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\begin{document}

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\section*{flat module}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{InTermsOfExactFunctorsAndTorFunctors}{In terms of exact functors and Tor-functors}\dotfill \pageref*{InTermsOfExactFunctorsAndTorFunctors} \linebreak
\noindent\hyperlink{InTermsOfIdentities}{Explicitly in terms of identities}\dotfill \pageref*{InTermsOfIdentities} \linebreak
\noindent\hyperlink{for_more_general_rings}{For more general rings}\dotfill \pageref*{for_more_general_rings} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{EquivalentCharacterizations}{Equivalent characterizations}\dotfill \pageref*{EquivalentCharacterizations} \linebreak
\noindent\hyperlink{relation_to_projective_modules}{Relation to projective modules}\dotfill \pageref*{relation_to_projective_modules} \linebreak
\noindent\hyperlink{relation_to_locally_free_modules}{Relation to (locally) free modules}\dotfill \pageref*{relation_to_locally_free_modules} \linebreak
\noindent\hyperlink{Examles}{Examples}\dotfill \pageref*{Examles} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A [[module]] over a [[ring]] $R$ is called \emph{flat} if its satisfies one of many equivalent conditions, the simplest to state of which is maybe: forming the [[tensor product of modules]] with $N$ preserves [[submodules]].

Under the dual geometric interpretation of \href{module#RelationToVectorBundlesInIntroduction}{modules as generalized vector bundles} over the space on which $R$ is the ring of functions, flatness of a module is essentially the \emph{local triviality} of these bundles, hence in particular the fact that the [[fibers]] of these bundles do not change, up to isomorphism. See prop. \ref{ForFinitelyGeneratedFlatIsLocallyFree} below for the precise statement. On the other hand there is \textbf{no} relation to ``flat'' as in [[flat connection]] on such a bundle.

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

We first state the definition and its equivalent reformulations over a [[commutative ring]] abstractly in \emph{\hyperlink{InTermsOfExactFunctorsAndTorFunctors}{In terms of exact functors and Tor-functors}}. Then we give an explicit element-wise characterization in \emph{\hyperlink{InTermsOfIdentities}{Explicitly in terms of identities}}.

Much of this discussion also works in the more general case where the ring is not-necessarily taken to be commutative and not necessarily required to be equipped with a unit, this we indicate in \emph{\hyperlink{spring}{For more general rings}}.

\hypertarget{InTermsOfExactFunctorsAndTorFunctors}{}\subsubsection*{{In terms of exact functors and Tor-functors}}\label{InTermsOfExactFunctorsAndTorFunctors}

Let $R$ be a [[commutative ring]].

\begin{defn}
\label{FlatModule}\hypertarget{FlatModule}{}
An $R$-[[module]] $N$ is \textbf{flat} if [[tensor product of modules|tensoring]] with $N$ over $R$ as a [[functor]] from $R$[[Mod]] to itself

\begin{displaymath}
(-)\otimes_R N : R Mod \to R Mod
\end{displaymath}
is an [[exact functor]] (sends [[short exact sequences]] to short exact sequences).

\end{defn}
\begin{defn}
\label{FaithfullyFlatModule}\hypertarget{FaithfullyFlatModule}{}
A module as above is \emph{faithfully flat} if it is flat and tensoring in addition \emph{reflects exactness}, hence if the tensored sequence is exact if and only if the original sequence was.

\end{defn}
\begin{remark}
\label{ImmediateReformulationOfFlatness}\hypertarget{ImmediateReformulationOfFlatness}{}
The condition in def. \ref{FlatModule} has the following immediate equivalent reformulations:

\begin{enumerate}%
\item $N$ is flat precisely if $(-)\otimes_R N$ is a [[left exact functor]],

because tensoring with any module is generally already a [[right exact functor]];


\item $N$ is flat precisely if $(-)\otimes_R N$ sends [[monomorphisms]] ([[injections]]) to monomorphisms,

because for a right exact functor to also be left exact the only remaining condition is that it preserves the monomorphisms on the left of a [[short exact sequence]];


\item $N$ is flat precisely if $(-)\otimes_R N$ is a [[flat functor]],

because [[Mod]] is [[finitely complete category|finitely complete]];


\item $N$ is flat precisely if the degree-1 [[Tor]]-[[functor]] $Tor_1^{R Mod}(-,N)$ is zero,

because by the general properties of [[derived functors in homological algebra]], $L_1 F$ is the obstruction to a [[right exact functor]] $F$ being left exact;


\item $N$ is flat precisely if all higher [[Tor]] functors $Tor_{\geq 1}(R Mod)(-,N)$ are zero,

because the higher [[derived functor in homological algebra|derived functors]] of an [[exact functor]] vanish;


\item $N$ is flat precisely if $N$ is an [[acyclic object]] with respect to the tensor product functor;

because the [[Tor]] functor is symmetric in both arguments and an object is called tensor-[[acyclic object]] if all its positive-degree $Tor$-groups vanish.



