# Abstract Nonsense

## Flat Modules (Pt. I)

Point of Post: In this post we discuss, in some detail, flat modules including equational characterizations and Lazard’s theorem.

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Motivation

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We have so far discussed two kinds of modules that make certain half-exact (i.e. right exact or left exact) functors exact. Namely, we have discussed projective modules as the modules that make the Hom functor $\text{Hom}_R(P,\bullet)$ exact, and we have discussed injective modules as the modules that make the contravariant Hom functor $\text{Hom}_R(\bullet,I)$ exact. Both of these are very important since projective and injective modules are the “approximating” modules used to define powerful notions like derived functors. That said, they aren’t the most “functional” modules that make certain functors exact. In particular, in basic module theory there are three main functors, two of which are the covariant and contravariant Hom functor. The last, of course, is the tensor functor $M\otimes_R\bullet$. The tensor functor shows up everywhere in mathematics and is, dare I say it, “more important” (whatever that really means) than the two Hom functors [note: this is just my limited opinion on the matter]. Thus, it would seem to behoove us to understand what kind of modules make the functor $M\otimes_R\bullet$ exact (it is already right exact).

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This naturally then leads us to considering flat modules, which are precisely those modules that make the relevant tensor functor exact. Flat modules show up in geometry in some pretty interesting ways (none of which I am really able to speak to at this point).

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Flat Modules

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Let $R$ be a (unital) ring and $M$ a left $R$-module. We say that $M$ is flat if the functor $\bullet\otimes_R M:\mathbf{Mod}\text{-}R\to\mathbf{Ab}$ is exact. Of course, we can view the domain and codomain of this functor in different lights depending upon the structures they have. Perhaps most generally an $(R,B)$-bimodule $M$ is flat if and only if the functor $\bullet\otimes M:(A,R)\text{-}\mathbf{Mod}\to(A,B)\text{-}\mathbf{Mod}$ is exact. Of course, despite whether we are thinking of $M$ as a bimodule and/or acting on bimodules, or not it is really the functor $\mathbf{Mod}\text{-}R\to\mathbf{Ab}$ that is central (for all the other functors are just obscured cases of this functor). Thus, we shall restrict our attention to this functor. For notational convenience let us denote $\bullet\otimes_R M:\mathbf{Mod}\text{-}R\to\mathbf{Ab}$ as $T_M$ for the remainder of this post.

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We know that $T_M$ is already right exact, and thus we see that the meat of the assertion that $M$ is flat comes from statement that if $0\to L\xrightarrow{f}N$ is an exact sequence of right $R$-modules and $R$-maps then $0\to L\otimes_R M\xrightarrow{f\otimes 1}N\otimes_R M$ is an exact sequence of abelian groups and group maps. Or, said more explicitly, $T_M$ takes injections to injections.

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For example, if $A$ is a non-zero torsion abelian group (i.e. every element of $A$ is torsion) then $A$ is not flat for while the inclusion $\mathbb{Z}\hookrightarrow\mathbb{Q}$ is injective $\mathbb{Z}\otimes_\mathbb{Z}A\to\mathbb{Q}\otimes_\mathbb{Z}A$ cannot possibly be injective since $\mathbb{Z}\otimes_\mathbb{Z}A\cong A$ and $\mathbb{Q}\otimes_\mathbb{Z}A\cong0$.

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An example of a flat $R$-module is $R$ for we know that $T_R$ is naturally isomorphic to the forgetful functor which is definitively exact.

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We shall greatly be able to expand our repetoire of flat modules by considering the following theorem:

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Theorem: Let $\{M_\alpha\}_{\alpha\in\mathcal{A}}$ be a directed system of flat left $R$-modules where $\mathcal{A}$ is a directed set. Then, the direct limit $\varinjlim M_\alpha$ is a flat left $R$-module.

Proof: Let $N,N',N''$ be any three right $R$-modules and suppose that have an exact sequence

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$0\to N\xrightarrow{f}N'\xrightarrow{g}N''\to0$

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Then, for each $\alpha\in\mathcal{A}$ we have that

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$0\to N\otimes_R M_\alpha\xrightarrow{f\otimes 1_{M_\alpha}}N'\otimes_R M_\alpha\xrightarrow{g\otimes1_{M_\alpha}}0$

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is exact. Now, since $\mathcal{A}$ is directed we know that the direct limit functor is exact we may conclude that

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$0\to \varinjlim(N\otimes_R M_\alpha)\xrightarrow{\varinjlim(f\otimes1_{M_\alpha})}\varinjlim(N'\otimes_R M_\alpha)\xrightarrow{\varinjlim(g\otimes1_{M_\alpha})}\varinjlim(N''\otimes_R M_\alpha)\to0$

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is exact. But, we know that $\varinjlim(N\otimes_R M_\alpha)\cong N\otimes_R\varinjlim M_\alpha$ in such a way that the following diagram commutes

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$\begin{array}{ccccccccc}0 & \to & \varinjlim(N\otimes_R M_\alpha) & \xrightarrow{\varinjlim(f\otimes 1_{M_\alpha})} & \varinjlim(N''\otimes_R M_\alpha) & \xrightarrow{\varinjlim(g\otimes 1_{M_\alpha})} & \varinjlim(N''\otimes_R M_\alpha) & \to & 0\\ & & \big\downarrow\approx & & \big\downarrow\approx & & \big\downarrow\approx & & \\ 0 & \to & N\otimes_(\varinjlim M_\alpha) & \xrightarrow{f\otimes 1_{\varinjlim M_\alpha}} & N'\otimes_R(\varinjlim M_\alpha) & \xrightarrow{g\otimes 1_{\varinjlim M_\alpha}} & N''\otimes_R(\varinjlim M_\alpha) & \to & 0 \end{array}$

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and so the bottom row is exact as desired. $\blacksquare$

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In particular, we see that direct sum of flat modules is flat and so we get the following:

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Theorem: Free modules are flat.

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We can actually take the statement that free modules are flat a step further by noticing that since the functor $T_{\oplus_\alpha M}$ is naturally isomorphic to $\displaystyle \oplus_\alpha T_{M_\alpha}$ and the direct sum of functors is exact if and only if each functor is exact we have the following:

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Theorem: The module $\displaystyle \bigoplus_\alpha M_\alpha$ is exact if and only if each $M_\alpha$ is exact.

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Now, since projective modules are just the direct summands of free modules which are flat we get the following corollary:

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Theorem: Projective modules are flat.

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References:

[1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. Module Theory. Clarendon, 1990. Print.

[5] Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.