# Abstract Nonsense

## Projective Modules (Pt. I)

Point of Post: In this post we discuss and motivate the notion of projective modules, and give/prove the standard equivalences between the most common definitions.

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Motivation

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In this post we give the definition of  a type of module which, among others, shall occupy a fair amount of our attention in the math (specifically the homological algebra) to come. There are, as there were for tensor products, a multitude of (seemingly) disparate ways to motivate the usefulness/interestingness of projective modules, some of which I understand better than others. A motivation which is somewhat high on the usefulness scale but which is, perhaps, less so on the scale of interestingness is that projective modules are precisely the answer to the following question: “we know that given a module $M$ the covariant Hom functor $\text{Hom}_R(M,\bullet)$ is left-exact, for which modules $M$ is the functor actually exact?” Another question for which projective modules answer quite nicely is the question as to which modules $M$ have the property that any short exact sequence ending in them splits (for example, we have [perhaps it’s slightly opaque from the presentation in that post] seen that this is true for free modules).

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While the first of these two motivations is very important both of these motivational characterization of projective modules are still, well, not so motivating. So, let’s consider some of the other questions that these projective modules seem to answer.

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One of the more down-to-earth reasons why one would want to consider projective modules is concerned with a certain ‘lifting’ property of free modules. Namely, suppose that we have modules $M,N$ and a free module $F$. Then, given any module epimorphism $h:M\to N$ and $g:F\to N$ we can create a module homomorphism $f:F\to M$ such that $h\circ f=g$. For free modules it is clear how to do this–we just do it (we define the mapping how we want on a basis and just extend). Of course, this universal ability to ‘factor’ doesn’t always hold for non-free modules. For example, we shall see that if $F=\mathbb{Z}_2$ then $F$ does NOT satisfy this property in the category of abelian groups. Projective modules are precisely the modules for which we can always lift such maps.

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Ok, let’s discuss yet another important question for which the class of projective modules is the answer. Suppose that we made a decision to try to understand module theory by attempting to construct all modules, going through them systematically. Being the good little math students that we are, we construct the most obvious modules first–free modules. I mean, these are just the answer to “what’s the module I can construct that requires the ‘least thought'”. Once we are done looking at free modules the next obvious thing we can look at is submodules of free modules. But, once again, being the good little math students that we are, we wish to select the submodules of free modules which are “simplest”. By this, we mean that properties of the free module can be transferred to properties about the submodule. A way in which this can happen, for example, is if the submodule is actually an isomorphic image of the free module. But, the nicest way in which THAT can happen is if the submodule is actually a direct summand of the free module (i.e. there exists another module whose coproduct with the submodule is isomorphic to the ambient free module). Thus, we have decided that after free modules we should study submodules of free modules that are direct summands. Of course, the punchline should be clear by now, this class of modules is precisely the class of projective modules.

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As a penultimate usefulness I mention I somewhat throwaway fact about projective modules in relation to algebraic geometry and topology which, at this point, I don’t understand very well (but would very much like to in the future). Namely, projective modules can be thought of as an algebraic analogue of trivial vector bundles in topology. This is made more precise by the Serre-Swan theorem and by this MSE answer by the very knowledgeable Georges Elencwajg. While I don’t fully understand this correspondence (it sits comfortably in the realm of K-theory) it definitely gives a somewhat fruitful geometric intuition as to what projective modules are–they represent the simplest possible pieces that a space can decompose into. It tells us that projective modules are “locally free”.

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Now, for the final, and most important motivation for this (as should now be clear) ubiquitous class of modules. Namely, in the math to come (the homological algebra) we shall be very concerned with when certain left or right exact functors are actually exact. If one thinks about it, this is actually a pretty serious condition to prove. Namely, we’d theoretically have to check that every short exact sequence goes to a short exact sequence. It would be much easier if we were able to construct some kind of gadget that allowed us to determine the right or left exactness of a given left or right (respectively) exact functor just by computing one thing with this gadget. In fact, we shall be able (in most cases) to construct a gadget and for the case of starting with a right-exact functor. The integral part of this construction shall be construction a so called “projective resolution” which uses, you guessed it, projective modules. Thus, if we wish to make this construction we better have a solid understand of these strange beasts. While this aspect of motivation scores high on interestingness and usefulness it is slightly opaque as to how the previous ‘definitions’ of projective modules relate to this motivation–how we can construct such a gadget using them, and why we should expect this to be true. This is a matter better left for later discussion!

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Hopefully now, you don’t just believe that projective modules are interesting, but you believe that their discussion and use was inevitable all along.

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Projective Modules

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While, as was pointed out above, there a many different equivalent notions of projective modules, we need to fix one as our initial definition. Considering that the lifting property is a) closes to home as to how projective modules should be thought of (i.e. it best captures their true ‘character’) and b) it is the formulation which correctly generalizes, it is the definition we shall choose. Namely, let $R$ be some ring and let $P$ be a left $R$-module. Then, $P$ is called projective if whenever there is an $R$-epimorphism $h:M\to N$ and a map $k:P\to N$ there exists a unique map $j:P\to M$ such that $h\circ j=k$. Projectivity is usually expressed by the following diagram

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$\displaystyle \begin{matrix} & & & & P & & & \\ & & & \overset{\exists j}{\swarrow} & \big\downarrow{^k} & & \\ & & M & \underset{h}{\longrightarrow} & N & \underset{}{\longrightarrow} & 0\end{matrix}\quad\mathbf{(1)}$

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So, let’s think about what this says. This says that any time we have a mapping $k:P\to N$ and a mapping $h:M\to N$ we can basically define a map $P\to M$ which ‘intuivitely’ takes $x$ to the preimage under $h$ of $k(x)$. As per usual in algebra, it’s clear that there is zero set-theoretic difficulty here, the meat of the problem is to whether or not we can construct a map which is a homomorphism. In some cases where issues of going from set-wise notions to homomorphisms notions are easy (e.g. for free modules) it’s clear we have projectivity. Indeed, suppose that $F$ is free with basis $\{x_\alpha\}$ and we have maps $h:M\twoheadrightarrow N$ and $k:F\to N$. For each basis vector $x_\alpha$ choose any element $y_\alpha\in M$ such that $h(y_\alpha)=k(x_\alpha)$. By the universal property of free modules the set map $\{x_\alpha\}\to M$ extends to an $R$-map $j:F\to M$ with the property that $j(x_\alpha)=y_\alpha$ for all $\alpha$. Note then that since $h(j(x_\alpha))=h(y_\alpha)=k(x_\alpha)$ for all $\alpha$ that $h\circ j$ and $k$ are $R$-maps on $F$ which agree on a basis and so they must be equal, i.e. $h\circ j=k$. Since everything in this argument was arbitrary we can confidently state that:

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Theorem: Free modules are projective.

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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.