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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61 views

Mistake in the proof that a domain is flat as a module over any subring

Where is the mistake in the following argument? I feel that there has to be one, for example by the very existence of this article. Let $R$ be an integral domain and $S \subseteq R$ be a subring ...
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0answers
58 views

Working with $\bigotimes_{\mathbb{Z}_X}$?

What is the meaning of the tensor product sign in a formula such as $$ A \bigotimes_{\mathbb{Z}_X} B $$ where X is a real manifold of arbitrary dimension? B is a orientation sheaf (ultimately a matrix ...
4
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1answer
88 views

Determining whether a certain element in a tensor product is zero

Let $I = (x,y) \subset k[x,y]$, where $k$ is a field. Prove that a) $x \otimes y - y \otimes x =0$ inside $k[x,y] \otimes_{k[x,y]} k[x,y]$. b) $x \otimes y - y \otimes x \not= 0$ inside ...
5
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1answer
583 views

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product ...
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1answer
52 views

problem with permutation symbol

Given $\varepsilon_{ijk}T_{ij} = 0$. Prove that $T_{ij} = T_{ji}$ I can prove it by expanding summation. It is very cumbersome. May be there is more compact solution?
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1answer
181 views

Tensor product, wedge product, Hodge product, dyad, or what?

Suppose I have two vectors $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$ in $\mathbb{R}^3$. I can regard $\mathbf{u}$ as a $3 \times 1$ matrix, and $\mathbf{v}$ as a $1 \times 3$ ...
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2answers
185 views

Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV

Let V be a real n-dimensional vector space. Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV. Note that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is a real vector space and is ...
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103 views

Isomorphism of Tensor Product over a Group Ring.

Let $\mathbb{Q}$ be the rationals and $\mathbb{Z}$ integers. Let further $p$ be prime and $t\in \mathbb{Z}$ such that $p \mid t$. Then $\mathbb{Z}_{(p)}$ is the local ring. Let $G < H$ be groups, ...
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0answers
69 views

Complete reducibility of tensor product

Let $L$ be a Lie algebra (over a algebraically closed field, not sure if it is relevant). If $V$ and $W$ are two completely reducible $L$-modules, can anyone give a hint on how to show that $V\otimes ...
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27 views

On the Tensor product of spaces

Let $\Bbb F$ be real field. Partition surface of a sphere $\Bbb S_n\in\Bbb F^n$ of radius $\sqrt{n}$ at an arbitrary center into $k$ equal divisions each holding $m=\frac{n}{k}$ orthonormal vectors ...
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641 views

What is a Tensor Product?

If you were to explain the concept of a tensor product to an undergraduate(post linear algebra), how would you do so? I would like to hear your definition, your take, on the definition of a tensor ...
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1answer
118 views

Component-free formula for the determinant of a tensor

Consider a unit vector $\mathbf{a}\in\mathbb{R}^3$ and the associated second-order tensor $\mathbb{A}=\mathbf{a}\otimes\mathbf{a}$. Is there a component-free formula for the determinant of this ...
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1answer
142 views

Defining an ideal in the tensor algebra

In the wikipedia article about exterior algebra: The exterior algebra $Λ(V)$ over a vector space $V$ over a field $K$ is defined as the Quotient algebra of the tensor algebra by the two-sided ...
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77 views

Extension of multiplication to the tensor algebra.

In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction We construct $T(V)$ as the direct sum of vector spaces $T^kV$ for $k=0,1,2,…$ $$ T(V)= ...
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2answers
63 views

Noncommutativity of tensor algebra

My question is simple. Let $M$ be an $A$-module and let $T(M)$ be its tensor algebra. I saw that it is noncommutative in general... but I can' understand this fact... I think that by ...
6
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2answers
227 views

Idempotents in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$

Letting $\mathbb{Z}[i]=\left\{a+bi:a,b \in \mathbb{Z} \right\}$ be the ring of Gaussian integers, how many idempotents are there in $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$? I came ...
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56 views

Comparison between ordinary products and tensor products

My question is quite simple, I didn't understand this comparison, why the first lines in this example implies the red ones: Thanks
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730 views

An 'obvious' property of algebraic integers?

