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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64 views

$\chi : \text{Hom}(M,A) \otimes \text{Hom}(A,N) \to \text{Hom}(M,N) $

Let $M,N$ be $A$-modules. Consider the map. $$\chi : \text{Hom}(M,A) \otimes \text{Hom}(A,N) \to \text{Hom}(M,N) $$ $$f \otimes g \mapsto g \circ f $$ If $M,N$ are projective I do know that $\chi$...
1
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1answer
56 views

Can we embed $X'\otimes Y$ into the space of bounded, linear operators $X\to Y$?

Let $\mathbb F\in\left\{\mathbb C,\mathbb R\right\}$ $X$ and $Y$ be normed $\mathbb F$-vector spaces $X'$ denote the topological dual space of $X$ $\mathfrak L(X,Y)$ denote the space of bounded, ...
2
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1answer
29 views

Are $X'\otimes Y$ and $\mathfrak L(X,Y)$ isomorphic?

Let $\mathbb F\in\left\{\mathbb C,\mathbb R\right\}$ $X$ and $Y$ be normed $\mathbb F$-vector spaces $X'$ be the topological dual space of $X$ $\mathfrak L(X,Y)$ be the set of bounded, linear ...
1
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1answer
41 views

An example of tensor product

Let $$ \otimes:R\times R\rightarrow W $$ $$ f:R\times R\rightarrow R~~,~f(X,Y)=XY $$ $\otimes$ is tensor product, $W$ is a vector space, and $f$ is a bilinear may. As I know , we need to find a ...
4
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2answers
85 views

The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category $\...
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0answers
37 views

Minimal presentation of a tensor product of modules

Let $(R,m)$ be a local commutative noetherian ring. Suppose I have two modules $M$ and $N$ over $R$ given in terms of minimal presentations $$ M = \operatorname{coker}A, $$ and $$ N = \operatorname{...
0
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3answers
44 views

Is $Z/mZ\otimes Z \cong Z/mZ$?

I'm reading a Homological Algebra book that states this in some point without proving. I was trying to prove it and it seems to me that the first module is infinite and the second is not.
1
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1answer
12 views

stability of $K_0$ ($C^*$-algebras), question about the tensor product $K(H)\otimes A$.

I have a small question about stability of $K_0$: If $A$ is a $C^*$-algebra and $H$ is a separable infinite dimensional Hilbert space then $$K_0(A)\cong K_0(K(H)\otimes A),$$ where $K(H)$ denotes the ...
0
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1answer
31 views

Kernel of the map given by $j(m)=1\otimes m$ is in torsion module

Let $R$ be an integral domain, $K$ its field of fractions and $M$ an $R$-module. I want to show that the kernel of the map $j:M\rightarrow K\otimes M, m\mapsto 1\otimes m$ is contained in the torsion ...
0
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0answers
15 views

Linear independence in a tensor product space

Consider two N-dimensional vector spaces $V$ and $W$ over the same field and its tensor product $V\otimes W$. Let $\{v_i\}_{i=1}^N$ and $\{w_i\}_{i=1}^N$ be bases in $V$ and $W$. I want to show that ...
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0answers
36 views

An algebra isomorphism between $\mathbb{C}\otimes\mathbb{C}$ and $\mathbb{C}\oplus\mathbb{C}$

Considering $\mathbb{C}$ as an algebra over $\mathbb{R}$, it is easy to find a vector space isomorphism between $\mathbb{C}\otimes\mathbb{C}$ and $\mathbb{C}\oplus\mathbb{C}$. I am struggling to ...
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0answers
17 views

Representation decomposition over $GL_2(\mathbb{C})$

I have found that $Sym^2(V) \otimes Sym^2(V)$ can be decomposed over the special linear group as follows: $Sym^2(V) \otimes Sym^2(V) \simeq Sym^4(V) \oplus Sym^2(V) \oplus 1$ This is done using the ...
3
votes
2answers
68 views

For $R$-modules M, is $M\cong R^{\oplus n}\otimes_RM\cong M^{\oplus n}$?

