Tensor products allow us to build "linear" objects from "multilinear" ones. It can refer to: basic ones from linear algebra/module theory, or more sophisticated versions from differential/algebraic geometry (bundles/sheaves), functional analysis (Hilbert/Banach/locally convex spaces), or in their ...

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221 views

On the regularity of the Laplace equations and tensor products and such

To start with, let me apologize for my ignorance as I know next to nothing about partial differential equations. My question is about the tensor product of Banach spaces but actually I do not ...
8
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2answers
329 views

Tensor product of monoids and arbitrary algebraic structures

Let $C$ be the category of algebraic structures of a certain type and let us denote by $|~|$ the underlying functor $C \to \mathsf{Set}$. For $M,N \in C$ we have a functor $\mathrm{BiHom}(M,N;-) : C ...
2
votes
2answers
136 views

Which graph products are categorical products?

There is a whole bunch of definitions of graph products, but only one of them - the tensor product - is the categorical product in the (standard) category of graphs with graph homomorphisms. I'd ...
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1answer
584 views

Tensor product and Kronecker Product

Is there any difference between tensor product and Kronecker Product?
2
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2answers
173 views

Does every element of tensor product look like this?

If $V\otimes W$ is the tensor product of vector spaces V and W, I know that for any basis $(v_i)_{i\in I}$ of V and $(w_j)_{j\in J}$ of W, $(v_i\otimes w_j)_{i\in I,j\in J}$ is a basis of $V\otimes ...
4
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1answer
293 views

Ideals of the tensor product $R\otimes_{k} S$?

Let $R$ and $S$ be commutative rings over a field $k$. Let $I$ be an ideal of the tensor ring $R\otimes_{k} S$. It is true that there exist ideals $I_{1}$ and $I_{2}$ of $R$ and $S$ respectively such ...
6
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0answers
128 views

Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
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1answer
48 views

If $\rho: G \to GL(V)$ is a representation with sub-representation $\tau$, is $\tau^{\otimes n}$ a subrepresenation of $\rho^{\otimes n}$?

I'm working over an algebraically closed field of characteristic $p>0$ so I'm not assuming that $\tau$ is a direct summand of $\rho$. I think I can prove this by looking at the Kronecker product of ...
2
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1answer
270 views

Endomorphism algebra of the tensor product of modules

Let $k$ be a commutative ring, and let $M$ and $N$ be $k$-modules. Let $\mathrm{End}(M) = \mathrm{Hom}_k (M,M)$ be the endomorphism algebra. Is it true that $\mathrm{End}(M) \otimes \mathrm{End}(N) ...
2
votes
1answer
129 views

Basis for a tensor product of group algebras

Let $G$ and $H$ be groups, and $R$ a commutative ring. Then elements of $RG$ look like finite sums $\sum\limits_{g\in G}r_g\,g$, and similarly for $RH$. So $RG$ and $RH$ are $R$-modules with bases $G$ ...
2
votes
1answer
148 views

Linearly disjoint vs. free field extensions

Consider two field extensions $K$ and $L$ of a common subfield $k$ and suppose $K$ and $L$ are both subfields of a field $\Omega$, algebraically closed. Lang defines $K$ and $L$ to be 'linearly ...
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280 views

Categorical definition of tensor product

It is standard to define the tensor product $M\otimes_R N$ of $R$-modules as a universal object of bilinear maps from $M\times N$. Now, suppose that $\mathscr{F}$, $\mathscr{G}$ are sheaves of ...
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1answer
126 views

Permutations of elements in a tensor product

Given a tensor product $A^{\otimes n}$ over a field $k$ (characteristic $\neq 2$) of $n$ copies of the $k$-algebra $A$, a premutation $\sigma \in S_n$ of order $2$ acts on the elements of $A^{\otimes ...
3
votes
1answer
211 views

Proving something is the basis of a quotient space

Let $k$ be a field which does not have characteristic 2. Let $M$ be the free $k$-vector space generated by two elements $\{ c, x \}$. Let $T(M)$ be the tensor algebra of $M$ and let $I$ be the ideal ...
5
votes
1answer
84 views

Is this a surjection of rings? What am I doing wrong?

