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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43 views

Different forms for the exterior power of a module

First I have defined the exterior algebra of a module $M$ as the quotient $T(M)/A(M)$ where $T(M)$ is the tensor algebra of $M$ and $A(M)$ is the ideal generated by all elements of the form $m\otimes ...
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2answers
64 views

Proving equivalent versions of faithfully flatness.

I was reading a proof of the the following theorem from Matsumura (p.47) There was something confusing about $(3) \implies (2)$ and $(2) \implies (1)$. Question 1 Here, it says $M \not= ...
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3answers
115 views

How to get matrix $A$ from $A^\top A=B$ with given symmetric matrix $b$?

Given a symmetric matrix $B \in \mathbb{C}^{n\times n}$. How many coefficients of $A \in \mathbb{C}^{n\times n}$ can you obtain from the following equation? $$A^\top A=B$$ I think this problem is ...
2
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1answer
106 views

Tensor products- balanced maps versus bilinear

When defining tensor products $M\otimes_R N$ over a commutative ring $R$ one can use a universal property with respect to bilinear maps $M\times N\rightarrow P$. On the other hand, in the general ...
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1answer
62 views

Definition of covariant derivative

Let $E \to M$ be a vector bundle (with fibres $V$) over a smooth manifold $M$. Define the covariant derivative $\nabla$ as a map $$\nabla : C^\infty(M,E) \to C^\infty(M,E \otimes T^*M),$$ where ...
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35 views

Compute $\mathbb{Z}/m\otimes\mathbb{Z}/n$ using exact sequence

I want to compute $\mathbb{Z}/m\otimes\mathbb{Z}/n$ using exact sequence as follows. Consider the exact sequence $$ \mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}/m\to 0. $$ Tensoring with $\mathbb{Z}/n$ gives ...
2
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1answer
89 views

A problem with tensor products

Let $K$ be a field, $R=K[x^2,x^3]$, $S=K[x]$, and consider $S$ as an $R$-module. Given $f: S \to R \oplus R$ so that $f:p \mapsto (x^3p,-x^2p)$, prove that $f\otimes 1: S \otimes_R S \to (R\oplus R) ...
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1answer
34 views

showing Module is simple

Given the following: let $C \subset \mathbb{H}$ be a subring of the real quarternion algebra such that it contains the center of $\mathbb{H}$ = $Z(\mathbb{H})$ Also C $\cong \mathbb{C}$ Then let R ...
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2answers
124 views

Examples of categories which naturally include End(O) as object

I want examples of categories $\textbf C$ which naturally include $End_{\textbf C}(O)$ as object for objects $O$ in the category. The set of all endomorphims is always a monoid under the composition ...
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1answer
217 views

Tensors of rank two in physics and mathematics

In physics one would speak of a tensor of second rank having nine components (in three dimensions) usually written as $$T = \begin{bmatrix} t_{11} & t_{12} & t_{13} \\ ...
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1answer
64 views

Lie algebra representations and tensor product decompositions.

Find the weights for $V_{L_1 - 2L_3}$, where $L_1, L_2, L_3$ are the weights for the standard representation of $\mathfrak{sl}_3 \Bbb{C}$ on $V \cong \Bbb{C}^3$. In order to find these weights, ...
2
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1answer
30 views

Show that a ring is simple

In the ring $R = \mathbb{H} \otimes_{\mathbb{R}} M_{2}(\mathbb{C})$ I have computed the center as $Z(R)= \mathbb{C}$. I am however struggling to show that $R$ is a simple ring and consequently find ...
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2answers
253 views

Tensor Product: Hilbert Spaces

This question has been modified... Problem Given Hilbert spaces. In general, their algebraic tensor product isn't complete: ...
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1answer
35 views

Natural homomorphism for tensor product ($\tau : G \mapsto $GL(Bilinear(V, W) )

I'm struggling to understand why the following function $\tau$ is a homomorphism: Given homomorphisms $\rho : G\rightarrow GL(V)$ and $\sigma : G\rightarrow GL(W)$ where $V$ and $W$ are ...
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1answer
230 views

When are two simple tensors $m' \otimes n'$ and $m \otimes n$ equal? (tensor product over modules)

Suppose that $M$ is a right R-module and $N$ is a left $R$-module. We can construct $M \underset{R}\otimes N$ and give it an Abelian group structure by considering the free R-module $K$ generated by ...
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209 views

Tensor products in general topology?

Let $(X,\tau)$ and $(Y,\sigma)$ be topological spaces and let $(X\times Y,\tau\times\sigma)$ be the space with the box topology. Since I never heard of it I guess that there is no space $X\otimes Y$ ...
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1answer
34 views

How to show that there is a natural isomorphism of rings $k\left(s\right)\otimes_{k}k\left(t\right)=S^{-1}\left(k\left(s\right)\left[t\right]\right)$?

