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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$\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$

In a proof (proof of theorem 4.3.36 in Liu's book) I need the equality $\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$. The hypothesis of the theorem are the following: $Y$ ...
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47 views

$(V^*)^{\otimes n} \cong (V^{\otimes n})^*$

We assume that $V$ is finite dimensional. Make $\theta: (V^*)^n\to (V^{\otimes n})^*$ by $$ \theta(\alpha_1,\cdots,\alpha_n)(v_1 \otimes \cdots \otimes v_n ) := \prod_{i=1}^n \alpha_i(v_i). $$ Then, ...
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31 views

Tensor Product over a field

This question appeared in my exam and I could not solve it. Let $L$ be a field, $K$ subfield of $L$. Assume that dim$_K(L)$=$m$, and $V$ be a $L-$vector space amd dim$_L(V)=n$. If as usual ...
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47 views

Distributivity of tensor products over direct sums for group representations

I'm sure that tensor products for group representations are defined such that the typical properties are satisfied, but it would be nice to have an explicit proof, for group representations that: $$A ...
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115 views

When does the tensor product consist of elementary tensors only?

The question is: Assume that $R$ is a (commutative) ring. Under what conditions on $R$-modules $M,N$ does the tensor product $M\otimes_RN$ consist of elementary tensors only? That is, every ...
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49 views

Tensor product of a vector space and a field

Let $F$ be a field and $V$ a vector space of finite dimension $n$ over $F$. Let $\overline{F}$ be the algebraic closure of $F$. and let $\overline{V}=\overline{F}\otimes_F V$ the tensor product over ...
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27 views

Brauer Group - A measure of complexity?

I have seen many authors state that the Brauer Group in some way measures the complexity of a field. I've convinced myself that the Brauer group of the reals is Z/2Z, and that the Brauer group of an ...
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12 views

Matrix calculus: rules for partial traces

I'm trying to understand a paper and have trouble seeing why the following can be written: $Tr_E\{[ \rho,V] \} = \sigma Tr_E\{\rho_E V\} - Tr_E\{ V \rho_E \} \sigma$, when we know the following ...
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46 views

Exercise from Rotman, Advanced modern algebra , $\wedge^2 (\mathbb{Z}_p \oplus \mathbb{Z}_p) \neq 0$

(i) Let $p$ be a prime,. Show that $\wedge^2 (\mathbb{Z}_p \oplus \mathbb{Z}_p) \neq 0$ , where $\mathbb{Z}_p \oplus \mathbb{Z}_p$ si viewed as $\mathbb{Z}$-module ( with $\mathbb{Z}_p $ I mean ...
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30 views

Construction of the Brauer Group

I've proven that the $K$ tensor product of two central simple $K$ algebras is itself central simple, and I've proven Wedderburn's theorem, but I now need to construct the Brauer group. I've been told ...
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$x \otimes y - y \otimes x \neq 0$ in $I \otimes_{R} I$

Let $R = k[x,y]$ , $I = (x,y)$ , $k$ is a field. I want to prove that : 1) $x \otimes y - y \otimes x \neq 0 $ in $I \otimes_{R} I$ 2) $x \otimes y - y \otimes x $ is a torsion element My ...
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41 views

An interesting phenomenon of $C^*$-tensor product

On the algebraic tensor product space of $C^*$-algebra, I try to find an example whose maximal $C^*$-norm is not the minimal $C^*$-norm, but it seems as it is impossible to do this because the finite ...
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55 views

Kruskal Tensor: sum of outer or Kronecker products?

I'm reading an ACL 2014 paper: Lei, Tao, et al. "Low-Rank Tensors for Scoring Dependency Structures.", ACL 2014. It defines the Kruskal form of a tensor as a sum of Kronecker products: However, ...
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1answer
148 views

Can one prove the existence of tensor product without explicitly constructing it? [duplicate]

R is a ring with 1. We construct tensor product $M \otimes N$ of right R-module $M$ and left R-module $N$ to basically be able to state its universal property that any R-bilinear map from $M\times ...
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41 views

Tensor Product of Algebras: Multiplication Definition

I've managed to get a weak grip on what the tensor product of two vector spaces is. I'm now trying to understand the tensor product of two algebras. I understand that we define $(v_1\otimes ...
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46 views

Embed a vector space into a tensor product

If $V$ is a $K$-vector space and $L$ is a field extension of $K$ then why is the map $v \to v\otimes 1$ an embedding of $V$ into $V\otimes_K L?$
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35 views

Universal Property of Tensor Product, Uniqueness

There is a universal property for free modules, where for any map $f:B\rightarrow S$ there is a map $g:F(B)\rightarrow S$ such $g\cdot b=f$ where $b$ is the canonical map from the basis set into the ...
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43 views

Number of generators in a tensor product

'Commutative Algebra' by Atiyah and Mcdonald, mentions if $ \lbrace x_{i} \rbrace_{i ∈ I}$ and $\lbrace y_{j} \rbrace_{j ∈ J}$ generate $M$ and $N$ as $A-$modules, respectively, then $x_{i} \otimes ...
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55 views

When is $N\otimes_A B \to N$ an isomorphism?