\end{enumerate}
\end{remark}
The condition in def. \ref{FlatModule} also has a number of not so immediate equivalent reformulations. These we discuss in detail below in \emph{\hyperlink{EquivalentCharacterizations}{Equivalent characterizations}}. One of them gives an explicit characterization of flat modules in terms of relations beween their elements. An exposition of this we give now in \emph{\hyperlink{InTermsOfIdentities}{In terms of identities}}.

\hypertarget{InTermsOfIdentities}{}\subsubsection*{{Explicitly in terms of identities}}\label{InTermsOfIdentities}

There is a characterisation of flatness that says that a left $A$-module $M$ is flat if and only if ``everything (that happens in $M$) happens for a reason (in $A$)''. We indicate now what this means. Below in prop. \ref{RelationsFromCharacterizationonIdealInclusion} it is shown how this is equivalent to def. \ref{FlatModule} above.

The meaning of this is akin to the existence of [[basis of a vector space|bases in vector spaces]]. In a [[vector space]], say $V$, if we have an identity of the form $\sum_i \alpha_i v_i = 0$ then we cannot necessarily assume that the $\alpha_i$ are all zero. However, if we choose a basis then we can write each $v_i$ in terms of the basis elements, say $v_i = \sum_j \beta_{i j} u_j$, and substitute in to get $\sum_{i j} \alpha_i \beta_{i j} u_j = 0$. Now as $\{u_j\}$ forms a basis, we can deduce from this that for each $j$, $\sum_i \alpha_i \beta_{i j} = 0$. These last identities happen in the coefficient field, which is standing in place of $A$ in the analogy.

When translating this into the language of modules we cannot use bases so we have to be a little more relaxed. The following statement is the right one.

Suppose there is some identity in $M$ of the form $\sum_i a_i m_i = 0$ with $m_i \in M$ and $a_i \in A$. Then there is a family $\{n_j\}$ in $M$ such that every $m_i$ can be written in the form $m_i = \sum_j b_{i j} n_j$ and the coefficients $b_{i j}$ have the property that $\sum_i a_i b_{i j} = 0$.

The module $M$ being flat is equivalent to being able always to do this.

There is an alternative way to phrase this which is less element-centric. The elements $m_i$ correspond to a [[morphism]] into $M$ from a [[free module]], say $m \colon F \to M$. The $a_i$ correspond to a morphism $a \colon F \to F$, multiplying the $i$th term by $a_i$. That we have the identity $\sum_i a_i m_i = 0$ says that the composition $m a$ is zero, or that $m \colon F \to M$ factors through the [[coequaliser]] of $a$ and $0$.

Now we consider the elements $n_j$. These define another morphism from a free module, say $n \colon E \to M$. That the $m_i$ can be expressed in terms of the $n_j$ says that the morphism $m$ factors through $n$. That is, there is a morphism $b \colon F \to M$ such that $m = n b$. We therefore have two factorisations of $m$: one through $n$ and one through the [[cokernel]] $\coker a$. The question is as to whether these have any relation to each other. In particular, does $\coker a \to M$ factor through $n$? We can represent all of this in the following diagram.

Saying that $M$ is flat says that this lift always occurs.

Taking this a step further, we consider the [[filtered category|filtered family]] of all finite [[subsets]] of $M$. This generates a filtered family of [[finitely generated module|finitely generated]] [[free modules]] with compatible morphisms to $M$. So there is a morphism from the [[colimit]] of this family to $M$. This morphism is [[surjection|surjective]] by construction. To show that it is [[injection|injective]], we need to show that any element in one of the terms in the family that dies by the time it reaches $M$ has actually died on the way. This is precisely what the above characterisation of flatness is saying: the element corresponding to $\sum_i a_i m_i$ that dies in $M$ is already dead by the time it reaches $E$.

We have thus arrived at the following result:

\begin{theorem}
\label{filtfree}\hypertarget{filtfree}{}
A module is flat if and only if it is a [[filtered colimit]] of [[free modules]].

\end{theorem}
This observation (Wraith, Blass) can be put into the more general context of modelling [[geometric theory|geometric theories]] by [[geometric morphism|geometric morphisms]] from their [[classifying topos|classifying toposes]], or equivalently, certain [[flat functor|flat functors]] from [[site|sites]] for such topoi.