Let $k$ be a field, $s$ and $t$ two indeterminates over $k$. Let $S$ be the set of non-zero elements of $k\left[t\right]$. How can it be shown that there is a natural isomorphism of rings ...
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0answers
75 views

Apparently meaningless computation with the Hodge star operator

In the last lecture we started speaking about hodge star operator. Let $E$ be a $n$ dimensional vector space with a non degenerate bilinear form $g$. $\mathcal{O}$ the orientation line of $E$, i.e. ...
3
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2answers
296 views

Algebraic operations with tensor products and direct sums.

When first learning about tensor products and direct sums of vector spaces, the analogy of multiplication and addition of vector spaces is sometimes used to help with the intuition. If we look at the ...
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1answer
71 views

Tensors, basic notation and components of

I'm trying to understand he basic notation(s) used to write out tensors, namely \begin{equation} T = T^{\nu_1,...,\nu_m}_{\mu_1,...,\mu_n} \frac{\partial}{\partial x^{\nu_1}} \otimes...\otimes ...
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1answer
43 views

Difference between to Tensor products with regards to modules

What would be the difference between $$ \otimes_B $$ and $$ \otimes $$ both in the following context and in general? Let A be a ring with $$ B \subset A $$ and M a B-Module. We can construct the ...
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1answer
49 views

Definition of Linear independence of algebraic $1$-forms

"Suppose that $a,b,c,d$ are linearly independent algebraic $1$-forms on $\mathbb{R}^n$". What does it mean for algebraic $1$-forms to be linearly independent? I have looked through my notes and ...
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1answer
148 views

Homotopy of double chain complexes

Consider complexes $(A,d_1), (A',d_1)$, $(C,d_2), (C',d_2)$ and morphisms $f_1,f_2: (A,d_1)\to (A',d_1)$ and $g_1,g_2: (C,d_2)\to (C',d_2)$ of degrees $0$. Consider the functor $(-\otimes-)$, then ...
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1answer
51 views

Show that $x\otimes x+2\otimes2$ is not an elementary tensor in $I\otimes_{A}I$ [duplicate]

Let $A=\mathbb{Z}[x]$ and $I=(2,x)\lhd A.$ Show that $x\otimes x+2\otimes2$ is not an elementary tensor in $I\otimes_{A}I$. I have $$I\otimes_A I = \frac{L_A(I\times I)}{T}$$ where $T$ is the ...
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1answer
83 views

Is $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\bigotimes_{\mathbb{C}}\mathbb{C}$? [duplicate]

I'm trying to see if for several cases changing the ring in a tensor product affects the result or doesn't. Now I'm trying to prove $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq ...
2
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1answer
82 views

tensor product and commutation, category theoretical argument

It is a well-known fact that taking direct limits commutes with tensor products in the following sense: Let $I$ be a directed set and suppose for every $i\in I$ we have modules $M_i,N_i$ over a ring ...
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0answers
25 views

Can I reform this to a tensor/matrix product?

so I have the following vector matrix product: $$v = A w$$ Now I have this $n$-times: $$v^{(n)} = A^{(n)} w^{(n)} \quad \forall n$$ Is there any way to write this without $\forall$. Maybe somthing ...
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2answers
262 views

Calculate the tensor product of two vectors

Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis ...
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1answer
46 views

Isomorphism with tensor product

Let $f$ : $\Bbb Z_2$ $\rightarrow$ $\Bbb Z_4$ given as $f$($a$ + $2$$\Bbb Z$) = $2a$ + $\Bbb Z_4$ is a monomorphism . And knowing that <0,2> as subgroup of $\Bbb Z_4$ is isomorphic to $\Bbb Z_2$ ...
2
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1answer
33 views

Taking complex conjugate of an element in $\mathfrak{su}(2)_\mathbb{C}$

I read from a book that the complexification of the Lie algebra $\mathfrak{su}(2)$, noted $\mathfrak{su}(2)_\mathbb{C}$, is in fact the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$, the reason being: ...
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1answer
47 views

Multiplication in symmetric product space

STATEMENT: Let $V=\mathbb{R}^2$.Take $Y:=\left\{x\cdot y: x,y\in V\right\}$ where $S_2(V)$ is the symmetric product of $V$. QUESTION: What is multiplication in the symmetric product space?
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1answer
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Is the map from representation ring to class functions a isomorphism?

I have a questions from representation theory. ($G$ is finite group) Fulton and Harris in "Representation Theory. A first Course" write that: the character defines a map $$\chi : R(G) \to ...
2
votes
1answer
78 views

When is the Tensor product of Modules itself a Module?

If $M$ is a right $R$ module and $N$ is a left $R$ module then $M \otimes_R N$ is an abelian group. Wikipedia says that if $M$ is an $R$ bimodule then $M \otimes_R N$ can take on the structure of a ...
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1answer
44 views

How is this expression well-defined?