Let $A, B$ be commutative (unital) rings and $f\colon A \to B$ an $A$-algebra. There then exists a canonical functor $f_*\colon \mathbf{Mod}_B \to \mathbf{Mod}_A$ such that, for every morphism of ...
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41 views

Basis of tensor product of vector spaces

Suppose $V,W$ are vector spaces with bases $\{e_1,e_2,\dots ,e_m\}$ and $\{f_1,f_2,\dots ,f_n\}$. I know that $V\otimes W=F(V\times W)/H$, where $F(V\times W)$ is the free vector space on $V\times W$ ...
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22 views

How to apply operators across parts of a tensor product.

Question I have a system which is the tensor product of two vectors from different vector spaces, I would like to now apply operations to both sub systems separately. The question is how can I do ...
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34 views

Unclear Construction of Basis for Tensor Product

My problem lies in page 363 of Steven Roman's Advanced Linear Algebra (Here's a link). The author says that for each ordered pair $(e_i,f_j)$ where $\left\{e_i\right\}_{i\in ...
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22 views

Tensor Algebra and Isomorphism

Consider the standard tensor algebra over a vector space: $$T(V)=\bigoplus_{n\in N}V^{\otimes n}$$, I am trying to define a linear map from $T(V)\times T(V)\rightarrow T(V)$. Up to this time, I have a ...
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37 views

Exactness of sequence of vector spaces with tensor product

Let $k$ be a field and $0\rightarrow V _n \xrightarrow{f_n} V_ {n − 1} \xrightarrow{f_{n−1}} ··· \xrightarrow{f_3} V_ 2 \xrightarrow{f_2} V_ 1 \xrightarrow{f_2} V_ 0 \rightarrow 0$ an exact sequence ...
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1answer
38 views

Is this element necessarily non-zero in the tensor product?

Suppose $R$ is an integral domain and $M$, $N$ are torsion-free $R$-modules. If $m$ and $n$ are nonzero elements of $M$ and $N$ respectively then does it follow that $m \otimes n \neq 0$ ? If either ...
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58 views

Tensor Product of Vector Spaces - Quotient Definition

I'm trying to figure out exactly what the tensor product of vector spaces is. This is what I understand so far: If $V, W$ are vector spaces over a field $R$ then the free vector space $C(V\times W)$ ...
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29 views

tensor derivative of a symmetric second order tensor mapped onto itself, with respect to itself

In a continuum mechanical context, let $\mathbf{A}$ be a symmetric second order tensor. How can I calculate the derivative $$\cfrac{\partial(\mathbf{AA})}{\partial\mathbf{A}}$$ using indicial ...
2
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1answer
57 views

$M \otimes_Z N =0$ if and only if $M \otimes_R N =0$?

If $R$ is a commutative ring, and $M$ and $N$ are $R$-modules, is $M \otimes_Z N =0$ if and only if $M \otimes_R N =0$? The forward direction seems clear. The backward direction seems like it should ...
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32 views

Tensor product and Brauer Group

For a project I'm doing, I've been briefed to describe the Brauer Group. I know it has something to do with tensor products of algebras, but my problem is with understanding the tensor product. The ...
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25 views

wedge product - distributivity over addition

Wedge and tensor algebra are very new concepts to me and I want to understand how to prove the following property of the wedge product: ...
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12 views

3D tensor multiplied with 2D Matrix

I have a nolinear system with it's taylor aproximation up to the second order so that: $\tilde{\omega} = f(\omega) : f = \sum_{j=1}^6 R_{ij}\omega_j + \sum_{j,k=1}^6 T_{ijk}\omega_j\omega_k + \dots ...
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Tensors and General Relativity

I've recently begun self studying general relativity, using mostly the material found in Robert Wald's "General Relativity", and almost right out of the gate one encounters the notion of a tensor. In ...
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57 views

Does tensoring with a module preserve this injection even if the module is not flat?

