nLab
residue field

This entry is about the notion of a residue field in algebraic geometry. There is another (related) notion of a residue field in constructive mathematics; see under field.

Given a local ring $R$, by the definition there is a maximal ideal $𝔪\subset R$. The quotient $R/𝔪$ is therefore a division ring, and in commutative case, therefore a field, called the residue field. In algebraic geometry, the residue field at a point $x$ of a scheme $X$ is the residue field of the corresponding stalk ${𝒪}_{X,x}$ of the structure sheaf, which is by the definition a local ring.