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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 RR, by the definition there is a maximal ideal 𝔪R\mathfrak{m}\subset R. The quotient R/𝔪R/\mathfrak{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 xx of a scheme XX is the residue field of the corresponding stalk 𝒪 X,x\mathcal{O}_{X,x} of the structure sheaf, which is by the definition a local ring.

Revised on September 1, 2012 20:56:35 by Toby Bartels (98.23.143.147)