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 ...

learn more… | top users | synonyms

1
vote
0answers
150 views

Does covariant derivative commute with “generalized contraction”_ About the proof of 2nd Bianchi identity

I am reading the proof of second Bianchi identity on wiki. In the proof, it says the following condition must satisfy: $$((D_X R) (Y,Z)) (W) + R (D_XY,Z) W + R(Y,D_XZ) W + R(Y,Z) D_X W = D_X (R(Y,Z)W)...
1
vote
1answer
43 views

$v \times w$ is a bilinear map, antisymmetic and $u \times w =0 \Leftrightarrow $ collinear in tensor product

This is my Attempt for part (b): Let's define: $$\Phi: \mathbb{R}^2 \times \mathbb{R}^2 \longrightarrow \mathbb{R}^2 \otimes \mathbb{R}^2 $$ by the following action: $$\Phi(v \times w) = v \otimes w -...
2
votes
1answer
70 views

Basic property of a tensor product

I think that this might follow from a basic properties of tensor products, but I am q bit stuck... Let $A$ be a $k$-algebra. Let $l/k$ be a finite field ext. of $k$. Suppose $A \otimes_k l$ is an ...
2
votes
1answer
66 views

How to find a non-surjective and non-injective tensor products of the formal completion?

Let $A$ be a commutative ring with unit endowed with $I$-adic topology where $I$ is the ideal of $A$. Let $\hat A$ be the formal completion of $A$ for the $I$-adic topology, and $M$ an $A$-module. Let ...
1
vote
0answers
161 views

Integration over a second order tensor

I would like to compute the mean value of a second order tensor $\mathbf{T}$ expressed in planar cylindrical coordinates. The mean value for any second order tensor is (reference [1] page 101) $$\...
1
vote
0answers
22 views

How to compute the inertial tensor ${\bf{J}} _{\Omega}$?

Suppose that $\Omega$ is a body of revolution of the function $y=f(x), a\le x \le b$ around the $x$-axis, where $f(x)>0$ is continuous. How to compute the inertial tensor ${\bf{J}} _{\Omega}$? ...
0
votes
1answer
68 views

How to prove some formulations aboult Kronecker product?

The Kronecker product has some properties as the wikipedia http://en.wikipedia.org/wiki/Kronecker_product. For the sake of simplicity, we denote $\mathbf{U}_{M}^T\otimes \cdots\mathbf{U}_{m+1}^T\...
2
votes
1answer
32 views

Tensor algebra of $Μ$ over $R$

Let $R$ be a commutative ring and $Μ$ an $R$-module. Consider the $R$-module $T_{R}(M)={\displaystyle \oplus_{n=0}^{\infty}}M^{\otimes n}$ and show that it can be endowed with the structure of an $R$-...
1
vote
0answers
53 views

Spatial tensor product of operator spaces

If $X$ and $Y$ are Banach spaces and $\otimes_\varepsilon$ denotes the injective tensor product, then in general $\otimes_\varepsilon$ does not respect quotients unless we map into $\mathscr{L}_\infty$...
0
votes
1answer
34 views

Tensor Product: Boundedness

This thread is just a note. Given Hilbert spaces. Then boundedness will be inherited: $$A,B\text{ bounded}\implies A\otimes B\text{ bounded}$$ Especially, the bounds multiply: $$\|A\otimes B\|=\|A\|\...
1
vote
1answer
33 views

Tensor Product: Denseness

This is thread is just a note. Given Hilbert spaces. Then denseness will be inherited on tensor products: $$\mathcal{D},\mathcal{E}\text{ dense}\implies\mathcal{D}\otimes\mathcal{E}\text{ dense}$$ ...
1
vote
1answer
27 views

Tensor Product: Closability

This was a real question of mine. Given Hilbert spaces. Then closability will be inherited on tensor products: $$A,B\text{ closable}\implies A\otimes B\text{ closable}$$ For simple tensors this is ...
0
votes
1answer
68 views

What is ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$?