I wanted to explicitly give a bilinear map $R^{\oplus n}\times M\longrightarrow M^{\oplus n}$ when trying to prove that $R^{\oplus n}\otimes_RM\cong M^{\oplus n}$ and ended up with $(r_1,...,r_n,m)\...
0
votes
2answers
48 views

For all $R$-modules $N$, $R\otimes_R N\cong N$

Why is this so? This statement is from Aluffi's book Algebra: Chapter 0 and seemingly so trivial that it deserves no proof. So working with the universal property of tensor products, how exactly does ...
0
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2answers
33 views

Representing a linear transformation as a tensor

I understand that a linear transformation from a vector space $V$ to a vector space $W$ is a rank-$2$ tensor. What I would like some help with is how exactly to represent specific linear ...
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0answers
28 views

Does $\phi: A \otimes \mathbb{Q} \to B \otimes \mathbb{Q}$ surj. imply that for $b \in B$, $b = n \phi(a)$ for some $n \in \mathbb{Z}$, $a \in A$?

Let $A$ and $B$ be abelian groups. Suppose that we have a morphism $\phi: A \to B$ and that $\phi \otimes_\mathbb{Z} \mathbb{Q}: A \otimes_\mathbb{Z} \mathbb{Q} \to B \otimes_\mathbb{Z} \mathbb{Q}$ is ...
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3answers
98 views

$\mathbb{Z}[i]\otimes_{\mathbb{Z}}\mathbb{R}$ is isomorphic to the complex numbers

I am new to tensor poducts (of modules over a commutative ring with identity) and need to understand the following example to continue with the actual exercises in my material. Namely, I need to ...
1
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1answer
33 views

Prove the image of basis elements is linearly independent

I was wondering if someone could give me a quick proof or counterexample to the following statement. Let $f:V \rightarrow W$ be a linear map between finite dimensional vector spaces $V$ and $W$, both ...
0
votes
1answer
42 views

Components of the metric tensor and its inverse

Let $g$ denote a metric tensor. Then Wald writes (in his book on general relativity): "The inverse of $g_{ab}$...is a tensor of type $(2,0)$ and could be denoted as $(g^{-1})^{ab}$. It is convenient, ...
0
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1answer
39 views

Tensor Product of $\mathbb{Q}[\sqrt{2}]$.

How can one show that $\mathbb{Q}[\sqrt{2}] \otimes_{\mathbb{Q}[\sqrt{2}]} \mathbb{Q}[\sqrt{2}] \simeq \mathbb{Q}[\sqrt{2}]$ (which is a $2$ dimension vector space over $\mathbb{Q}$) and $\mathbb{Q}[\...
0
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1answer
79 views

Confusion with abstract tensor notation

I am currently going through the abstract tensor notation in Wald's "General Relativity". I understand the purpose of it, but I need help understanding some of the conventions and definitions. So, ...
3
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2answers
30 views

why is $f\otimes g:A\otimes_{\min}C\to B \otimes_{\min} D$ injective?

If $f:A\to B$, $g:C\to D$ are injective $\ast$-homomorphisms between $C^*$-algebras $A, B, C, D$, is the induced map on the spatial tensor product $$f\otimes g:A\otimes_{\min}C\to B \otimes_{\min} D$$...
2
votes
1answer
44 views

Tensor product of dual vectors and vectors

I am reading "General Relativity" by Wald. At first he defines a tensor of type $(k,l)$ to be a multilinear map $T: V^* \times \cdots \times V^* \times V \times \cdots \times V \rightarrow \mathbb{R}$,...
2
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1answer
28 views

Tensor product with vectors

I just started reading Wald's "General Relativity" and I am on his section regarding tensors. He defines the outer product as an operation on tensors of type of $(k,l)$ and $(k', l')$ which gives a ...
1
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1answer
26 views

Foiling two tensor products

I have this problem in exterior algebra where I have a function B and B is defined in the following ways. $$ B\left( \left( \begin{array}{c} u \\ v \\ w \end{array} \right), \left( \begin{array}{c} x\\...
2
votes
0answers
48 views

Irreducible components of tensor product representations.

