## Flat Modules (Pt. I)

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

*Motivation*

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 exact, and we have discussed injective modules as the modules that make the contravariant Hom functor 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 . 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 exact (it is already right exact).

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).

**Flat Modules**

Let be a (unital) ring and a left -module. We say that is *flat *if the functor 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 -bimodule is flat if and only if the functor is exact. Of course, despite whether we are thinking of as a bimodule and/or acting on bimodules, or not it is really the functor 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 as for the remainder of this post.

We know that is already right exact, and thus we see that the meat of the assertion that is flat comes from statement that if is an exact sequence of right -modules and -maps then is an exact sequence of abelian groups and group maps. Or, said more explicitly, takes injections to injections.

For example, if is a non-zero torsion abelian group (i.e. every element of is torsion) then is not flat for while the inclusion is injective cannot possibly be injective since and .

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

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

**Theorem: ***Let be a directed system of flat left -modules where is a directed set. Then, the direct limit is a flat left -module.*

**Proof: **Let be any three right -modules and suppose that have an exact sequence

Then, for each we have that

is exact. Now, since is directed we know that the direct limit functor is exact we may conclude that

is exact. But, we know that in such a way that the following diagram commutes

and so the bottom row is exact as desired.

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

**Theorem: ***Free modules are flat.*

We can actually take the statement that free modules are flat a step further by noticing that since the functor is naturally isomorphic to and the direct sum of functors is exact if and only if each functor is exact we have the following:

**Theorem: ***The module is exact if and only if each is exact. *

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

**Theorem: ***Projective modules are flat. *

**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.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

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

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