T-structure

In the branch of mathematics called homological algebra, a t-structure is a way to axiomatize the properties of an of a . A t-structure on consists of two subcategories of a or stable 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. 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[1] 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 (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. It is clear that, to define a t-structure, it suffices to fix integers m and n and specify and .