Let $Z\newcommand{\df}{:=}\df\newcommand{\C}{\mathbb C}\C$ and $T\df\C^\times$. Then, the coordinate ring of $Z$ is $\C[z]$ and that of $T$ is $\C[t,t^{-1}]$. Consider another copy of $T$ with ...
3
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0answers
104 views

Defining entanglement in subspaces of tensor product

Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where ...
0
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1answer
46 views

Binomial/Tensor Identity

Let $k$ be a a field and consider the space $k[x] \otimes_k k[x]$. I would like to verify the equation $$ \sum_{k=0}^{m+n} {m+n \choose k} x^k \otimes x^{(n+m)-k}= \sum_{i=0}^n \sum_{j=0}^m{n \choose ...
5
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2answers
625 views

Universal property of tensor product [duplicate]

Possible Duplicate: Equality of two notions of tensor products over a commutative ring Let $A$ be a commutative ring. There are two common definitions of tensor product of two $A$-modules ...
4
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2answers
135 views

Tensor product of faithful modules

In commutative algebra, is it true that the tensor product of two faithful modules is a faithful module? I have written for myself a proof for the case of finitely generated modules over reduced ...
8
votes
1answer
304 views

Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. My questions are: How can we describe $M \otimes_R N$ explicitly? Well, I guess that it is a quotient of $R$ by a sum of ...
2
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0answers
146 views

A doubt about tensor product on Hilbert Spaces

An operator is a bounded (i.e., continuous) linear transformation between Hilbert spaces. Let $\mathcal{B}[\mathcal{H}]$ be the set of all operators in the Hilbert space $\mathcal{H}$. Let ...
3
votes
1answer
206 views

An $(R,S)$-bimodule is a left $R \otimes_k S^{\text{op}}$-module

Let $k$ be a commutative ring, and let $R,S$ be $k$-algebras. To me "$R$ is a $k$-algebra" means that $R$ is a $k$-module such that $a(rs)=(ar)s=r(as)$ for all $a\in k$ and $r,s \in R$. Let $M$ be a ...
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3answers
181 views

Tensor-commutative abelian groups

Say that an abelian group $A$ is tensor-commutative if the equality $x\otimes y=y\otimes x$ holds in $A\otimes_{\mathbb Z}A$ for all $x,y$ in $A$. The first question is somewhat vague: Question 1. ...
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4answers
611 views

Are bimodules over a commutative ring always modules?

Let $R$ be a commutative ring. It is true that every module over $R$ is an $(R,R)$-bimodule. Is the converse true? In other words is it possible that there is an $R$-module where left multiplication ...
9
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4answers
452 views

Equality of two notions of tensor products over a commutative ring

Let $R$ be a ring (not necessarily commutative), let $M$ be a right $R$-module and let $N$ be a left $R$-module. Then the tensor product $M \otimes_R N$ is an abelian group satisfying the universal ...
3
votes
2answers
171 views

Why do you need tensors of rank $>2$?

Question from someone just starting to study tensors (sorry if it's silly): So I understand (maybe?) that tensors are basically about coordinate transformations (and things that are invariant under ...
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2answers
1k views

Tensor product algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$

I want to understand the tensor product $\mathbb C$-algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$. Of course it must be isomorphic to $\mathbb{C}\times\mathbb{C}.$ How can one construct an ...
1
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0answers
42 views

Radial and angular part of the space of compactly supported smooth functions

On page 124 of Thaller's The Dirac equation the following space is mentioned : $$C^{\infty}_0(0, \infty)\otimes C^\infty(\mathbb{S}^2)\subset L^2(0, \infty)\otimes L^2(\mathbb{S}^2),$$ where the ...
1
vote
1answer
110 views

“Simplifying” an extension of scalars

Let $A$ be a commutative $\mathbb Z$-algebra and $M$ be a $\mathbb Z\oplus \mathbb Z$-module. Then $A\otimes_{\mathbb Z} M$ is an $A\oplus A$-module. Is it true that $(A\oplus A)\otimes_{\mathbb ...
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5answers
319 views

Question about proof of $A[X] \otimes_A A[Y] \cong A[X, Y] $

As far as I understand universal properties, one can prove $A[X] \otimes_A A[Y] \cong A[X, Y] $ where $A$ is a commutative unital ring in two ways: (i) by showing that $A[X,Y]$ satisfies the ...
3
votes
2answers
120 views

$B \otimes_A A[X] \cong B[X]$

Let $A$ be a subring of a commutative unital ring $B$. Can you tell me if my proof of the following claim is correct? Claim: $B \otimes_A A[X] \cong B[X]$ Proof: It's enough to show that $B[X]$ ...
2
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0answers
231 views

Completion and Tensor Product of Algebras

Let $A$ be a commutative ring with 1, $I$ an ideal in $A$, $B$ an $A$-algebra. I am trying to prove the following isomorphism of $A$-algebras: $$ \big( A^* \otimes _A B \big) ^* \cong B^* $$ "$^*$" ...
5
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0answers
286 views

understanding this differential operator on a tensor product

I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
22
votes
3answers
406 views

$\operatorname{Ann}(M\otimes_A N)=\operatorname{Ann}M+\operatorname{Ann}N$?