\hypertarget{for_more_general_rings}{}\subsubsection*{{For more general rings}}\label{for_more_general_rings}

Even if the [[ring]] $R$ is not necessarily commutative and not necessarily unital, we can say:

A left $R$-[[module]] is flat precisely if the [[tensor product of modules|tensoring]] functor

\begin{displaymath}
(-)\otimes_R \colon Mod_R \to Ab
\end{displaymath}
from \emph{right} $R$ modules to [[abelian groups]] is an [[exact functor]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{EquivalentCharacterizations}{}\subsubsection*{{Equivalent characterizations}}\label{EquivalentCharacterizations}

By def. \ref{FlatModule}, or its immediate consequence, remark \ref{ImmediateReformulationOfFlatness}. $N \in R Mod$ is flat if for every injection $i \colon A \hookrightarrow$ also $i \otimes_R N \colon A \otimes_R N \to B \otimes_R N$ is an injection. The following proposition says that this may already be checked on just a very small subclass of injections.

\begin{theorem}
\label{CharacterizationOnIdealInclusions}\hypertarget{CharacterizationOnIdealInclusions}{}
An $R$-module $N$ is flat already if for all inclusions $I \hookrightarrow R$ of a [[finitely generated module|finitely generated]] [[ideal]] into $R$, regarded as a module over itself, the induced morphism

\begin{displaymath}
I \otimes_R N \to R \otimes_R N \simeq N
\end{displaymath}
is an injection.

\end{theorem}
\begin{proof}
(\ldots{})

\end{proof}
\begin{prop}
\label{RelationsFromCharacterizationonIdealInclusion}\hypertarget{RelationsFromCharacterizationonIdealInclusion}{}
A module $N$ is flat precisely if for every finite linear combination of zero, $\sum_i r_i n_i  = 0\in N$ with $\{r_i \in R\}_i$, $\{n_i \in N\}$ there are elements $\{\tilde n_j \in N\}_j$ and linear combinations

\begin{displaymath}
n_i = \sum_j b_{i j} \tilde n_j \;\;\in  N
\end{displaymath}
with $\{b_{i j} \in R\}_{i,j}$ such that for all $j$ we have

\begin{displaymath}
\sum_i r_i b_{i j}  = 0\;\;\; \in R
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
A finite set $\{r_i \in R\}_i$ corresponds to the inclusion of a finitely generated ideal $I \hookrightarrow R$.

By theorem \ref{CharacterizationOnIdealInclusions} $N$ is flat precisely if $I \otimes_R N \to N$ is an injection. This in turn is the case precisely if the only element of the tensor product $I \otimes_R R$ that is 0 in $R \otimes_R N = N$ is already 0 on $I \otimes_R N$.

Now by definition of [[tensor product of modules]] an element of $I \otimes_R N$ is of the form $\sum_i (r_i ,n_i)$ for some $\{n_i \in N\}$. Under the inclusion $I \otimes_R N \to N$ this maps to the actual linear combination $\sum_i r_i n_i$. This map is injective if whenever this linear combination is 0, already $\sum_i (r_i, n_i)$ is 0.

But the latter is the case precisely if this is equal to a combination $\sum_j (\tilde r_j  , \tilde n_j)$ where all the $\tilde r_j$ are 0. This implies the claim.

\end{proof}
\hypertarget{relation_to_projective_modules}{}\subsubsection*{{Relation to projective modules}}\label{relation_to_projective_modules}

\begin{prop}
\label{ProjectiveModulesAreFlat}\hypertarget{ProjectiveModulesAreFlat}{}
Every [[projective module]] is flat.

\end{prop}
\begin{proof}
Clearly every ring $R$ is a flat module over itself, and [[direct sums]] as well as direct summands of flat modules are flat. Hence direct summands of [[free modules]] are flat, and these are precisely the projective modules (\href{projective+module#ProjectiveIsPreciselyDirectSummandOfFreeModule}{prop.})

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{([[Lazard's criterion]])}

A module is flat if and only if it is a [[filtered colimit]] of [[free modules]].

