Abstract Nonsense

Crushing one theorem at a time

Some Natural Identifications (Pt. III)


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

Continue reading

August 14, 2012 Posted by | Algebra, Linear Algebra | , , , , , , , | 1 Comment

The Tensor Algebra and Exterior Algebra (Pt. V)


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

Continue reading

May 10, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , , | 1 Comment

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.

\text{ }

Motivation

\text{ }

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.

\text{ }

Continue reading

January 25, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , | 1 Comment

Extension of Scalars (Pt. I)


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

\text{ }

Motivation

\text{ }

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 R-module M and some superring S\supseteq R and would like to see to what extent we can consider M to be an S-module. In other words, are are considering the problem which is dual to taking an S-module and considering it as an R-module by just “restricting” scalars. Unfortunately, this is often impossible to do. For example, \mathbb{Z} can’t be made into a \mathbb{Q}-space since, if it could, whatever \frac{1}{2}\cdot 1 would be (and note, it has to be an INTEGER since the multiplication is a map \mathbb{Q}\times\mathbb{Z}\to\mathbb{Z}) it would satisfy 2(\frac{1}{2}\cdot1)=1 which is impossible! So, the next best thing one could hope to do is perhaps extend the module M in some “minimal” way so that it can be naturally imbued with an S-module structure. In other words, we want to find some S-module N for which M embeds into N as an R-module, and doing this in some minimal sort of way.  For example, while \mathbb{Z} cannot be given the structure of an \mathbb{Q}-space it can surely be \mathbb{Z}-embedded into such a space, namely \mathbb{Q} itself. Once again, this may not always be possible, for example if A is a finite abelian group (e.g. \mathbb{Z}-module) then A can never be \mathbb{Z}-embedded into a \mathbb{Q}-space since (as can be easily proven!) every element of a \mathbb{Q}-space has infinite order. Thus, we can really only hope to ask for a “best case scenario”. What S-module maximizes both the ability to faithfully (to some degree) embed M and is minimal in some sense. In the case of our abelian group A it’s clear that we’re going to have to take 0 to be our \mathbb{Q}-space since this is the only such space in which A can be “embedded” (albeit very unfaithfully). This is what we mean by extension of scalars, such an S-module N.

\text{ }

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 G and some subgroup H\leqslant G. Roughly then what we wish to do is pass from an H-representation to a G-representation, which can be thought of as extending an \mathcal{A}(H)-module (where \mathcal{A}(H) is the group algebra) to an \mathcal{A}(G)-module.

\text{ }

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 R-module M into an S-module in the most brutish way. Namely, let’s define a “formal multiplication” of S and M elements. Namely, given s\in S and m\in M let s\star m just be  formal symbol, our “multiplication”. We then see that if this “multiplication” is to create a valid S-module structure extending that of M‘s preexisting R-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 \star is linear in each entry (this is because we should have that (s+s')\star m=s\star m+s'\star m, etc.). Moreover, since we want r\star m=rm (since we are extending the R-module structure) and (ss')\star m=s\star(s'\star m) we see that we are going to have (sr)\star m=s\star(r\star m)=s\star(rm) for all s\in S, r\in R, and m\in M. Thus, we see that \star is an R-biadditive map S\times M\to S\star M. Therefore if we’d like to consider a “universal” way to define an S-module structure on M it seems that we should be looking for a “universal” R-biadditive map S\times M\to S\star M and so really we want S\star M to just be S\otimes_R M and that this should be our extension of scalars.

\text{ }

Continue reading

January 24, 2012 Posted by | Uncategorized | , , , , | 1 Comment

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.

\text{ }

Motivation

\text{ }

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 A\to B\to C\to0 then we know that C is the cokernel of the initial, right exactness tells us then that A\otimes D\to B\otimes D\to C\otimes D\to0 is exact and so C\otimes D 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.

\text{ }

Continue reading

January 20, 2012 Posted by | Algebra, Uncategorized | , , , , , , , , , | 4 Comments

Tensor Products Naturally Commute with Direct Limits


Point of Post: In this post we give a proof that, roughly, \varinjlim (M_\alpha\otimes_R N)\cong (\varinjlim M_\alpha)\otimes_R N, and moreover we show that this isomorphism is natural.

\text{ }

Motivation

\text{ }

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).

\text{ }

Continue reading

January 19, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , | 5 Comments

R-Algebras


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

\text{ }

Motivation

\text{ }

Very often when we run into R-modules (where we assume that R is commutative and unital) there is more structure involved, namely the modules are also rings which interact nicely with the R-module structure. Namely, we have a (left, right)R-module A along with a bilinear map \cdot:A\times A\to A which makes A into a unital ring A such that r(x\cdot y)=(rx)\cdot y=x\cdot (ry). We have already run into algebras before, in the context of endomorphism algebras of vector spaces. More generally, any ring of matrices \text{Mat}_n(R) is given the structure of an R-algebra. In fact, it’s easy to see that algebras generalize ring theory since, as we shall see, the category \mathbb{Z}\text{-}\mathbf{Alg} of all \mathbb{Z}-algebras is isomorphic to the category \mathbf{Rng} of rings. Other examples of algebras are polynomial algebras and \mathbb{C} is a 2-dimensional \mathbb{R}-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.

Continue reading

January 10, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , | 5 Comments

Basic Properties of Tensor Products (Pt. II)


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

\text{ }

Continue reading

January 5, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , , , , | Leave a comment

Functorial Properties of the Tensor Product (Pt. II)


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

\text{ }

Continue reading

January 4, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , | 2 Comments

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.

\text{ }

Motivation

\text{ }

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.

\text{ }

Continue reading

January 4, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , | 4 Comments