right "t"-exact if and only if formula_201 is left "t"-exact, and in this case, formula_204 are a pair of adjoint functors formula_205.
Constructions of "t"-structures.
Let formula_2 be a "t"-structure on formula_1. If "n" is an integer, then the translation by "n" "t"-structure is formula_208. The dual "t"-structure is the "t"-structure on the opposite category formula_209 defined by formula_210.
Let formula_211 be a triangulated subcategory of a triangulated category formula_1. If formula_2 is a "t"-structure on formula_1, then
is a "t"-structure on formula_211 if and only if formula_211 is stable under the truncation functor formula_123. When this condition holds, the "t"-structure formula_219 is called the induced "t"-structure. The truncation and cohomology functors for the induced "t"-structure are the restriction to formula_211 of those on formula_1. Consequently, the inclusion of formula_211 in formula_1 is "t"-exact, and formula_224.
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.
In this case, given "t"-structures formula_234 and formula_235 on formula_236 and formula_237, respectively, there is a "t"-structure on formula_1 defined by
This "t"-structure is said to be the gluing of the "t"-structures on "U" and "F". The intended use cases are when formula_1, formula_236, and formula_237 are bounded below derived categories of sheaves on a space "X", an open subset "U", and the closed complement "F" of "U". The functors formula_230 and formula_228 are the usual pullback and pushforward functors. This works, in particular, when the sheaves in question are left modules over a sheaf of rings formula_245 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, and a set formula_246 of compact objects in formula_1, the subcategories
can be shown to be a t-structure. The resulting "t"-structure is said to be generated by formula_246.
Given an abelian subcategory formula_83 of a triangulated category formula_1, it is possible to construct a subcategory of formula_1 and a "t"-structure on that subcategory whose heart is formula_83.
On stable ∞-categories.
The elementary theory of "t"-structures carries over to the case of ∞-categories with few changes. Let formula_1 be a stable ∞-category. A "t"-structure on formula_1 is defined to be a "t"-structure on its homotopy category formula_256 (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.
Suppose that formula_1 is an ∞-category with homotopy category formula_256 and that formula_259 is a "t"-structure on formula_256. Then, for each integer "n", we define formula_261 and formula_262 to be the full subcategories of formula_1 spanned by the objects in formula_264 and formula_265, 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"-structure on formula_1 is defined to be the ∞-subcategory formula_269. The category formula_42 is equivalent to the nerve of its homotopy category formula_271. The cohomology functor formula_146 is defined to be formula_273, or equivalently formula_274.
The existence of formula_275 means that formula_276 is, by definition, a localization functor. In fact, there is a bijection between "t"-structures on formula_1 and certain kinds of localization functors called "t"-localizations. These are localization functors "L" whose essential image is closed under extension, meaning that if formula_278 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 ∞-category formula_1 is quasisaturated if it contains all equivalences, if any 2-simplex in formula_1 with two of its non-degenerate edges in "S" has its third non-degenerate edge in "S", and if it is stable under pushouts. If formula_282 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" is the smallest quasisaturated set of morphisms containing all morphisms formula_283.
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,
These are stable subcategories of formula_1. One says that formula_1 is left bounded (with respect to the given "t"-structure) if formula_287, right bounded if formula_288, and bounded if formula_289.
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 formula_290 of formula_1 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"-structure. There is a canonical map from formula_1 to either of its completions, and this map is "t"-exact. We say that formula_1 is left complete or right complete if the canonical map to its left or right completion, respectively, is an equivalence.
Related concepts.
If the requirement formula_295, formula_296 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".