nLab
null system

Idea

A null system in a triangulated category is a triangulated subcategory whose objects may consistently be regarded as being equivalent to the zero object. Null systems give a convenient means for encoding and computing localization of triangulated categories.

Definition

A null system of a triangulated category CC is a full subcategory NCN \subset C such that

  • NN is saturated: every object XX in CC which is isomorphic in CC to an object in NN is in NN;

  • the zero object is in NN;

  • XX is in NN precisely if TXT X is in NN;

  • if XYZTXX \to Y \to Z \to T X is a distinguished triangle in CC with X,ZNX, Z \in N, then also YNY \in N.

Properties

The point about null systems is the following:

for NN a null system, let NQN Q be the collection of all morphisms in CC whose “mapping cone” is in NN, precisely: set

NQ:={XfY|dist.tri.XYZinCwithZN}. N Q := \{ X \stackrel{f}{\to} Y | \exists dist. tri. X \to Y \to Z in C with Z \in N\} \,.

Then NQN Q admits a left and right calculus of fractions in CC.

Examples

David Roberts: Would Serre class?es fit in here? Perhaps that’s one step back.

References

For instance section 10.2 of

Revised on April 24, 2009 21:37:53 by Toby Bartels (71.104.234.95)