Let $(\rho,V)$ be an irreducible representation of a finite group $G$, and let $W$ be a vector space. Then clearly $(\rho\otimes\text{Id}_{W},V\otimes W)$ is also a representation of $G$. I would like ...
2
votes
1answer
15 views

Dipole-Coupling Tensor: Electrostatic Dipole Moments

I've been struggling with this problem today. Here's an image of the question I'm attempting to answer. I'm relatively new to tensor algebra (I've been studying it for about a week or two), and I've ...
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0answers
32 views

Minimality of the Spatial C*-Norm

Given C*-Algebras $A,B$ and $x \in A \otimes_* B$ show that: (a) If $(\tau \otimes_* \rho)(x)=0$ for all $\tau \in A_+^*$ and $\rho \in B_+^*$, then $x=0$. PS: $\tau \in A_+^*$ means that $\tau$ is ...
2
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0answers
29 views

What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
2
votes
1answer
39 views

Basic question about the metric tensor

So I am reading Barrett O'Neil's Elementary Differential Geometry, which defines the metric tensor $g_p$ on a surface $M$ to be a bilinear symmetric positive-definite function on the set of ordered ...
2
votes
1answer
59 views

About the $1$ of ring

I could not find neither a proof nor a counterexample, can anyone solve this? Let $A$ be a finite dimensional $k$-algebra. (It not necessarily has $1$.) If $$\mu:A\otimes A \rightarrow A,\ x\...
2
votes
2answers
31 views

Elementary tensors

Let $G,H$ be $R$-modules, and $G \otimes H$ be it's tensor product. I can't prove it and I suspect it's false that any element $\tau \in G \otimes H$ can be written as $\tau = g \otimes h$ for some $...
0
votes
2answers
43 views

$M \otimes_\mathbb{Z} N \cong \mathbb{Z}$ for cyclic modules $M,N$?

Is it true that $M \otimes_\mathbb{Z} N \cong \mathbb{Z}$ (considering $\mathbb{Z}$ as a $\mathbb{Z}$-module) if $M,N$ are cyclic $\mathbb{Z}$-modules (i. e. generated by one element)? I would say ...
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votes
2answers
39 views

Showing $R\otimes M \cong M$ for $R$-modules $R,M$

How to see that $R \otimes M \cong M$ if $R$ and $M$ are $R$-modules (with $R$ being a commutative ring with unity)? I thought about defining $f: R\otimes M \rightarrow M$ by $(r,m) \mapsto rm$. ...
1
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0answers
16 views

Rank of a locally free $\mathbb Z[G]$- module

Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$. Here $\mathbb Z_p[G]=\mathbb Z_p\otimes_\mathbb ...
1
vote
1answer
21 views

What are the main differences between Tensor Products of Vector Space vs Modules vs Algebras

What I understand so far: I understand the definitions of vector space, modules, algebras. I am also acquainted with basic properties of Tensor Product of Vector spaces, especially the "universal ...
0
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1answer
18 views

Query on Tensor Product of Quaternion Algebras

In this proof of isomorphism of tensor product of quaternion algebras I have two queries: 1) Why does one need to "check $xy=yx$ for all $x\in C$ and $y\in D$? 2) How is the product $CD$ defined? ...
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0answers
15 views

How to make $S\otimes_R M$ into a left $S[G]$-module

I have a basic question on tensor products which might have been asked before-sorry if this is the case. Let $G$ be a finite group and $R\subset S$ any (unital) ring extension. Let $M$ be a finitely ...
0
votes
1answer
39 views

When is tensor product isomorphic to product?

