nLab Gabriel multiplication



Category theory

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




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.

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


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.

category: algebra

Last revised on October 7, 2022 at 16:04:53. See the history of this page for a list of all contributions to it.