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. It allows one to construct an abelian category, namely the heart of the t-structure, from a triangulated category.
Contents
Definition
The derived category D of an abelian category A contains, for each n, the full subcategories
This prototypical basic example gives rise to the following definition: a t-structure on a triangulated category consists of full subcategories
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).
Constructing t-structures
Many t-structures arise by means of the following fact: in a triangulated category with arbitrary direct sums, and a set
can be shown to be a t-structure. It is called the t-structure generated by
Consequences of the definition
The core or heart (the original French word is "cœur") of a t-structure is the category
The two subcategories
The objects
The n-th cohomology functor
It is in fact a cohomological functor: for any triangle
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
where
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
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
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.
Related concepts
If the requirement
(and the other two axioms kept the same), the resulting notion is called a co-t-structure or weight structure.