## The Tensor Algebra and Exterior Algebra (Pt. III)

**Point of Post: **This is a continuation of this post.

## The Tensor Algebra and the Exterior Algebra (Pt. II)

**Point of Post: **This is a continuation of this post.

## The Tensor Algebra and the Exterior Algebra (Pt. I)

**Point of Post: **In this post we introduce the basic notions of the tensor algebra, graded algebras/rings, and the exterior algebra.

**Motivation**

If there is anyone (and I do mean *anyone) *who* *reads my blog regularly, they should know that I have started blogging about complex analysis. It was going fine and dandy until–blam!–I hit Cauchy’s Theorem. Why was this a problem? Well, I was going to discuss how Cauchy’s theorem is simple for holomorphic functions (as I have defined them, as being ) since we could apply Green’s theorem. But, then it hit me, I didn’t really remember how the proof of Green’s/Stokes theorem went. Thus, I decided that I should review all of the basic differential forms/Stokes theorem stuff and post it on my blog. In preparation for this I started looking at books, upon books to brush up on this stuff. One of the things that I found most confusing was the lack of algebraic rigour concerning the machinery involved. In particular, books either do one of two things. They either don’t talk about the more conceptually difficult algebra they are using, which while easier to understand (to someone not algebraically inclined!) makes certain constructions/operations seem forced, unnatural. Or, they use the higher level stuff and make identifications with the lower level stuff without explaining properly what this identification is, and why it makes sense.

Thus, before I actually start talking about differential forms I want to discuss the algebra that is going to be needed. This post is devoted to the higher level stuff–the tensor algebra and exterior powers. The tensor algebra is a very natural constructions since it is the “free algebra” over a given module–it’s just what you get when you just multiply vectors in a module, without any restrictions. One can think about the tensor product as being noncommutative polynomials where the indeterminates are the elements of the module you started with. The exterior algebra can be thought about algebraically as the freest algebra on a module where every indeterminate squares to zero. While this is slightly less interesting than the tensor algebra, its smaller pieces (the exterior powers of the module) is what is really interesting for they are to alternating multilinear maps what the tensor product is to just plain old multilinear maps. In other words, the exterior powers will be the modules that let us trade in alternating multiliner maps for linear ones in the “freest” possible way.

I will be following heavily from [6], making altercations and additions as I see fit.

## 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).

## Injective Modules (Pt. I)

**Point of Post: **In this post we discuss the notion of injective modules and show that the category has “enough injectives”.

Motivation

We are going to discuss now the “dual” notion to projective modules which, as one would expect, are just the modules one gets by dualizing the lifting axioms for projective modules. Of course, it should also follow then that we can dualize the other properties of projective modules to get other characterizations, is exact, etc.

Our main use for injective modules, similar to the case of projective modules, is that injective modules are “nice” modules for which we can effectively approximate any other module (i.e. we can find some long exact sequence beginning with a given module and with the rest of the terms in the sequence being injective modules). This will be key for when we discuss the notion of derived functors.