Editing Flat module (section)

== Faithful flatness ==
A module is ''faithfully flat'' if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Although the concept is defined for modules over a non-necessary commutative ring, it is used mainly for [[commutative algebra (structure)|commutative algebras]]. So, this is the only case that is considered here, even if some results can be generalized to the case of modules over a non-commutaive ring.

In this section, <math>f\colon R \to S</math> is a [[ring homomorphism]] of commutative rings, which gives to <math>S</math> the structures of an <math>R</math>-algebra and an <math>R</math>-module. If <math>S</math> is a <math>R</math>-module flat (or faithfully flat), one says commonly that <math>S</math> is flat (or faithfully flat) over <math>R, </math> and that <math>f</math> is flat (or faithfully flat).

If <math>S</math> is flat over <math>R,</math> the following conditions are equivalent.
* <math>S</math> is faithfully flat.
* For each maximal ideal <math>\mathfrak{m}</math> of <math>R</math>, one has <math>\mathfrak{m}S \ne S.</math>
* If <math>M</math> is a nonzero <math>R</math>-module, then <math>M \otimes_R S \ne 0.</math>
* For every prime ideal <math>\mathfrak{p}</math> of <math>R,</math> there is a prime ideal <math>\mathfrak{P}</math> of <math>S</math> such that <math>\mathfrak{p} = f^{-1}(\mathfrak P).</math> In other words, the map <math>f^*\colon \operatorname{Spec}(S) \to \operatorname{Spec}(R)</math> induced by <math>f</math> on the [[spectrum of a ring|spectra]] is surjective.
* <math>f,</math> is injective, and <math>R</math> is a [[pure subring]] of <math>S;</math> that is, <math>M \to M \otimes_R S</math> is injective for every <math>R</math>-module <math>M</math>.{{efn|Proof: Suppose <math>f: R \to S</math> is faithfully flat. For an {{mvar|R}}-module <math>M,</math> the map <math>S = R \otimes_R S \to S \otimes_R S</math> exhibits <math>S</math> as a pure subring and so <math>M \otimes_R S \to M \otimes_R (S \otimes S) \simeq (M \otimes_R S) \otimes_R S</math> is injective. Hence, <math>M \to M \otimes_R S</math> is injective. Conversely, if <math>M \ne 0</math> is a module over <math>R</math>, then <math>0 \ne M \subset M \otimes_R S.</math>}}

The second condition implies that a flat local homomorphism of [[local ring]]s is faithfully flat. It follows from the last condition that <math>I = I S \cap R</math> for every ideal <math>I</math> of <math>R</math> (take <math>M = R/I</math>). In particular, if <math>S</math> is a Noetherian ring, then <math>R</math> is also Noetherian.

The last but one condition can be stated in the following strengthened form: <math>\operatorname{Spec}(S) \to \operatorname{Spec}(R)</math> is ''submersive'', which means that the [[Zariski topology]] of <math>\operatorname{Spec}(R)</math> is the [[quotient topology]] of  that of <math>\operatorname{Spec}(S)</math> (this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property.{{sfn|SGA I|loc=Exposé VIII., Corollay 4.3|ps=none}}). See also ''{{slink|Flat morphism#Properties of flat morphisms}}''.

=== Examples ===
* A ring homomorphism <math>R\to S</math> such that <math>S</math> is a nonzero free {{mvar|R}}-module is faithfully flat. For example:
** Every [[field extension]] is faithfully flat. This property is implicitly behind the use of [[complexification]] for proving results on real vector spaces.
** A [[polynomial ring]] is a faithfully flat extension of its ring of coefficients.
** If <math>p\in R[x]</math> is a [[monic polynomial]], the inclusion <math>R \hookrightarrow R[t]/\langle p \rangle</math> is faithfully flat.
* Let <math>t_1, \ldots, t_k\in R.</math> The [[direct product]] <math>\textstyle\prod_i R[t_i^{-1}]</math> of the [[localization (commutative algebra)|localizations]] at the <math>t_i</math> is faithfully flat over <math>R</math> if and only if <math>t_1, \ldots, t_k</math> generate the [[unit ideal]] of <math>R</math> (that is, if <math>1</math> is a [[linear combination]] of the <math>t_i</math>).{{sfn|Artin|1999|loc=Exercise (3) after Proposition III.5.2|ps=none}}
* The [[direct sum]] of the localizations <math>R_\mathfrak p</math> of <math>R</math> at all its prime ideals is a faithfully flat module that is not an algebra, except if there are finitely many prime ideals.

The two last examples are implicitly behind the wide use of localization in commutative algebra and algebraic geometry.

* For a given ring homomorphism <math>f: A \to B,</math> there is an associated complex called the [[Amitsur complex]]:<ref>{{cite web |url=https://ncatlab.org/nlab/show/Amitsur+complex |title=Amitsur Complex |website=ncatlab.org}}</ref>
<math display="block">0 \to A \overset{f}\to B \overset{\delta^0}\to B \otimes_A B \overset{\delta^1}\to B \otimes_A B \otimes_A B \to \cdots</math>
where the coboundary operators <math>\delta^n</math> are the alternating sums of the maps obtained by inserting 1 in each spot; e.g., <math>\delta^0(b) = b \otimes 1-1 \otimes b</math>. Then (Grothendieck) this complex is exact if <math>f</math> is faithfully flat.

=== Faithfully flat local homomorphisms ===
Here is one characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism <math>(R, \mathfrak m) \hookrightarrow (S, \mathfrak n)</math> such that <math>\mathfrak{m} S</math> is an <math>\mathfrak{n}</math>-[[primary ideal]], the homomorphism <math>S \to B</math> is faithfully flat if and only if the [[theorem of transition]] holds for it; that is, for each <math>\mathfrak m</math>-primary ideal <math>\mathfrak q</math> of <math>R</math>, <math>\operatorname{length}_S (S/ \mathfrak q S) = \operatorname{length}_S (S/ \mathfrak{m} S) \operatorname{length}_R(R/\mathfrak q).</math>{{sfn|Matsumura|1986|loc=Ch. 8, Exercise 22.1|ps=none}}