I am looking at the book A Brief Guide to Algebraic Number Theory by H. P. F. Swinnerton-Dyer. I like the section on page 1 'the ring of integers' as it gives a motivation for choosing which elements ...
2
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1answer
63 views

Tensor product of modules

Let $R$ be a polynomial ring over $\mathbb{C}$. Let $R_1=R/I$ for some ideal $I \subset R$. Let $M_1, M_2$ be $R_1$-modules. So, they are $R$-modules as well. Is it true that $M_1 \otimes_{R_1} M_2 ...
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1answer
88 views

Decomposable Tensors over Rings

Suppose $R$ is a commutative ring and $M$ is a $R$-module. Then we can define the tensor product $M\otimes_R M$ and more generally the $k$-fold tensor powers $\otimes_R^kM$ for any $k\in\mathbb{N}$, ...
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3answers
144 views

When does there exist a commutative ring $C$ that contains rings $A$ and $B$ as a subring?

The statement I'm trying to prove is the following: Let $A$ and $B$ be commutative rings, both of characteristic $0$. Then there exists a commutative ring $C$ that contains both $A$ and $B$ as ...
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2answers
260 views

Tensor Product of the fraction field of the Gaussian Integers

Let $\mathbb{Z}[i]=\left\{a+bi:a,b \in \mathbb{Z}\right\}$ be the ring of Gaussian integers $(i^{2}=-1)$ and let $\mathbb{Q}[i]=\left\{a+bi:a,b \in \mathbb{Q}\right\}$ be its fraction field. 1) ...
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70 views

Tensor products of weakly compact sets

Let $X$ and $Y$ be Banach spaces. Denote by $X\otimes_\varepsilon Y$ the injective tensor product of $X$ and $Y$. Also, let $A\subset X, B\subset Y$ be any sets. Set $A\otimes B = \{a\otimes b\colon ...
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1answer
95 views

Tensor product of fields and homomorphisms

Question I'm pretty confused about changing the base field of a tensor product. For example, if $E/F,L/F$ are field extensions, the $E\otimes_F L$ could be seen as a vector space over $E$, by ...
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1answer
125 views

What is meant by $A\otimes \mathbb{Q}$ where $A$ is a torsion free abelian group?

I'm not sure what is meant by, $A\otimes \mathbb{Q}$ where $A$ is a torsion free abelian group? I would be helpful if someone can tell me why the tensor can be taken, as I only know the vector space ...
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2answers
212 views

About the injection $M \hookrightarrow \mathbb Q \otimes_{\mathbb Z} M$.

I want to prove that every abelian group can be embedded in a divisible abelian group. So I tried $M \rightarrow \mathbb Q \otimes_{\mathbb Z} M, m \mapsto 1 \otimes m$. It is obvious that $\mathbb Q ...
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1answer
483 views

Tensor product and Direct product.

Is the direct product of two vector spaces (over the same field) just the tensor product of two vector spaces over $\mathbb{Z}$? Am I right in thinking that essentially we would use the tensor ...
4
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1answer
96 views

Is injectivity of algebras preserved by tensor products?

Suppose $R' \subset R$, $S'\subset S$ are inclusion of $k$-algebras. Does it hold that $R'\otimes_kS' \rightarrow R \otimes_k S$ is injective ? I know there're counterexamples for modules, but ...
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274 views

Example of rings of the same positive characteristic that do not embed into their tensor product?

I'm overcoming my fear of tensor products, and the following exercise got me wondering: Give an example of commutative rings $A$ and $B$ with $\operatorname{char}A=\operatorname{char}B$ such that ...
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64 views

How to think about the object $A\otimes_kk'$.

Let $k'/k$ be a finite extension of fields, and let $A$ be a finitely generated commutative $k'$-algebra. Through $k\hookrightarrow k'$ we can consider $A$ to be a finitely generated commutative ...
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269 views

Help me understand the tensor product

I have several books and other literature that define the tensor product, but I understand none of them. Since this really concerns one topic, namely understanding the construction of and arithmetic ...
3
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0answers
172 views

When is the canonical extension of scalars map $M\to S\otimes_RM$ injective?

Let $\alpha:R\to S$ be a map of unital rings, and let $M$ be an $R$-module. We have a canonical map of $R$-modules: ...
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2answers
812 views

Identity tensor as a tensor product of two vectors

Any second order tensor in a given basis can be expressed as a matrix. Also, as any second order tensor can be expressed a tensor product of two first order tensors (or vectors), I would like to find ...
3
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3answers
299 views

Direct isomorphism between $\mathbb{C} \otimes_\mathbb{R} \mathbb{C}$ and direct sum.

I "know" that $\mathbb{C} \otimes_\mathbb{R} \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}$ as rings, but I don't really know it, what I mean with this is that I don't know any explicit isomorphism ...
3
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1answer
70 views

Help with notation in second order tensor.

I have seen recently this notation: $$F=F_{ij}\,e_i\otimes e_j $$ Where $F_{ij}=\frac{\partial x_i}{\partial X_j}$ is the tensor matrix: $$ F=\left[ {\begin{array}{cc} ...
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0answers
52 views

Is any tensor sum of decomposables?

