Residue field

Suppose that ${\displaystyle R}$  is a commutative local ring, with maximal ideal ${\displaystyle {\mathfrak {m}}}$ . Then the residue field is the quotient ring ${\displaystyle R/{\mathfrak {m}}}$ .

Now suppose that ${\displaystyle X}$  is a scheme and ${\displaystyle x}$  is a point of ${\displaystyle X}$ . By the definition of scheme, we may find an affine neighbourhood ${\displaystyle {\mathcal {U}}={\text{Spec}}(A)}$  of ${\displaystyle x}$ , with some commutative ring ${\displaystyle A}$ . Considered in the neighbourhood ${\displaystyle {\mathcal {U}}}$ , the point ${\displaystyle x}$  corresponds to a prime ideal ${\displaystyle {\mathfrak {p}}\subseteq A}$  (see Zariski topology). The local ring of ${\displaystyle X}$  at ${\displaystyle x}$  is by definition the localization ${\displaystyle A_{\mathfrak {p}}}$  of ${\displaystyle A}$  by ${\displaystyle A\setminus {\mathfrak {p}}}$ , and ${\displaystyle A_{\mathfrak {p}}}$  has maximal ideal ${\displaystyle {\mathfrak {m}}}$  = ${\displaystyle {\mathfrak {p}}A_{\mathfrak {p}}}$ . Applying the construction above, we obtain the residue field of the point ${\displaystyle x}$  :

${\displaystyle k(x):=A_{\mathfrak {p}}/{\mathfrak {p}}A_{\mathfrak {p}}}$ .

One can prove that this definition does not depend on the choice of the affine neighbourhood ${\displaystyle {\mathcal {U}}}$  .^[3]

A point is called ${\displaystyle \color {blue}k}$ -rational for a certain field ${\displaystyle k}$ , if ${\displaystyle k(x)=k}$ .^[4]

Consider the affine line ${\displaystyle \mathbb {A} ^{1}(k)={\text{Spec}}(k[t])}$  over a field ${\displaystyle k}$ . If ${\displaystyle k}$  is algebraically closed, there are exactly two types of prime ideals, namely

• ${\displaystyle (t-a),\,a\in k}$ 
• ${\displaystyle (0)}$  , the zero-ideal.

The residue fields are

• ${\displaystyle k[t]_{(t-a)}/(t-a)k[t]_{(t-a)}\cong k}$ 
• ${\displaystyle k[t]_{(0)}\cong k(t)}$ , the function field over k in one variable.

If ${\displaystyle k}$  is not algebraically closed, then more types arise, for example if ${\displaystyle k=\mathbb {R} }$ , then the prime ideal ${\displaystyle (x^{2}+1)}$  has residue field isomorphic to ${\displaystyle \mathbb {C} }$ .

• For a scheme locally of finite type over a field ${\displaystyle k}$ , a point ${\displaystyle x}$  is closed if and only if ${\displaystyle k(x)}$  is a finite extension of the base field ${\displaystyle k}$ . This is a geometric formulation of Hilbert's Nullstellensatz. In the above example, the points of the first kind are closed, having residue field ${\displaystyle k}$ , whereas the second point is the generic point, having transcendence degree 1 over ${\displaystyle k}$ .
• A morphism ${\displaystyle {\text{Spec}}(K)\rightarrow X}$  , ${\displaystyle K}$  some field, is equivalent to giving a point ${\displaystyle x\in X}$  and an extension ${\displaystyle K/k(x)}$ .
• The dimension of a scheme of finite type over a field is equal to the transcendence degree of the residue field of the generic point.

• Arithmetic zeta function

• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, section II.2