nLab Gabriel multiplication

Definition

For any two full subcategories $S$ and $T$ of an abelian category $A$, define their Gabriel product $S\bullet T$ as the full subcategory of $A$ generated by all objects $M$ such that $M$ fits in a short exact sequence of the form

$$0\to N\to M\to P\to 0$$

where $N$ is an object in $S$ and $P$ is an object in $T$.

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.

An analogous notion in the triangulated setup is the Verdier product.

Properties

When restricted to the class of topologizing subcategories, Gabriel multiplication is associative; if $A$ 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.

category: algebra