In the course of working on an exercise in Atiyah-MacDonald (exercise 3 on p. 31), I've come to the belief that, for $A$ an arbitrary commutative ring and $M,N$ arbitrary $A$-modules, ...
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0answers
153 views

Tensor Product over $\Bbb Q$

I am trying to find an example of rings $A\subset B$ and module $N$ over $B$ such that $N \otimes_A B$ over $A$ (by restriction of scalars on $N$) is not same as $N$ as $B$-modules. I think I have ...
3
votes
1answer
260 views

As a $\mathbb{Z}$-module, is $\mathbb{R}\otimes_\mathbb{Z} \mathbb{R}$ isomorphic to $\mathbb{R}$?

Is it true that $\mathbb{R}\otimes_\mathbb{Z} \mathbb{R}$ (the tensor product of $\mathbb{R}$ and $\mathbb{R}$ over $\mathbb{Z}$) is not isomorphic to $\mathbb{R}$ as a $\mathbb{Z}$-module? Please ...
0
votes
1answer
370 views

Tensor contraction with multiple indices

I think I missed a rule somewhere, because I can contract the following expression in multiple ways. $\epsilon^{\alpha \beta} \sigma_{\dot{\alpha} \alpha} \epsilon^{\dot{\alpha} \dot{\beta}} = ...
8
votes
2answers
594 views

Prove tensor product of two multilinear forms is commutative only if one of them is zero

Prove $L \otimes M = M \otimes L$ only if either $L=0$ or $M=0$ I saw this statement on Linear Algebra (2ed) written by Hoffman and Kunze. I can't figure out how to prove it. The multilinear forms ...
2
votes
1answer
118 views

Eigenvalue For A 2-Fold Tensor

This is problem 4 from page 258 of Curtis's Linear Algebra: An Introductory Approach. I seem to be having trouble understanding something needed to solve the problem, which reads Suppose A and B ...
1
vote
1answer
299 views

Tensor product of sets

The cartesian product of two sets $A$ and $B$ can be seen as a tensor product. Are there examples for the tensor product of two sets $A$ and $B$ other than the usual cartesian product ? The context ...
4
votes
1answer
250 views

How do I do “calculations” with tensors?

I just started to read about tensor products and tensors and I understand that a tensor product $V \otimes W$ is a space used to replace bilinear maps $V \times W \to U$ with linear maps $V \otimes W ...
3
votes
2answers
522 views

Matrix Representation of the Tensor Product of Linear Maps

I'm trying to work out some examples of applying the tensor product in some concrete cases to get a better understanding of it. Within this context, let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a ...
2
votes
1answer
293 views

Proving $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without the universal property

Let $F$ be a commutative field, and let $U$, $V$, and $W$ be finite dimensional vector spaces over $F$. How can one prove $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without using the ...
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vote
1answer
73 views

Graded Vector Spaces and the Interchange Law

I'm a little confused about how to correctly interchange factors in tensor products on graded vector spaces. In particular let $V:= \bigoplus_{n \in \mathbb{N}} V_n$ be a $\mathbb{N}$-graded vector ...
3
votes
2answers
253 views

$C \otimes A \cong C \otimes B$ does not imply $A \cong B$

Let $R$ be a commutative unital ring and let $M$ be an $R$-module and let $S$ be a multiplicative subset of $R$. Today I proved both of the following: $$ S^{-1} R\otimes_R S^{-1}M \cong S^{-1} M$$ ...
0
votes
2answers
231 views

Proof that $A \otimes B \cong B \otimes A$ [duplicate]

Possible Duplicate: There exists a unique isomorphism $M \otimes N \to N \otimes M$ I want to show that for Abelian groups $A$ and $B$ that the tensor product $A \otimes B$ is isomorphic to ...
3
votes
2answers
372 views

Tensor product of $R$-algebras

Let $f: R \to S$ and $g: R \to T$ be two $R$-algebras. To show that $S \otimes_R T$ is an $R$-algebra I need to define a ring structure (multiplication) on it and a ring homomorphism $h : R \to S ...
3
votes
3answers
323 views

$R \otimes_R M \cong M$

Let $R$ be a commutative unital ring and $M$ an $R$-module. I'm trying to prove $R \otimes_R M \cong M$ but I'm stuck. If $(R \otimes M, b)$ is the tensor product then I thought I could construct an ...
2
votes
1answer
133 views

Example computation of $\operatorname{Tor_i}{(M,N)}$

Let $M = \mathbb Z / 284 \mathbb Z$ and $N = \mathbb Z / 2 \mathbb Z$. Can you tell me if my computation of $\operatorname{Tor_i}{(M,N)}$ is correct: (i) First we want a projective resolution of ...
2
votes
1answer
176 views

Question about a proof of $f$ injective $\implies$ $f \otimes \operatorname{id}$ injective

I'd like to prove (i) implies (ii) where: (i) Whenever $f: A \to B$ is injective and $A,B$ are finitely generated then $f \otimes \operatorname{id}: A \otimes P \to B \otimes P$ is injective. (ii) ...