\end{prop}
This is due to (\hyperlink{Lazard}{Lazard (1964)}).

\begin{proof}
(\ldots{}) For the moment see the above discussion. (\ldots{})

\end{proof}
\hypertarget{relation_to_locally_free_modules}{}\subsubsection*{{Relation to (locally) free modules}}\label{relation_to_locally_free_modules}

\begin{defn}
\label{LocallyFreeModule}\hypertarget{LocallyFreeModule}{}
An $R$-module $N$ over a [[Noetherian ring]] $R$ is called a \textbf{locally free module} if there is a [[Zariski topology|cover]] by [[prime ideals]] $I \hookrightarrow R$ such that the [[localization of a module|localization]] $N_I$ is a [[free module]] over the [[localization of a ring|localization]] $R_I$.

\end{defn}
\begin{prop}
\label{ForFinitelyGeneratedFlatIsLocallyFree}\hypertarget{ForFinitelyGeneratedFlatIsLocallyFree}{}
For $R$ a [[Noetherian ring]] and $N$ a [[finitely generated module]] over $R$, $N$ is flat precisely if it is [[locally free module]], def. \ref{LocallyFreeModule}.

\end{prop}
By \hyperlink{RaynaudGruson}{Raynaud-Gruson, 3.4.6 (part I)}

\begin{prop}
\label{}\hypertarget{}{}
If

\begin{enumerate}%
\item $R$ is a [[local ring]],


\item $N$ is a [[finitely generated module]],


\item $N$ is a flat module



\end{enumerate}
then $N$ is a [[free module]].

\end{prop}
This is \hyperlink{Matsumara}{Matsumara, Theorem 7.10}

\hypertarget{Examles}{}\subsection*{{Examples}}\label{Examles}

\begin{example}
\label{}\hypertarget{}{}
An [[abelian group]] is flat (regarded as a $\mathbb{Z}$-module) precisely if it is [[torsion subgroup|torsion-free]].

\end{example}
\begin{proof}
By the general discussion at [[derived functor in homological algebra]], the obstruction to $A \in Ab$ being flat are the first [[Tor]]-groups $Tor_1^{\mathbb{Z}}(-,A)$. By the discussion at \emph{\href{Tor#RelationToTorsionGroups}{Tor -- relation to torsion subgroups}} these a [[filtered colimits]] and [[direct sums]] of the [[torsion subgroups]] of $A$. In particular for $Tor_1^\mathbb{Z}(\mathbb{Z}_n,A)$ is the $n$-torsion subgroup of $A$. Hence $Tor_1^\mathbb{Z}(-,A)$ vanishes and hence $A$ is flat precisely if all torsion subgroups of $A$ are trivial.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[flat morphism of schemes]]


\item [[flat resolution lemma]]


\item [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]


\item [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

\begin{itemize}%
\item [[injective module]]

\end{itemize}

\item [[free object]], [[free resolution]]


\item flat object, [[flat resolution]]


\item [[free module]] $\Rightarrow$ [[projective module]] $\Rightarrow$ \textbf{flat module} $\Rightarrow$ [[torsion-free module]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Original articles include

\begin{itemize}%
\item Shizuo Endo, \emph{On flat modules over commutative rings}, J. Math. Soc. Japan Volume 14, Number 3 (1962), 284-291. (\href{http://projecteuclid.org/euclid.jmsj/1261060583}{EUCLID})


\item Michel Raynaud, Laurent Gruson, \emph{Crit\`e{}res de platitude et de projectivit\'e{}}, Techniques de ``platification'' d'un module. Invent. Math. 13 (1971), 1--89.



\end{itemize}
\begin{itemize}%
\item S. Jondrup, \emph{Flat and projective modules}, Math, Scand. 43 (1978) (\href{http://www.mscand.dk/article.php?id=2456}{pdf})

\end{itemize}
The characterization of flat modules as filtered colimits of [[projective modules]] is due to

\begin{itemize}%
\item [[Daniel Lazard]], \emph{Sur les modules plats} C. R. Acad. Sci. Paris 258, 6313--6316 (1964)

\end{itemize}
For a general account see for instance section 3.2 of

\begin{itemize}%
\item [[Charles Weibel]], \emph{[[An Introduction to Homological Algebra]]}

\end{itemize}
For more details see

\begin{itemize}%
\item Matsumura's CRT book

\end{itemize}
Lecture notes include

\begin{itemize}%
\item Arthur Ogus, \emph{Flatness -- a brief overview} (\href{http://math.berkeley.edu/~ogus/Math%20_256B--09/Supplements/flat.pdf}{pdf})

\end{itemize}
Further resources include

\begin{itemize}%
\item MO discussion \emph{\href{http://mathoverflow.net/questions/33522/flatness-and-local-freeness}{flatness and local freeness}}

\end{itemize}
[[!redirects flat modules]]

[[!redirects faithfully flat module]] [[!redirects faithfully flat modules]]



\end{document}