What does it mean to categorise projective modules? How would you go about it?

**feyomi** · December 5th 2011, 11:01 AM

Hi, I am doing a project where I have to explain what a projective module is.
My supervisor said one of the things I should do is show ways of categorising projective modules? What does this mean? Is this category theory? Are there different types of projective modules? Please help to explain, thank you, any links and help much appreciated

**Drexel28** · December 5th 2011, 11:17 AM

Originally Posted by feyomi

Hi, I am doing a project where I have to explain what a projective module is.
My supervisor said one of the things I should do is show ways of categorising projective modules? What does this mean? Is this category theory? Are there different types of projective modules? Please help to explain, thank you, any links and help much appreciated

If I had to guess, they just mean phrasing projective modules in the language of category theory. In particular, suppose $\mathcal{A}$ is an abelian category so that we can speak of short exact sequences $0\to A\to B\to C\to 0$ in $\text{Ch}(\mathcal{A})$ (the category of chains in $\mathcal{A}$ ). We can then define a covariant endofunctor $F:\mathcal{A}\to\mathcal{A}$ to be "exact" if a short exact sequence $0\to A\to B\to C\to0$ is carried to a short exact sequence $0\to F(A)\to F(B)\to F(C)\to 0$ . Or, equivalently we are saying that the induced functor $\widehat{F}:\text{Ch}(\mathcal{A})\to\text{Ch}( \mathcal{A})$ carres SES to SES.

Now, note that $R\text{-}\mathbf{Mod}$ is an abelian category. One can then prove that $M\in\text{obj}\left(R\text{-}\mathbf{Mod}\right)$ is projective if and only if the (covariant) Hom functor $\text{Hom}_R(M,\bullet):R\text{-}\mathbf{Mod}\to R\text{-}\mathbf{Mod}$ is exact where (as I'm sure you know) $\text{Hom}_R(M,\bullet)$ acts on the objects of $R\text{-}\mathbf{Mod}$ by sending $N\mapsto \text{Hom}_R(M,N)$ and acts on morphisms by sending $f:N\to L$ to $f^\ast:\text{Hom}_R(M,N)\to\text{Hom}_R(M,L)$ where $f^\ast(g)=f\circ g$ .

You can find more information on this in any good algebra book, and in any good homological algebra book. Perhaps the most self-contained, elementary discussion of this can be found here in Rotman.

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