t-categories

There is a rather useful notion in the theory of triangulated categories which was introduced by Beilinson, Bernstein, Deligne (1982).

A t-category is a triangulated category D endowed with two full subcategories D≤o and D≥o; which are closed under isomorphisms and such that for D≥o=n(D<Rsup>≥o) and Do=n(D≤o) the following three conditions are satisfied:

1. For X∈D≤o and Y∈D≤1 we have that Hom(X,Y) = 0.

2. D≤o⊃D≤1 and D≥1⊃D≤o.

3. For X ∈ V there is a triangle B'→X→Brdquo; →TB' such that B' ∈ D≤o and B” ∈ D≥1.

Under these conditions, we say that the pair (D≤o,D≥o) is a t-structure on D. Denote by H the full subcategory (D≤o∩D≥oH is called the heart of the t-structure. The following fact (which we will not use) is shown in Beilinson, Bernstein, Deligne (1982): the heart H of a t-structure is an abelian category.

We include an example. Let A be an abelian category and Db(A) be the derived category of A. Let D≤o(resp. D≥o) be the full subcategory of D (A) formed by the objects X· such that Hi(X·) = 0 for i>o (i<o respectively). The properties (1) and (2) are easily verified. For the third property we denote by τ≤ox· the subcomplex of X·=(Xi,diX) with (τ≤ox·)i= Xi for i<o, (τ≤ox·)o= ker doX and zero otherwise. Set τ≥1X·= X·/τ≥oX· Clearly τ≤oX·∈D≤o and τ≥1X·∈D≥1.The short exact sequence of complexes 0→τ≤oX·→X·→τ≥1X·→0 yields the required triangle.

A second example will be given in the next section.