nLab
Gabriel multiplication

Context

Category theory

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

For any two full subcategories SS and TT of an abelian category AA, define their Gabriel product STS\bullet T as the full subcategory of AA generated by all objects MM such that MM fits in a short exact sequence of the form

0NMP0 0\to N\to M\to P\to 0

where NN is an object in SS and PP is an object in TT.

In the case of the abelian category of modules over a ring, the Gabriel multiplication is sometimes expressed as Gabriel composition of filters of ideals, rather than in terms of abelian subcategories.

Properties

When restricted to the class of topologizing subcategories, Gabriel multiplication is associative; if AA is small then the topologizing subcategories make a semiring with respect to the commutative operation \cap and Gabriel multiplication; in particular \bullet is left and right distributive with respect to intersection of topologizing subcategories.

Revised on May 5, 2011 12:57:52 by Urs Schreiber (89.204.137.116)