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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262 views

How to prove tensor product is exact when acted on split short exact sequence?

I know tensor product is right exact, but I can't figure out why it's exact when it is acted on a split short exact sequence. In addition, can you give an example that tensor product acts on a short ...
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115 views

Kernel of a map on tensor product of modules

Let $M,N, P$ be $A$-modules, and let $f:M \otimes N \to P$ be an $A$-homomorphism. If $m \otimes n \in \ker f$ implies $m\otimes n =0$ for all $m\in M, n\in N$, does it follow that $\ker f=0?$ For ...
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27 views

use universality property to prove the existence of isomorphism

Suppose $V$ and $W$ are finite-dimensional vector spaces over a field $K$. Use the universality property of tensor products to show that there exists a canonical isomorphism $f:V^{*}\otimes W\to Hom_K(...
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115 views

Adjoint functor to an R-algebra only “remembering” itself as a ring

I have been wondering this question while trying to comprehend adjoint functors and the various definitions. If you let $$F:\mathbf {R\text - Alg}\to \mathbf {Ring}$$ be the functor that sends ...
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39 views

Why the following is not a tensor?

Say we have an arbitrary coordinate system, in which a position vector is represented by: $$\vec V=Z^i\vec Z_i$$ Where $\vec Z_i$ is a covariant basis. Now $\partial {\vec Z_i}/\partial Z^j$ term has ...
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107 views

Prove that $Ker(g \otimes k)= Im(f \otimes 1_{N}) + Im (1_{M} \otimes h)$

Suppose we have two short exact sequences: $$0 \to M' \mathrel{\overset{f}{\to}} M \mathrel{\overset{g}{\to}} M'' \to 0 $$ in Mod-R $$0 \to N' \mathrel{\overset{h}{\to}} N \mathrel{\overset{k}{\to}} ...
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26 views

Prove that $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup of $A$

Let $A$ be a finite abelian group of order $n$ and let $p^k$ be the largest power of the prime $p$ dividing $n$. Then $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup ...
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60 views

Action of universal R-matrix of U_q(sl_2)

My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let $q \in \mathbb{C}$ with $q \neq 0, \pm 1$. ...
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1answer
23 views

The rank of a tensor is not invariant under isomorphism

Let $V,W$ be two $K$-vector spaces; for $a\in V\otimes W$ define \begin{equation*} rk(a)=\min \left\{r|a=\sum_{i=1}^r\alpha_iv_i\otimes w_i\text{ for some }\alpha_i\in K,v_i\in V,w_i\in W\right\}. \...
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97 views

Partial Trace of Density Operator

Before stating my question I present my motivation: to learn more about the tensor product. Now, quantum mechanics assigns a Hilbert space to each physical system as a postulate of the theory. ...
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48 views

Natural Ordering of the Class of Hermitian Preserving Maps

I am trying to understand Man-Duen Choi's remark 3 in his paper Completely Positive Linear Maps on Complex Matrices: For a linear map $\Phi : \mathcal{M}_{n} \to \mathcal{M}_{m}$, it is obvious that $...
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31 views

Tensor product of algebras and generating sets.

Let $A$ be a module over $k$ generated by $x$ and $y$. The generating set for $A \otimes_k A$ is $\{x \otimes x, x \otimes y, y \otimes x, y \otimes y\}$. But does this still hold if $A$ is an algebra ...
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62 views

What is this notation $\odot$ for?

(Note that symmetric algebra and symmetric tensor do not coincide when the characteristic is not $0$.) I'm reading this aricle:http://en.m.wikipedia.org/wiki/Symmetric_tensor And here it defines $...
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1answer
37 views

Testing for decomposable tensors

Let $V$ be a vector space of dimension 3 with the basis {$e_1,e_2,e_3$} and let $W$ be a vector space of dimension 2 with basis {$f_1,f_2$}. Which of the following tensors are decomposable (i.e. of ...
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1answer
22 views

Is it okay to define k-th symmetric power of $M$ in this way?

I want to define the tensor algebra and related algebras in a very formal way. I will illustrate how I tried to define algebras below. Let $R$ be a commutative ring and $M$ be an $R$-module. ...
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38 views

Universal property of tensor product of $R$-algebras

Let $R$ be a commutative ring and $A_1,...,A_{n+1}$ be $R$-algebras. Let $A_1\otimes_R\cdots\otimes_R A_{n+1}$ be equipped with the natural $R$-algebra structure. Let $N$ be an $R$-algebra. Let $\...
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66 views

Matrix into a 4-tensor

I've been trying to figure out a way to add a dimension to a matrix with a certain rule: Let our matrix X have dimensions NxM, L be some positive number, and the desired resulting 4-tensor (3d matrix)...
3
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1answer
52 views

How do I use Einstein notation to propose a rule for $MX=0$ and proving it's linear?