Let $V$ be (not necessary finite dimensional) $\mathbb{R}$-vector space and $TV$ is tensor power. (The vector space underlaying the tensor algebra) Is any tensor $v\in TV$ a linear combination of ...
3
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1answer
78 views

Show that $\mathbb{Z}[G\times H]\cong \mathbb{Z}[G]\otimes_\mathbb Z \mathbb{Z}[H]$

This is exercise 5.5 of Rotman's 'Introduction to Homological Algebra'. Let $G$ and $H$ be groups. We must show that $$\mathbb{Z}[G\times H]\cong \mathbb{Z}[G]\otimes_\mathbb Z \mathbb{Z}[H]$$ as ...
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1answer
121 views

Linear System where Coefficient Matrix is a Kronecker Product

I have a system of linear equations where the coefficient matrix and right hand side is given by a Kronecker product: $(A_1 \otimes A_2) u = f_1 \otimes f_2$ My question: Is the solution simply ...
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1answer
111 views

Question about adding two tensor products

I am trying to calculate $\sum \limits_{\pi \in S_3} f_{\pi(1)} \otimes f_{\pi(2)} \otimes f_{\pi(3)}$, where $S_3$ is the set of all permutations $\pi$ of $3$ objects. I know that \begin{align*} ...
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1answer
53 views

A Tensor Product Identification

Hi folks just looking at Timmermann's "Introduction to Quantum Groups and Duality" and looking at the algebra of representative functions on a compact topological group $G$. I take off in Example ...
2
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1answer
451 views

Structure of Parity Check Matrix of Non-Systematic Tensor Product Codes

Let $[n_i,k_i,d_i]_q$ for $i=1,2,\dots,r$ be a set of $r$ non-systematic linear codes over $\Bbb F_q$ with $k_i \times n_i$ generator matrix $G_i$ each and $n_i \times (n_i - k_i)$ parity check matrix ...
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1answer
73 views

Image of the tensor product of strict maps of Banach spaces

Let $f:A\to C$ and $g:B\to D$ be bounded linear maps of Banach spaces with closed image. Will $f\widehat{\otimes}g:A\widehat{\otimes}_\pi B\to C\widehat{\otimes}_\pi D$ also have closed image? What ...
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1k views

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have ...
3
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1answer
95 views

$(\beta_1 \otimes \beta_2)(\alpha_1 \otimes \alpha_2)=(\beta_1 \alpha_1)\otimes(\beta_2 \alpha_2)$

Let $V_1, V_2, W_1, W_2, U_1, U_2 \in$ K-Vect, $V_1 \xrightarrow{\;\; \alpha_1 \;\; }W_1 \xrightarrow{\;\; \beta_1 \;\; }U_1, V_2 \xrightarrow{\;\; \alpha_2 \;\; }W_2 \xrightarrow{\;\; \beta_2 \;\; ...
3
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1answer
98 views

Non commutative tensor products / simple example?

In O'Neil's "Semi-Riemannian Geometry" it is stated that if $A$ is a $(r,0)$-tensor field and $B$ is $(0,s)$-tensor field then it is ""from the definition'' that $A \otimes B = B \otimes A$. I still ...
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1answer
131 views

O'Neil's problem 2.10 on Tensor Derivations. Literature on the subject.

I am having real trouble trying to understand a problem in O'Neil's "Semi-Riemannian Geometry" and I can't find much literature on the subject. I will expose the problem and I will be grateful to ...
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1answer
138 views

How can this linear map be injective?

Studying the Peter-Weyl theorem, I've come across the following linear maps: $\theta_E: E'\otimes E \rightarrow$ Hom$(E,E)$ Where $ E'\otimes E$ denotes the tensor product of a finite-dimensional ...
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2answers
166 views

Tensor Product of Submodules

In Keith Conrad's text on Tensor Products, he states on Page 8 of the text that for a fixed ring $R$ and $R$-modules $M$, $M'$, $N$ and $N'$, and maps $f:M\to M'$, $g:N\to N'$ that ...
2
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1answer
332 views

Universal Property of Tensor Algebra

I am trying to prove the Universal property of the Tensor Algebra $T(V)$, which states that given any unital associative algebra $\mathcal{A}$ and a linear transformation $\varphi:V\rightarrow ...
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2answers
138 views

Embedding a module into its quotient module

I've got a very basic question on tensor products. Let $R$ be a commutative integral domain, $K$ its quotient field and let $M$ be a $R$-module. Is the map $M \rightarrow K\otimes_R M$ given by ...