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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Understanding the Definition of the Tensor Product of Chain Complexes

The tensor product of chain complexes (of $R$ modules) $C_\bullet ,D_\bullet$ is defined as $$(C_\bullet \otimes D_\bullet )_n = \bigoplus_{i+j=n} C_i \otimes_R D_{j}$$ I understand this definition ...
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Tensor-Hom Adjunction In Monoidal Categories?

Is there a generalization of the tensor-hom adjunction to monoidal categories, or is it a special property of $\mathsf{Mod}$-$R$?
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Definition of rank of an element in tensor algebra

Let $V$ be a vector space over a field $K$ and let $T(V)$ be its tensor algebra; that is, $$T(V)=\displaystyle\bigoplus_{k=0}^{\infty}V^{\otimes k}=K\oplus V\oplus V\oplus(V\otimes V)\oplus(V\otimes ...
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27 views

tesnsor product of two copies of $\Bbb{R}$ over $\Bbb{R}$

I would like to know what would be tensor product of set of reals over reals would be? That is, $\Bbb{R} \otimes_\Bbb{R} \Bbb{R}$ I think it should be $\Bbb{R}^{2}$ as tensor product combines two ...
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Proof of the properties of tensor product

On page 25 of Atiyah-Macdonald "Introduction to commutative algebra", the author says that "We shall never again need to use the construction of the tensor product given above and the reader may ...
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Matrix operation repeat matrix members

I am going to use C++ Armadillo library which handles matrices to generate matrix $B$ and $C$ from matrix $A$. $$ A=[M_0,M_1,\ldots,M_{n-1}]^T $$ $$ ...
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56 views

Vector Spaces: Tensor Product

Reference Foundation for: Hilbert Spaces: Tensor Product Problem Given a vector spaces $V$ and $W$. Take its algebraic tensor product: $\tau:V\times W\to V\otimes W$ How to prove that the image ...
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Prove that this element is nonzero in a tensor product

I want to solve the following problem: show that the element $1\otimes (1,1,....)$ is not the zero element in $$\mathbb{Q}\otimes_{\mathbb{Z}} \prod^{\infty}_{n\geq 2}\mathbb{Z}/n\mathbb{Z}$$. My ...
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+50

covariant and contravariant components and change of basis

I encountered the following in reading about covariant and contravariant: In those discussions, you may see words to the effect that covariant components transform in the same way as basis ...
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Understanding the definition of tensor product as a quotient of a free abelian group

I've been give the Definition: Let F be a free abelian group with a basis $X$ such that. $$F = \langle A\times B\mid \emptyset \rangle $$ Let $f$ be a subgroup of $F$ generated by the ...
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Nakayama's lemma, second version

Let $R$ be a commutative ring with identity, $J$ an ideal that is contained in every maximal ideal of $R$, and $A$ is finitely generated $R-$ module. If $R/J\otimes _R A=0$, then $A=0$. ...
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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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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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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 ...
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1answer
45 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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30 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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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 ...
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73 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
29 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
105 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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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
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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, ...
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1answer
23 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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94 views

Hilbert Spaces: Tensor Product

Attention The question has been modified! (Previous answers were perfectly correct, then.) Reference Build-up on: Vector Spaces: Tensor Product Problem Given Hilbert spaces $\mathcal{H}$ and ...
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23 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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Defining miscellaneous products in a miscellaneous mathematical structure

This is a question about elementary sets, functions and relations, and about a functor $F$ that maps functions $f\subseteq X\times Y$ to relations $F(f)\subseteq F(X)\times F(Y)$. The miscellaneous ...
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1answer
61 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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108 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
31 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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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. ...
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141 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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42 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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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
39 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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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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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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72 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
56 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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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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33 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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36 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 $\Bbb Z_2$ ...
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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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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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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 ...
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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
28 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 ...
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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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53 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 \\ ...
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1answer
62 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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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}) ...