Editing Flat module (section)

=== Flat morphisms of schemes ===
The definition of a [[flat morphism]] of [[scheme (mathematics)|schemes]] results immediately from the local property of flatness.

A morphism <math>f: X \to Y</math> of [[scheme (mathematics)|schemes]] is a [[flat morphism]] if the induced map on local rings
: <math>\mathcal O_{Y, f(x)} \to \mathcal O_{X,x}</math>
is a flat ring homomorphism for any point {{mvar|''x''}} in {{mvar|''X''}}. 

Thus, properties of flat (or faithfully flat) ring homomorphisms extends naturally to geometric properties of flat morphisms in algebraic geometry.
 
For example, consider the flat <math>\mathbb{C}[t]</math>-algebra <math>R = \mathbb{C}[t,x,y]/(xy-t)</math> (see below). The inclusion <math>\mathbb{C}[t] \hookrightarrow R</math> induces the flat morphism
: <math>\pi : \operatorname{Spec}(R) \to \operatorname{Spec}(\mathbb C[t]).</math>
Each (geometric) fiber <math>\pi^{-1}(t)</math> is the curve of equation <math>xy = t.</math> (See also [[flat degeneration]] and [[deformation to normal cone]].)

Let <math>S = R[x_1, \dots, x_r]</math> be a polynomial ring over a commutative Noetherian ring <math>R</math> and <math>f \in S</math> a nonzerodivisor. Then <math>S/fS</math> is flat over <math>R</math> if and only if <math>f</math> is [[primitive polynomial (ring theory)|primitive]] (the coefficients generate the unit ideal).{{sfn|Eisenbud|1995|loc=Exercise 6.4|ps=none}} An example is{{sfn|Artin|p=3|ps=none}} <math>\mathbb{C}[t,x,y]/(xy-t),</math> which is flat (and even free) over <math>\mathbb{C}[t]</math> (see also below for the geometric meaning). Such flat extensions can be used to yield examples of flat modules that are not free and do not result from a localization.