Let $$M = \begin{bmatrix}a^1_1 & a^1_2&....& a^1_k\\a^2_1 & a^2_2 & .... & a^2_k \\ : & : & .... &: \\ a^r_1 & a^r_2 & .... & a^r_k\end{bmatrix} \quad\...
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165 views

Eigenvectors of operators on a tensor product Hilbert Space

Suppose I have finite dimensional Hilbert spaces $V$, $W$, and an operator $A$ acting on vectors in $V$ such that it has eigenvectors/values $Ax_a=\lambda_ax_a$. In the tensor product space I want to ...
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156 views

When is the tensor product of a separable field extension with itself a domain?

I'm reading Algebraic Geometry and Arithmetic Curves by Qing Liu. On page 92, in the proof of Corollary 3.2.14 d), he states that if $K \otimes_k K$ is a domain, then $K = k$. Here $K$ is a separable (...
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32 views

Systems of equations of the form $\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a$

Is there any theory that deals (directly or not) with systems of equations of the form $$\sum_{i \in I} \sum_{j \in J_i} v_i \times v_j = a,$$ where $a \in \mathbb{R}^3$ is known, $v_i, v_j \in \...
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47 views

Checking alternating tensors

How do I check that $$f(x,y)=x_1y_2-x_2y_1+x_1y_1$$ is an alternating tensor? I did check that f is a tensor, but how do I know if it is alternating by direct computation? Thanks in advance!
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Is it true that every element of $V \otimes W$ is a simple tensor $v \otimes w$?

I know that every vector in a tensor product $V \otimes W$ is a sum of simple tensors $v \otimes w$ with $v \in V$ and $w \in W$. In other words, any $u \in V \otimes W$ can be expressed in the form$$...
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1answer
37 views

Zero Tensor Product

Suppose we have a space $|\psi_1\rangle \otimes |\psi_2\rangle \otimes |\psi_3\rangle$, and operators (matrices) A ⊗ B ⊗ C acting on this Hilbert space (like in quantum mechanics). I'm trying to ...
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119 views

How do we wedge the complex differentials $\mathrm{d}z^i$ and $\mathrm{d}\bar z^{\bar j}$?

By the standard definition of the wedge product as an alternated tensor product, I would think we have $$\tag{1}\mathrm{d}z^i\wedge\mathrm{d}\bar z^{\bar j}=\mathrm{d}z^i\otimes\mathrm{d}\bar z^{\bar ...
4
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1answer
87 views

Show an ideal is a finitely generated projective module via a split exact sequence

Let $I$ be an ideal of $R$ such that the mapping $f:I\otimes_R\operatorname{Hom}_R (I,R)→R$ defined (on the generators) by $f(i\otimes α)=α(i)$ for all $i∈I$ and $α∈\operatorname{Hom}_R (I,R)$ is onto....
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232 views

Does a long exact sequence of flat modules remain exact after tensoring with an arbitrary module?

In Liu's Algebraic Geometry and Arithmetic Curves, Proposition 1.2.6 states that given any short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ with $M''$ flat, taking ...
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35 views

Is this tensor identity true?

If We have two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ and a symmetric positive definite Matrix $\boldsymbol{M}$ I was wondering if the expression $((\boldsymbol{a}\times \boldsymbol{b}) \cdot \...
2
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1answer
63 views

Extending scalars from $k$ to $K$: how find $K$ linear maps?

If I extend scalars from a ring $R$ to a ring $S$ by a homomorphism $f:R \to S$, then starting with an $R$ module $M$, I get an $S$ module $S \otimes_R M$. Given $\sigma \in \text{End}_R(M)$, I know ...
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63 views

Does tensoring the integers $\Bbb Z$ with any field $F$ give that field itself? How to prove that?

Does tensoring the integers $\Bbb Z$ with any field $F$ give that field itself? How to prove that? I know for $\Bbb Q$ it is true.
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520 views

Notion of contraction in tensor algebra

Assuming a vector space V and it's basis set $\{\vec{e}_\nu\}$. A vector $\vec{v}$ can be written as: $\vec{v}=x^\nu\vec{e}_\nu$ where $x^\nu$ is the corresponding contravariant coordinate. We can ...
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1answer
22 views

Definition of the tensor product of finite sequence of modules

I have posted several questions about the tensor product of modules before and this post would be the final one. I have read wikipedia,mathSE,Dummit&Foote and Bourbaki for the definition of the ...
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1answer
107 views

Definition of tensor product

Here is the standard dedonition of the tensor product of two modules: Definition for tensor products of two modules: Let $R$ be a ring and $M$ be a right module and $N$ be a left module. ...
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92 views

