Basic Properties of Tensor Products (Pt. II)

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

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Theorem(Associativity): Let $M,N,L$ be $(A,R),(R,S)$ and $(S,B)$ bimodules respectively. Then, there is an isomorphism

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$M\otimes_R(N\otimes_S L)\cong (M\otimes_R N)\otimes_S L$

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of $(A,B)$ -bimodules such that $m\otimes (n\otimes \ell)\mapsto (m\otimes n)\otimes\ell$

Proof: Fix $\ell\in L$ and define

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$\displaystyle f_\ell:M\times N\to M\otimes_R(N\otimes_S L):(m,n)\mapsto m\otimes(n\otimes\ell)$

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We claim that $f_\ell$ is an $R$ -biadditive $A$ -map. Indeed,

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$\displaystyle \begin{aligned}&f_\ell(mr,n)=mr\otimes(n\otimes\ell)=m\otimes r(n\otimes\ell)=m\otimes(rn\otimes\ell)=f_\ell(m,rn)\\ &f_\ell(m+m',n)=(m+m')\otimes(n\otimes\ell)=m\otimes(n\otimes_\ell)+m'\otimes(n\otimes\ell)=f_\ell(m,n)+f_\ell(m',n)\\&f_\ell(m,n+n')=m\otimes((n+n')\otimes\ell)=m\otimes(n\otimes\ell+n'\otimes\ell)=m\otimes(n\otimes\ell)+m\otimes(n'\otimes\ell)=f_\ell(m,n)+f_\ell(m,n')\\& f_\ell(am,n)=am\otimes(n\otimes\ell)=a(m\otimes(n\otimes\ell))=af_\ell(m,n)\end{aligned}$

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Thus, by the universal properties of tensor products we are afforded an $A$ -map $\widetilde{f}_\ell:M\otimes_R\to M\otimes_R(N\otimes_S L)$ with the property that $\widetilde{f}_\ell(m\otimes n)=m\otimes(n\otimes\ell)$ . We now define a map

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$g:(M\otimes_R N)\times L\to M\otimes_R(N\otimes_S L):(x,\ell)\mapsto \widetilde{f}_\ell(x)$

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We claim that $g$ is $S$ -biadditive and $B$ -linear. Indeed,

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$g(x+x',\ell)=\widetilde{f}_\ell(x+x')=\widetilde{f}_\ell(x)+\widetilde{f}_\ell(x')=g(x,\ell)+g(x',\ell)$

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To prove that $\widetilde{f}_{\ell}(xs)=g(xs,\ell)=g(x,s\ell)=\widetilde{f}_{s\ell}(x)$ we merely prove that for each fixed $s$ one has that the maps $x\to \widetilde{f}_{\ell}(xs)$ and $x\mapsto \widetilde{f}_{s\ell}(x)$ are equal. Since both of these maps are $R$ -biadditive, it suffices to check that they agree on simple tensors. To do this we note that $\widetilde{f}_\ell((m\otimes n)s)=\widetilde{f}_\ell(m\otimes (ns))=m\otimes(ns\otimes\ell)=m\otimes(n\otimes s\ell)$ and $\widetilde{f}_{s\ell}(m\otimes n)=m\otimes(n\otimes s\ell)$ . To prove that $\widetilde{f}_{\ell+\ell'}(x)=g(x,\ell+\ell')=g(x,\ell)+g(x,\ell')=\widetilde{f}_\ell(x)+\widetilde{f}_{\ell'}(x)$ we once again, show that $\widetilde{f}_{\ell+\ell'}$ and $\widetilde{f}_\ell+\widetilde{f}_{\ell'}$ agree on simple tensors, but this is trivial. Lastly, we want to show that $\widetilde{f}_{\ell b}(x)=g(x,\ell b)=g(x,\ell)g=\widetilde{f}_\ell(x)b$ but this merely requires, once again, checking on simple tensors, which is simple enough.

