I’ve been reading the Beilinson-Bernstein-Deligne (BBD) paper “Faisceaux pervers” as part of my summer project. To help myself understand it, I thought I would try to do a few blog posts about the material. The paper is ultimately about perverse sheaves, which are suitable complexes of sheaves on a (stratified) topological space satisfying certain cohomological conditions. The surprise is that perverse sheaves actually form an *abelian* category. The reason behind this is a bit of homological algebra, which I would like to discuss today to start.

**1. t-structures**

The first important notion in BBD is that of a -structure. If is an abelian category, and its derived category, we know that there is a notion of an object in *positive degrees* and one in *negative degrees*. There are two subcategories . Namely, a complex can be said to be in if it has no cohomology in negative degrees, and similarly in if it has no cohomology in positive degrees. This is equivalent to saying that is isomorphic to a complex all of whose nonzero terms are in nonnegative (resp. nonpositive) degrees because of the truncation functors .

A -structure is an axiomatization of this observation for the derived category.

Let be a triangulated category. A **-structure** on is the data of two full subcategories . We can define as and similarly , in accordance with what happens for the derived category.

The following axioms are required:

- is closed under translation .
- If , , then .
- If , there is a triangle
such that and .

The axioms for a -structure are simple, but they turn out to be quite powerful, because the axioms for a triangulated category are themselves quite powerful. (more…)