Let $A$ and $B$ be algebras. When do we have $A\otimes B\cong AB$, where $$AB=\{\sum a_ib_i\mid a_i\in A, b_i\in B\}$$ Is commutativity $ab=ba$ for $a\in A$, $b\in B$ a sufficient condition? Thanks ...
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0answers
31 views

Is $l^2 \cong l^2 \otimes l^2 \cong l^2 \oplus l^2$?

I'm trying to learn about tensor products of Hilbert spaces and started to wonder if $l^2 \cong l^2 \otimes l^2 \cong l^2 \oplus l^2$? If $(e_n)$ denotes the standard basis, in the first case, it ...
3
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0answers
47 views

Tensor Product of Modules over R and a subring S

If we have a commutative Ring R with a subring S and two R-Modules (or S-Modules). What can we say about the correlations between $$ M \otimes_R N \quad and \quad M \otimes_S N \quad ?$$ Wikipedia ...
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0answers
27 views

vanishing tensor product, geometric meaning

We can derive the following property directly. " $\mathbb{Z}_m \otimes_{\mathbb{Z}} \mathbb{Z}_n = O$. where $m,n$ are relatively prime. But, this is just a simple computation. I want to know the ...
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1answer
70 views

A question about tensor product and multiplication?

Let $k$ be a commutative ring, $A$ be an algebra over $k$. The tensor product $A\otimes A$ is over $k$. If $\sum_{i} a_i \otimes b_i=\sum_j c_j\otimes d_j$, where $a_i,b_i,c_j,d_j\in A$, I wonder if $\...
0
votes
1answer
14 views

Suppose $N \cong R^n $ then for every non-zero $R$ module $M$ show that $M \otimes N$ can be written uniquely by $\sum m_i \otimes e_i $

Suppose $N \cong R^n $ be free $R$ module of rank $n$ with basis $\{e_1,...,e_n\}$ and $R$ is commutative then for every non-zero $R$ module $M$ show that $M \otimes N$ can be written uniquely by $\...
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votes
2answers
18 views

If $G$ be a abelian Then $G\otimes_{\mathbb{Z}}\mathbb{Z}_m\simeq G/mG$ [closed]

Let $G$ be a abelian group. Then $G\otimes_{\mathbb{Z}}\mathbb{Z}_m\simeq G/mG$ suggestion please.
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0answers
9 views

Proof of a vector identity for adjoint heat transfer

I have a domain $\Omega$ and the expression below. S, T and $\alpha$ are scalar fields. $$\int_\Omega \frac{\partial}{\partial \alpha}T\nabla\cdot(\kappa(\alpha)\nabla S)\;d\Omega$$ I have read in a ...
1
vote
1answer
29 views

How do outer products differ from tensor products?

From my naive understanding, an outer product appears to be a particular case of a tensor product, but applied to the "simple" case of tensor products of vectors. However, in quantum mechanics there ...
2
votes
2answers
72 views

Fibers of extension of scalars to algebraic closure of an affine variety is of dimension 0.

Let $X$ be an affine variety over $k$ and $\overline{k}$ be the algebraic closure of $k$. Then the projection morphism $X_\overline{k} = X \times _k \overline{k} \to X$ has dimension 0 fibers. Say $...
1
vote
2answers
88 views

How does extension of restriction of $M$ relate to $M$?

Let $A,B$ be rings, $f:B\to A$ be a ring homomorphism, and $M$ be an $A$-module. We can view $M$ as a $B$-module via restriction, and we may then extend the restriction of $M$ to an $A$-module by ...
2
votes
1answer
13 views

Is $p\in B(\mathbb{C}^4)$ a s.o.t-limit of a sequence $(a_n\otimes b_n)_{n\in\mathbb{N}}\subseteq B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)$?

Let $L(H)$ the bounded linear operators on a hilbert space $H$. I proved that the inclusion $$i:B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)\hookrightarrow B(\mathbb{C}^4)$$ is not surjective: take $p=\...