Editing Flat module (section)

== Local property==
In this section, {{mvar|R}} denotes a [[commutative ring]]. If <math>\mathfrak p</math> is a [[prime ideal]] of {{mvar|R}}, the [[localization (commutative algebra)#Localization at primes|localization]] at <math>\mathfrak p</math> is, as usual, denoted with <math>\mathfrak p</math> as an index. That is, <math>R_{\mathfrak p} = (R\setminus \mathfrak p)^{-1}R,</math> and, if {{mvar|M}} is an {{mvar|R}}-module, <math>M_{\mathfrak p} = (R\setminus \mathfrak p)^{-1}M = R_{\mathfrak p}\otimes_R M.</math>

If {{mvar|M}} is an {{mvar|R}}-module the three following conditions are equivalent:
*<math>M</math> is a flat <math>R</math>-module;
*<math>M_\mathfrak p</math> is a flat <math>R_\mathfrak p</math>-module for every prime ideal <math>\mathfrak p;</math>
*<math>M_\mathfrak m</math> is a flat <math>R_\mathfrak m</math>-module for every [[maximal ideal]] <math>\mathfrak m.</math>

This property is fundamental in commutative algebra and algebraic geometry, since it reduces the study of  flatness to the case of [[local ring]]s. They are often expressed by saying that flatness is a [[local property]].<!-- 
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If ''R'' is a local (commutative) ring and either ''M'' is finitely generated or the maximal ideal of ''R'' is nilpotent (e.g., an [[artinian local ring]]) then the standard implication "free implies flat" can be reversed: in this case ''M'' is flat if and if only if its free.{{sfn|Matsumura|loc=Prop. 3.G|ps=none}}

The '''[[local criterion for flatness]]''' states:{{sfn|Eisenbud|1994|loc=Theorem 6.8|ps=none}}
: Let ''R'' be a local noetherian ring, ''S'' a local noetherian ''R''-algebra with <math>\mathfrak{m}_R S \subset \mathfrak{m}_S</math>, and ''M'' a finitely generated ''S''-module. Then ''M'' is flat over ''R'' if and only if <math>\operatorname{Tor}_1^R(M, R/\mathfrak{m}_R) = 0.</math>

The significance of this is that ''S'' need not be finite over ''R'' and we only need to consider the maximal ideal of ''R'' instead of an arbitrary ideal of ''R''.

The next criterion is also useful for testing flatness:{{sfn|Eisenbud|1994|loc=Theorem 18.16|ps=none}}
: Let ''R'', ''S'' be as in the local criterion for flatness. Assume ''S'' is [[Cohen–Macaulay ring|Cohen–Macaulay]] and ''R'' is [[regular local ring|regular]]. Then ''S'' is flat over ''R'' if and only if <math>\dim S = \dim R + \dim S/\mathfrak{m}_R S.</math>
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=== 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.