Grothendieck category [SEP] discoverer or inventor
Grothendieck category
In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Peter Gabriel's seminal thèse in 1962.
To every algebraic variety formula_1 one can associate a Grothendieck category formula_2, consisting of the quasi-coherent sheaves on formula_1. This category encodes all the relevant geometric information about formula_1, and formula_1 can
space formula_17, the category of all sheaves of abelian groups on formula_17 is a Grothendieck category. (More generally: the category of all sheaves of right formula_24-modules on formula_17 is a Grothendieck category for any ring formula_24.)
- Given a ringed space formula_34, the category of sheaves of "O"-modules is a Grothendieck category.
- Given an (affine or projective) algebraic variety formula_1 (or more generally: a quasi-compact quasi-separated scheme), the category formula_2 of quasi-coherent sheaves on formula_1 is