# 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 $C$ is a full subcategory $N\subset C$ such that

• $N$ is saturated: every object $X$ in $C$ which is isomorphic in $C$ to an object in $N$ is in $N$;

• the zero object is in $N$;

• $X$ is in $N$ precisely if $TX$ is in $N$;

• if $X\to Y\to Z\to TX$ is a distinguished triangle in $C$ with $X,Z\in N$, then also $Y\in N$.

# Properties

The point about null systems is the following:

for $N$ a null system, let $NQ$ be the collection of all morphisms in $C$ whose “mapping cone” is in $N$, precisely: set

$NQ:=\left\{X\stackrel{f}{\to }Y\mid \exists \mathrm{dist}.\mathrm{tri}.X\to Y\to Z\mathrm{in}C\mathrm{with}Z\in N\right\}\phantom{\rule{thinmathspace}{0ex}}.$N Q := \{ X \stackrel{f}{\to} Y | \exists dist. tri. X \to Y \to Z in C with Z \in N\} \,.

Then $NQ$ admits a left and right calculus of fractions in $C$.

# Examples

• For $K\left(C\right)$ the category of chain complexes modulo chain homotopy of an abelian category, the full subcategory of $K\left(C\right)$ of chain complexes $V$ whose homology vanishes, $H\left(V\right)\simeq 0$ is a null system. Then$D\left(C\right):=K\left(C\right)/N$ is the derived category of $C$.

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)