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From all of this we see that $g$ lifts to an $(A,B)$ -map $\widetilde{g}:(M\otimes_R N)\otimes_S L\to M\otimes_R (N\otimes_S L)$ such that $(m\otimes n)\otimes\ell=m\otimes(n\otimes\ell)$ . Now, it’s clear that we could have repeated this same procedure, in the opposite direction, to produce a map

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$q:M\otimes_R(N\otimes_S L)\to (M\otimes_R N)\otimes_S L$

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with the property that $m\otimes(n\otimes\ell)=(m\otimes n)\otimes\ell)$ . It’s easy to see then that $\widetilde{g}$ and $q$ are inverses, and so $\widetilde{g}$ is an isomorphism. $\blacksquare$

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Of course, by induction the above theorem allows us to freely associate any collection of tensor products and unambiguously denote a tensor product without parenthesising. Of course, the theoretical interpretation of what the tensor product of more than two modules means can be interpreted two-by-two, in the sense that $M\otimes_R (N\otimes_S L)$ $M\otimes_R (N\otimes_L)$ can be seen as the module for which all $R$ -biadditive maps $M\times N\otimes_S L$ factor through. But, as we can see we can “further factor” this to see that we are really looking at modules which are universal for maps out of $M\times N\times L$ which are additive in each variable (with the others fixed) and which the appropriate scalars switch between adjacent entries. To be more explicit, let $M_1,\cdots,M_n$ be $(A,R_1),(R_1,R_2),\cdots,(R_n,B)$ modules respectively. Then, we call an $n$ -additive $(A,B)$ -bimap to be a map $f:M_1\times\cdots\times M_n\to L$ such that $f$ is additive in each variable, and such that

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$f(x_1,\cdots,x_{j-1}r,x_j,\cdots,x_n)=f(x_1,\cdots,x_{j-1},rx_j,\cdots,x_n)$

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for all $j$ and $r\in R_j$ . We see then $M_1\otimes\cdots\otimes M_n$ , which in literal constructive terms is any associate of $M_1\otimes(M_2\otimes(\cdots(\otimes M_{n-1}\otimes M_n)))))$ , is universal with respect to $n$ -additive $(A,B)$ -bimaps out of $M_1\times\cdots\times M_n$ .

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References:

[1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. Module Theory. Clarendon, 1990. Print.

[4] Lang, Serge. Algebra. Reading, MA: Addison-Wesley Pub., 1965. Print.

[5] Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.

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January 5, 2012 - Posted by Alex Youcis | Algebra, Module Theory, Ring Theory | Algebra, associativity, Basic, commutativity, Identities, identity, Module, Module Theory, Modules, Properties, Ring, Ring Theory, Rings, Tensor, Tensor Product

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About me

My name is Alex Youcis. I am currently a junior at the University of Maryland, College park getting my bachelor’s in mathematics. Before I came to UMD I was a math major at Drexel University in Philadelphia, Pennsylvania. I was born and raised in Lititz, Pennsylvania, a small town near Lancaster.

As is probably clear, I really enjoy doing math. It’s a passion for me, and I don’t means this in the weekend-hobby sense, I can’t imagine doing anything else other than mathematics. That said, I’m not one of those space cadets one often sees wandering around math departments, trying to solve others problems, or perpetrating some other equally creepy shenanigans. No, while math is a gargantuan aspect of my life, there are many, many other things that interest me. Perhaps the most consuming of these is my love of music. I find it difficult to go even one day without my iPhone, or grooveshark.com If you want to say something to me in the car, you have about eight seconds before my iPhone is plugged in and we’ve set sail on a musical adventure. I have hundreds, upon hundreds of bands that I intensely love, but at the time of writing I can emphatically say that Regina Spektor is my favorite artist–the woman is a musical Gauss. I also love dogs, friends, conversations about things, craft beer and the occasional online comic.

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Current Course List:

Macroeconomics

History of the Medieval World

Homological Algebra–uses Weibel, Rotman, and the course notes by Schapira.

Graduate Algebra II (covers advanced field theory and the rudiments of homological algebra, representation theory, etc.)–uses Dummit and Foote, Lang, Hungerford and Serre

I am also sitting in on the following courses (when possible):

Graduate Complex Analysis-Uses Conway

Differential Topology (Taught by Novikov!)

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