Suppose $R$ is a domain and let $Q$ be its field of fractions. Then $ 0 \to R \to Q $ is exact. Now suppose that $M$ is a torsion free $R$ module. Is it necessary that $ 0 \to M \to Q \otimes M $ ...
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1answer
66 views

Dual of injective tensor norm is not projective tensor norm

Let $A$, $B$ are two Banach space, on the algebraic tensor space $A$ $\odot$ $B$, we can define the projection(maximal) tensor norm $\gamma$ and injective(minimal) tensor norm $\lambda$. For the ...
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1answer
47 views

wedge product and tensor operation problem

Let $\{e_{1},\ldots,e_{n}\}$ be the usual basis for $\mathbb{R}^{n}$ and let $\{\varphi_{i},\ldots,\varphi_{n}\}$ be the dual basis. Show that $$\varphi_{i_{1}}\wedge\cdots\wedge ...
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48 views

Representing natural numbers as matrices by use of $\otimes$

What I am wanting to do is to find a unique matrix representations for Natural numbers. Say I have the number $n$, how can I represent this number as a matrix in which I can do matrix multiplication ...
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1answer
78 views

$\mathbb{C}\otimes_\mathbb{C} \mathbb{C} \cong \mathbb{R}\otimes _\mathbb{R} \mathbb{C}$

I am trying to show that $\mathbb{C}\otimes_\mathbb{C} \mathbb{C} \cong \mathbb{R}\otimes _\mathbb{R} \mathbb{C}$ as abelian groups. I've tried to come up with various maps but gotten nowhere. ...
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1answer
78 views

$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q}\ne 0$

I've found this claim $$\biggl( \prod_{i \in \mathbb{N}}\mathbb{Z}/p^i\mathbb{Z}\biggr)\otimes_{\mathbb{Z}} \mathbb{Q} \not\cong \prod_{i \in \mathbb{N}}\biggl( ...
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25 views

tensor product of infinite dimensional vector spaces [duplicate]

Let $V,W$ be vector space over field $k$. Then \begin{eqnarray*} V^*\otimes W &=&V^*(\oplus_{i\in I} k_i)\\ &=&\oplus_{i\in I}(V^*\otimes k_i)\\ &=& \oplus_{i\in I}V^*_i\\ ...
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The derivative of the determinant of a Kronecker product

For an invertible matrix $A$, we have the identity \begin{align} \dfrac{\partial \det A}{\partial A} = \det A (A^{-1})^T \end{align} where the $T$ denotes the transpose operation. How does this ...
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1answer
31 views

Constructing nontrivial $\mathbb{Z}$-bilinear map from $\mathbb{R} \times (\mathbb{R} / \mathbb{Z})$

I'm trying to show $\mathbb{R} \otimes_\mathbb{Z} (\mathbb{R} / \mathbb{Z})$ is nontrivial, where my tensor product is defined using the universal property. This problem can be easily reduced ...
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Tensorial product, a simple question

I need to find the components of $D$: $$D=a\otimes a$$ where $a$ is a tensor of order 2. Thanks!
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48 views

Product of two geometric series

I have used the Product of two power series and find out the below results. But it is to some extend strange for me, could you please confirm the results? Let $A=\sum_{i=0}^{\infty}(\frac{L}{a})^i$ ...
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46 views

Tensor Product over a ring

Given Two Fields $F,K$, and two vector spaces $V,W$ over $F$, what does tensor product $$V\otimes_{K} W$$ mean? I am not certain whether this is defined in general. I came across it in cases wheh ...
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50 views

Tensor product of two simple modules

Let $k$ be a field of arbitrary characteristic. Suppose that $A$ and $B$ are finite dimensional $k$-algebras. Let $S$ be a finite dimensional simple $A$-module and let $T$ be a finite dimensional ...
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1answer
50 views

Another matrix of a given operator $A \otimes A$

Let $V$ be a $4$-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $(e_{1}, ...
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62 views

About the definition of the tensor product of modules

I was reading something about tensor product (balanced product) of modules (over an arbitrary ring $R$), but I cannot realize why we need a left and a right module. Would it be the same with two right ...
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1answer
49 views

What is a wedge product?

What is actually a wedge product ? How does it differ from a tensor product ? What is the intuition to invoke such a product, kind of like tensor product is invoked to simplify multilinear maps ?
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1answer
113 views

$k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$

This is part of an exercise from Eisenbud: $k$ is a field, describe as explicitly as possible a) $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$ b) $k[x] \otimes_{k} k[y]$ Any hint ?
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69 views

Tensor product is o?

$K$ generated by $(ar,rb)$ for every $a\in A$, $b\in B$, $r\in R$. $F$ generated by $r(a,b)$ for every $a\in A$, $b\in B$, $r\in R$. $(a,rb)=(ar,b)=r(a,b)$,then $K=F$, $F/K=0$ Apparently I ...