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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8
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1answer
439 views

Tensor product of Noetherian modules

Let $L$ and $N$ be two Noetherian $R$-modules ($R$ is a commutative ring with 1). Is it right that $L \otimes_R N$ is Noetherian? If not, what additional conditions on $L$ and $N$ are required in ...
2
votes
1answer
245 views

Extension of scalars

Let $A\rightarrow B$ be a ring homomorphism, $M$ and $N$ - modules over $A$. How to prove, that $$ (M \otimes_A N) \otimes _A B = (M \otimes_A B) \otimes_B (N\otimes_A B) $$ as $B$-modules in the ...
3
votes
0answers
288 views

Convolution theorem in 3D

Suppose to have a 3-dimensional discrete grid. I would like to convolve it with a 3-dimensional tensor (a 3x3x3 "cube"), applying the convolution theorem. Hence, I should apply a Fourier transform to ...
3
votes
2answers
274 views

Tensor product of fields

Suppose $D$ is a finite dimensional skew field over the field $K$. Futher, take $x \in D\setminus K$ and let $L=K(x)$. My question: is $D\otimes_K L$ a field? I think not. However I can't seem to ...
0
votes
2answers
138 views

How to compute $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$? [duplicate]

Possible Duplicate: Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ How to compute $\mathbb{Z}/m\mathbb{Z} ...
3
votes
2answers
86 views

Proving $\otimes_{i=1}^{i=n}\mathcal{B}_{X_{i}}=\mathcal{B}_{\Pi X_{i}}$

I am given the following exercise: Let $X_{\alpha}$ be a measureable space with $\sigma$-algebra $M_{\alpha}$ , mark $X\triangleq{\displaystyle \prod_{\alpha\in A}X_{\alpha}}$ and ...
3
votes
1answer
484 views

Is the tensor product of two torsion-free modules always non-zero?

Let $R$ be a commutative domain and let $M$ and $N$ be torsion-free $R$ modules. I would like to know whether or not $M\otimes_{R}{N}$ is always non-zero. Now, I know this is true in the finitely ...
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0answers
128 views

Explicit computations of tensor and wedge product

Let $f\colon K^3\to K^3$ be a map in Jordan canonical form having matrix $$ f=\begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0& -1 \end{bmatrix} $$ What is $f\otimes f$? What ...
1
vote
0answers
157 views

Relating tensor product with exterior power and symmetric product

Note: I have updated my work section of this since I asked the question yesterday to reflect my understanding to this point. Let $V$ and $W$ be two vector spaces. We denote by $S_2V$ the second ...
1
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0answers
67 views

Isomorphism between tensor product and set of all tensors

I'm new to tensor and quite confused. First of all, could anyone provide any friendly reference about tensors which explains the idea behind those boring-looking definition? Then is my problem. Let ...
2
votes
1answer
146 views

A question regarding tensor product and isogenies of elliptic curves

Let $E_1$ and $E_2$ be elliptic curves and $T_l(E_i)\cong \mathbb{Z}_l \oplus \mathbb{Z}_l$ the $l$-adic Tate module. Given $ \varphi \in Hom(E_1,E_2)$ this induces $\varphi_l \in ...
6
votes
0answers
152 views

Mnemonic device for relationships between Hom and Tensor

Probably this is a stupid question, but nevertheless... Let $A$, $B$, $C$ and $D$ be rings, and $M$, $N$ and $K$ be appropriate bimodules between them. There are extremely well-known canonical ...
1
vote
1answer
71 views

$\mathbb{Q}$-dimension of f. g. $\mathbb{Q}[\mathbb{Z}/p^l]$-modules.

My question arises from the previous question Let $M$ be a finitely generated $\mathbb{Q}[\mathbb{Z}/p^l]$-module, where $p$ is a prime number. Is it true that \begin{equation}\dim_\mathbb{Q} ...
2
votes
3answers
222 views

Is it true that $\mathbb{C}\otimes_\mathbb{Z}\mathbb{C}=\mathbb{C}$?

Is it true that $\mathbb{C}\otimes_\mathbb{Z}\mathbb{C}=\mathbb{C}$? This is not the homework. I just want to know what it is and I cannot find it any text I have.
1
vote
1answer
54 views

Is this valid reasoning? Exactness question

I am wondering if this is valid reasoning: Let $N$ and $N'$ be $R$-modules. I know that $0\rightarrow N\rightarrow N'$ is exact and want to show for a given multiplicative subset $S$ and that ...
2
votes
1answer
96 views

Simple Tensor Product Question about Well-definedness

If I want to define a homomorphism, $f$, from $A\otimes_R B$ into some $R$ module $M$. If I defined it on simple tensors $a\otimes b$ what are the conditions I need to check to make this is well ...
6
votes
1answer
900 views

Postitive Definiteness of Kronecker Product of Two Positive Definite Matrix

Let $A$ and $B$ both be positive definite matrices. How do I show that their Kronecker product is also positive definite? I know we can use the fact that the eigenvalues of the Kronecker product is ...
1
vote
1answer
132 views

Conceptual explanation for $tr(A\otimes B)=tr(A)tr(B)$?

Let $A\otimes B$ denote the tensor product of two matrices, $A$ and $B$. I can show the trace of it is the same as the product of the traces of $A$ and $B$, which follows from computation. Is there ...
3
votes
1answer
224 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
votes
2answers
336 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 ...
3
votes
2answers
141 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 ...
5
votes
1answer
595 views

Tensor product and Kronecker Product

Is there any difference between tensor product and Kronecker Product?
2
votes
2answers
176 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 ...
5
votes
1answer
303 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
votes
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 ...
0
votes
1answer
49 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
votes
1answer
282 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
131 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
156 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 ...
5
votes
1answer
286 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 ...
1
vote
1answer
129 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
212 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
votes
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
votes
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
votes
2answers
635 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
votes
2answers
140 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
307 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
votes
0answers
147 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
207 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 ...
8
votes
3answers
182 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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votes
4answers
622 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
votes
4answers
468 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
174 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 ...
11
votes
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
vote
0answers
43 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
111 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 ...
6
votes
5answers
321 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
121 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
votes
0answers
235 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: $$ \big( A^* \otimes _A B \big) ^* \cong B^* $$ "$^*$" ...