I am going through the book "Introduction to Tensor Product of Banach Spaces" by Raymond Ryan. The tensor product of vector spaces is introduced in the first chapter which I briefly outline now. Let ...
2
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2answers
48 views

Understand the meaning of tensor product of modules

I am reading Atiyah's Introduction to Commutative Algebra. I have difficulty understanding the meaning of free A module $A^{(M\times N)}$. Here M and N are both A modules. In this book, the free A ...
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1answer
147 views

Tensor product of function space and vector space

In one book I found that they treat vector valued functions as tensor product of some function space and vector space. So for example $$ C(\mathbb{R},V) \simeq C(\mathbb{R})\otimes V \\ ...
2
votes
1answer
83 views

Decomposition of $\mathbb{Q}(\sqrt{2})\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{3})$ and $\textbf{F}_q(t)\otimes_{\textbf{F}_q(t^q)}\textbf{F}_q(t)$

I have two questions about splitting of the tensor product into the product of fields How can one find a decomposition of $$\mathbb{Q}(\sqrt{2})\otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt{2})$$ and ...
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1answer
35 views

evaluation of $\nabla \cdot ( \boldsymbol{B}(\boldsymbol{x}) \cdot \boldsymbol{B}^{T}(\boldsymbol{x}) )$

i have a symmetric positive matrix $\boldsymbol{D}(\boldsymbol{x})$ which can be decomposed as: $\boldsymbol{D}(\boldsymbol{x})$ = $\boldsymbol{B}(\boldsymbol{x}) ...
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1answer
62 views

If A , B are finitely generated R-algebras then $A\otimes_RB$ is a finitely generated $R$-algebra.

$A$, $B$ are finitely generated $R$-algebras. $R$ is a commutative ring with $1$. Then how can I show that $A\otimes_RB$ is finitely generated $R$-algebra? What I have tried: First I have to show ...
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2answers
58 views

Extension of scalars $M_B=B\otimes_A M$

Most textbooks say that the $B$-module structure on $M_B$ (for $A\rightarrow B$ a ring morphism and $M$ an $A$-module) is "defined" by $b'(b\otimes m)=b'b\otimes m$. How is this a proper definition? ...
3
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1answer
265 views

Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
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1answer
45 views

Basis of a tensor

I'm new to tensors, and I need to understand what a certain basis actually is, how to visualise it. Say we have the $r$-dimensional vector space $T_p M$ and $n$-dimensional dual space $T^{*}_p M$. ...
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1answer
129 views

Complex conjugate and tensor product

Let $V$ be a real vector space and $f : V \rightarrow V$ a linear endomorphism. Also, let $\sigma : \mathbb{C} \rightarrow \mathbb{C}$ be complex conjugation. If $A$ is a real matrix, then it is ...
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1answer
164 views

Tensor product of the fraction field of a domain and a module over the domain

Given a fraction field $k(x)$ of the polynomial ring $k[x]$ over a field $k$ and an integral domain $R$ that is also a $k[x]$-module, is it true that $k(x) \otimes_{k[x]} R \cong Frac(R)$? I ...
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1answer
29 views

What are zeroes in $\mathbb{Q}\otimes M$

If regard $\mathbb{Q}$ as $\mathbb{Z}$-module, and M another $\mathbb{Z}$-module, then $q\otimes m$ is zero in $\mathbb{Q}\otimes M$ iff q or m is zero, how to proof it?
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1answer
52 views

Functorial isomorphism involving tensor products

Let $R$ be a commutative ring and $E', E, F', F$ be free, f.g. $R$-modules of equal rank. For $f\in L(E',E):={\rm Hom}_R(E',E)$ and $g\in L(F',F)$, let $T(f.g)\in L(E'\otimes_R F', E\otimes_R F)$ be ...
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0answers
60 views

Universal property of the Tensor Algebra

Let M be an A-module over a commutative ring A. For any A-algebra N and A-module homomorphism $\phi : M \rightarrow N$ there is a unique A-algebra homomorphism $\Phi : T(M) \rightarrow N$ (where T(M) ...
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1answer
60 views

Simple question on tensoring by a quotient ring

$A \subset B$ is an extension of commutative rings s.t. $B$ is a f.g. free $A$-module of rank $n$, so I have $A^n \stackrel{\sim}{\longrightarrow} B$ as $A$-modules. Let $\mathfrak a$ be an ideal of ...
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1answer
83 views

Tensor products isomorphic to hom-sets with a structure

In which cases the tensor product of objects, say A and B, is (isomorphic with) the objects with the carrier set Hom(A,B) and a corresponding structure?
2
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1answer
49 views

using a symmetric matrix to simplify expression involving antisymmetric matrices

I am trying to simplify the following expression --$$(\vec{A} \times \vec{B})\cdot\underline{\underline{S}}\cdot(\vec{C} \times \vec{D}) = (A_jB_k\epsilon_{jki})S_{im}(\epsilon_{mnp}C_nD_p)$$ $S$ is a ...