Index notation confusion in tensor algebra

I have some confusions regarding index notation in tensor algebra. Let's assume $\vec{v}$ is a vector belonging to vector space $V$. Choosing a basis set $\{\vec{e}_\nu\}$, $\vec{v}=x^\nu\vec{e}_\nu$ ...
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453 views

Basis for Tensor Product of Infinite Dimensional Vector Spaces

If V and W are vector spaces over a common field with bases $V_B = ${$v_i : i \in I$} and $W_B = ${$w_j : j \in J$}, then is {$v_i \otimes w_j: i \in I, j \in J$} a basis for $V \otimes W$ ? I have ...
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47 views

Defining multiplication on the tensor product of $R$-algebras.

If $M$ and $N$ are $R$-algebras, then one can define a multiplication of elementary tensors as follows; $(m \otimes n) \cdot (m' \otimes n') = mm' \otimes nn'$. My question is how can we show, using ...
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50 views

Tensor product over a tensor product

Suppose $M$ is an $R$-module and $N$ is an $S$-module. Then $M\otimes_{\mathbb{Z}} N$ is an $R\otimes_{\mathbb{Z}} S$-module given by $r\otimes s\cdot m\otimes n:= rm\otimes sn$. My question is ...
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82 views

Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let $...
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59 views

When is a pure tensor equal to $0$?

Let $R$ be a commutative unital ring and let $A$ and $B$ be $R$-modules. For any $b\!\in\!B$ the set $$\{a\!\in\!A;\, a\!\otimes\!b\!=\!0\text{ in }A\!\otimes_R\!B\}$$ is a submodule of $A$ that ...
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94 views

Tensor product of $C^*$- algebras

We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal $\mathcal{I}$ of $\mathcal{A}(C(T^2))$, the $C^*$-algebra generated by Toeplitz operators with ...
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76 views

sections of tensor product of sheaves of modules

I am confused about notation concerning tensor products of sheaves of modules. I know that given a ringed space $X$ and $\mathcal{O}_X$-Modules $\mathcal{F}$ and $\mathcal{G}$ their tensor product is ...
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1answer
61 views

Definition of connection on vector bundle

A connection on a vector bundle $E$ is a map $ D:\Gamma(E)\rightarrow \Gamma(T^*(M)\otimes E)$ satisfying 1) For any $s_1,s_2\in \Gamma(E)$, $D(s_1+s_2)=Ds_1+Ds_2$ 2) For $s\in \Gamma(E)$ and $\...
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1answer
33 views

What is this map between tensor spaces called? (Change of coordinates).

Let $V$ and $W$ be finite dimensional vector spaces. Let $A: V \to W$ be linear. Define the map $A^* : (W^*)^{\otimes r} \to (V^*)^{\otimes r}$ by $(A^*\alpha)(w_1, w_2, \ldots, w_r) = \alpha(Aw_1, ...
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63 views

Natural Isomorphism between $V^*\otimes W^*$ and $\mathcal L^2(V,W; F)$.

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\...
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3answers
57 views

Basic Question Regarding the Universal Property of the Tensor Product.

(All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space $V\...
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66 views

Natural Transformaton $\text{Hom}(V,W)$ and $W\otimes V^*$

Something of this form has already been answered here: Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? I'm starting introductory category theory stuff, and I'm looking for some help. I ...
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2answers
63 views

Derivation of tensor (with einstein summation)

Let $f$ be a scalar defined by: $$f = a_{iik}a_{kjj} (1 \leq i, j, k \leq 3)$$ in which $a$ is a third-rank tensor $a_{ijk} = a_{jik}$ and the summation convention for repeated indices is employed. ...
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1answer
64 views

Injective tensor product

We know that $c_0\check\otimes c_0=c_0(c_0)$ where $\check\otimes$ is the injective tensor product. is the following still true? $$c_0\check\otimes l^\infty = l^\infty(c_0).$$ Thank you for your help
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1answer
40 views

Showing $S^{-1}(M \otimes_{A} N) \cong S^{1}M \otimes_{S^{-1}A} S^{-1}N$

One of the propositions in Atiyah-MacDonald's Commutative Algebra states $$S^{-1}(M \otimes_{A} N) \cong S^{-1}M \otimes_{S^{-1}A} S^{-1}N.$$ The proof in the text states that one should use that $S^{-...
4
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1answer
125 views

Tensor product notation for $(x,y)\mapsto f(x)g(y)$

I often stumbled across (variations of) the notation $f\otimes g$ for $(x,y)\mapsto f(x)g(x)$. If $f\in V^*$, $g\in W^*$ for vector spaces $V,W$ over ${\bf K}$ then $f\otimes g:x\otimes y\mapsto f(x)...