## Some Natural Identifications (Pt. III)

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

## Tensor Product of Free Modules

**Point of Post: **In this post we discuss more particularly about the tensor product of free modules, and some of the consequences of this.

*Motivation*

We have proved theorems before to the effect that the tensor product of free modules are free. In this post we’d like to solidify this by discussing precisely how, given bases for two free modules, construct a basis on their tensor product. This will enable us to give an explicit example of elements of a tensor product which are not simple tensors.

## Extension of Scalars (Pt. I)

**Point of Post: **In this post we discuss some of the fundamental ideas concerning extension of scalars.

*Motivation*

We finally can now get around to discussing a use of tensor products that I touted in my original introduction: extension of scalars. Namely, the idea is that we are handed some -module and some superring and would like to see to what extent we can consider to be an -module. In other words, are are considering the problem which is dual to taking an -module and considering it as an -module by just “restricting” scalars. Unfortunately, this is often impossible to do. For example, can’t be made into a -space since, if it could, whatever would be (and note, it has to be an INTEGER since the multiplication is a map ) it would satisfy which is impossible! So, the next best thing one could hope to do is perhaps extend the module in some “minimal” way so that it can be naturally imbued with an -module structure. In other words, we want to find some -module for which embeds into as an -module, and doing this in some minimal sort of way. For example, while cannot be given the structure of an -space it can surely be -embedded into such a space, namely itself. Once again, this may not always be possible, for example if is a finite abelian group (e.g. -module) then can never be -embedded into a -space since (as can be easily proven!) every element of a -space has infinite order. Thus, we can really only hope to ask for a “best case scenario”. What -module maximizes both the ability to faithfully (to some degree) embed and is minimal in some sense. In the case of our abelian group it’s clear that we’re going to have to take to be our -space since this is the only such space in which can be “embedded” (albeit very unfaithfully). This is what we mean by extension of scalars, such an -module .

If the obvious intellectual curiosity isn’t enough to motivate this problem I can mention that it has many uses. For example, I have in the past discussed the notion of induced representations which can be seen as extension of scalars problem. Namely, we suppose that we have some group and some subgroup . Roughly then what we wish to do is pass from an -representation to a -representation, which can be thought of as extending an -module (where is the group algebra) to an -module.

So, why might we expect that the tensor product is the correct route for such an extension of scalars? There is actually a quite natural way one might realize this. The first is the naive attempt that one might actually try to make a given -module into an -module in the most brutish way. Namely, let’s define a “formal multiplication” of and elements. Namely, given and let just be formal symbol, our “multiplication”. We then see that if this “multiplication” is to create a valid -module structure extending that of ‘s preexisting -module structure, we’re going to need certain identities to hold. For example, by mere definition of a module we are going to need that is linear in each entry (this is because we should have that , etc.). Moreover, since we want (since we are extending the -module structure) and we see that we are going to have for all , , and . Thus, we see that is an -biadditive map . Therefore if we’d like to consider a “universal” way to define an -module structure on it seems that we should be looking for a “universal” -biadditive map and so really we want to just be and that this should be our extension of scalars.

## Using Partial Exactness to Compute Things (Pt. I)

**Point of Post: **In this post we show how one can use partial exactness to actually compute, explicitly, the isomorphism type of certain tensor products.

*Motivation*

Up until this point I have committed a sort of embarrassing crime–I have yet to explicitly compute a tensor product! This is not good, because I strongly believe that if one doesn’t stop and compute some things, one can get a little lost in the abstraction. So, this post is devoted primarily to discussing a way in which we can use partial exactness to explicitly compute the isomorphism type of some actual examples. Roughly the idea is that if we can express a module in terms of an exact sequence of the form then we know that is the cokernel of the initial, right exactness tells us then that is exact and so will be the cokernel of the first map. After showing some examples where this is useful we shall describe a general technique for computing tensor products by using “free presentations” which ultimately amount to expressing a module in terms of generators and relations.

## Tensor Products Naturally Commute with Direct Limits

**Point of Post: **In this post we give a proof that, roughly, , and moreover we show that this isomorphism is natural.

*Motivation*

In this post we begin the long succession of instances where the adjointness of Hom and tensor in conjunction with Yoneda’s lemma will be useful. In particular, we will show that tensor and direct limits naturally commute, a fact that shall be supremely useful in calculations involving direct limits (most prominently, coproducts).

## R-Algebras

**Point of Post: **In this post we discuss the notion of -algebras where is some commutative unital ring, and the associated categories.

*Motivation*

Very often when we run into -modules (where we assume that is commutative and unital) there is more structure involved, namely the modules are also rings which interact nicely with the -module structure. Namely, we have a (left, right)-module along with a bilinear map which makes into a unital ring such that . We have already run into algebras before, in the context of endomorphism algebras of vector spaces. More generally, any ring of matrices is given the structure of an -algebra. In fact, it’s easy to see that algebras generalize ring theory since, as we shall see, the category of all -algebras is isomorphic to the category of rings. Other examples of algebras are polynomial algebras and is a -dimensional -algebra. Algebras play an important role in a lot of algebraic subjects, perhaps most notably with their appearance in differential geometry in the form of tensor algebras, and their appearance in commutative algebra.

## Functorial Properties of the Tensor Product (Pt. II)

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

## Functorial Properties of the Tensor Product (Pt. I)

**Point of Post: **In this post we discuss some of the more “functorial” properties of the tensor product–namely that the tensor product defines a bifunctor, additive in each entry between certain categories.

*Motivation*

Now that we have defined tensor products we’d like to discuss some of the more category theoretic aspects of the construction. The tensor product functor shall serve as one of the prime examples of an additive functor.