Tensor Products - Dummit and Foote Section 10.4 Corollary 9

[Deveno]

Thanks Deveno.

I am now working through, and reflecting upon, your post.

Just a quick clarification:

You write:
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"Since we know that at least for SOME $R$ and an $R$-module $N$, and some extension $S$ and an $S$-module $M$ we can't injectively map $N$ to $M$ (there have been several examples shown), it is a legitimate question to ask:

"What characterizes the largest possible quotient of $N$ that can possibly work (given $R,S,N, M$)"? '

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In what you are saying, is M actually the L of Dummit and Foote in this set up.

So, in Corollary 9, we are actually seeking to map a quotient of N into L? ... and further, the tensor product is only a step on the path to do this ...? Further, is the S-module where we have actually succeeded in "extending the scalars" actually L?

Yes.

Also, when you write:

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"The tensor product is two things:

1) A bilinear map $\otimes: S \times N \to S \otimes_R N$

which takes $(s,n) \mapsto s\otimes n$

2) the $S$-module that is the "target" of this bilinear map."

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In this case, the S-module that is the "target" is \(\displaystyle S \otimes_R N \) - or is it L? (I think it is \(\displaystyle S \otimes_R N\) ... but can you confirm?

Peter

by target, I mean the co-domain of the tensor product bilinear mapping.

PS Just thinking that some of my confusion over Theorem 8 and Corollary 9 is due to D&F introducing extension of the scalars, in a sense, before tensor products (or at least at the same time as) - instead of introducing the extension of the scalars after tensor products - some of the confusion anyway :)

My thought is that they were trying to introduce the tensor product in a limited setting first, as an abelian group construction ($\Bbb Z$-module) before tensoring over a ring. Other structures (most notably vector spaces and their duals) have tensor products as well.

I think Dummit and Foote assume one has some familiarity already with linear algebra (at least with the notions of basis, linear combination and matrix representation), and therefore some of their exposition on various topics is rather brief. Don't get me wrong, it's a darn good algebra book, but it's also known for being a challenge to work through.

 

[Math Amateur]

Yes.
by target, I mean the co-domain of the tensor product bilinear mapping.
My thought is that they were trying to introduce the tensor product in a limited setting first, as an abelian group construction ($\Bbb Z$-module) before tensoring over a ring. Other structures (most notably vector spaces and their duals) have tensor products as well.

I think Dummit and Foote assume one has some familiarity already with linear algebra (at least with the notions of basis, linear combination and matrix representation), and therefore some of their exposition on various topics is rather brief. Don't get me wrong, it's a darn good algebra book, but it's also known for being a challenge to work through.

Thanks Deveno.

You write:

"by target, I mean the co-domain of the tensor product bilinear mapping."

... since as you write: "bilinear map $\otimes: S \times N \to S \otimes_R N$ which takes $(s,n) \mapsto s\otimes n$ ...

that co-domain is \(\displaystyle S \otimes_R N \). Is that right? Can you confirm?

Peter

 

[Deveno]

Yes, that is indeed what I intended. The module $S \otimes_R N$ doesn't mean anything without the bilinear map $\otimes$.

In fact, most of the time, one can just think of $\otimes$ as "the most general bilinear map possible." Most of the theorems concerning the tensor product are to reinforce the utility of this view.