Grothendieck category

In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957^[1] 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 Pierre Gabriel's 1962 thesis.^[2]

To every algebraic variety ${\displaystyle V}$ one can associate a Grothendieck category ${\displaystyle \operatorname {Qcoh} (V)}$, consisting of the quasi-coherent sheaves on ${\displaystyle V}$. This category encodes all the relevant geometric information about ${\displaystyle V}$, and ${\displaystyle V}$ can be recovered from ${\displaystyle \operatorname {Qcoh} (V)}$ (the Gabriel–Rosenberg reconstruction theorem). This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of (certain) Grothendieck categories.^[3]