How exactly can I prove that ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$ equals the linear space $M$ spanned by 'simple fucntions' $\phi\in{\cal D}(\Omega_0\times\Omega_1)$ of the form $\phi(x,y)=\...
0
votes
1answer
31 views

Tensor Product: Identification

This is meant as note. Given a measure space and a Hilbert space. Then there's an identification: $$\mathcal{L}^2(\mu)\hat{\otimes}\mathcal{H}\cong\mathcal{L}^2_\mathcal{H}(\mu):\quad \varphi\otimes\...
0
votes
1answer
37 views

Tensor Product: ONB

This thread is just a note. Given Hilbert spaces. Consider their hilbertian tensor product: $$\mathcal{H}\hat{\otimes}\mathcal{K}:\quad\langle\varphi\otimes\psi,\varphi'\otimes\psi'\rangle:=\langle\...
1
vote
0answers
51 views

Can someone show why this tensor product of matrices is correct?

I am trying to understand tensor products better. I decided to find an example and try to understand it. I read somewhere the the following is a correct application of the tensor product for these ...
2
votes
1answer
91 views

Evaluating contractions of a tensor product

Consider $T = \delta \otimes \gamma$ where $\delta$ is the $(1,1)$ Kronecker delta tensor and $\gamma \in T_p^*(M)$, the co-tangent space over some manifold $M$. Evaluate all possible contractions of ...
4
votes
2answers
48 views

What is the module structure here?

Let $A$ be a local ring with maximal ideal $\mathfrak{m}$, and let $M$ be an $A$-module. I want to turn the following object into an $A/\mathfrak{m}$-module: $$A/\mathfrak{m} \otimes_A M$$ I ...
4
votes
2answers
186 views

What does it mean to tensor with $\mathbb{Q}$?

At our algebraic geometry seminar I often hear that something is 'tensored with $\mathbb{Q}$', e.g. a ring of endomorphisms. This phrase seems to have some intuitive meaning that I don't know. What ...
15
votes
4answers
457 views

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$?

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$? Here tensor product is over the ring $\mathbb Z$ and $\mathbb Z[[X]] $ denotes formal power series over $\mathbb Z$. I think this ...
-1
votes
1answer
86 views

Canonical map is injective

Let $A$ and $B$ be commutative rings, and let $f:A\to B$ be a faithfully flat ring homomorphism. How can I show that for any $A$-module $M$, the canonical map $M\to M\otimes_AB$ is injective? I was ...
2
votes
0answers
125 views

How to prove the formulation of mode-$n$ matricization and preclusive mode-$n$ product?

The mode-$n$ product of a tensor $\mathcal{X}=[x_{i_1,\ldots,i_M}]\in \mathbb{R}^{I_1\times \cdots \times I_M}$ and a matrix $\mathbf{U}=[u_{i_m,j}]\in \mathbb{R}^{I_m\times J}$ is denoted by $\...
0
votes
2answers
64 views

Inclusion $\mathbb{Q} \hookrightarrow \mathbb{R}$ induces an injection in scalar extensions

Question Given a torsion-free $\mathbb{Z}$-module (aka. abelian group) $G$, let $i: \mathbb{Q} \hookrightarrow \mathbb{R}$ be the inclusion. I want to show that $$ i \otimes \mathrm{id}: \mathbb{Q} \...
5
votes
0answers
80 views

Tensor of tensored categories

Given two $V$-categories $C$ and $D$ tensored over a symmetric monoidal category $V$, could I form the "tensor" of $C$ and $D$? More precisely, is there a $V$-category $T(C,D)$ such that $V$-functors ...
1
vote
1answer
78 views

Confusion in the tensor product homomorphism in representations

Let $k$ be a field and $G$ a group. Let $V$ and $W$ be two representations. And $V \otimes _k W$ be their tensor product which itself a representation with $G$ act on $V \otimes _k W$ by $g ( v \...
2
votes
1answer
71 views

Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: $$S_\pm\left(\bigotimes_{i=1}^kv_i\right):=\frac{1}{k!}\sum_{\sigma\in\...
1
vote
1answer
88 views

Show that the trace of the operator $S \wedge T$ is zero

I have some difficulties with the following problem: Let $V$ be a finite dimensional vector space over $\mathbb{K}$. Let $S,T \in L(V,V)$. Show that the trace of the operator $S \wedge T$...
1
vote
2answers
91 views

Question about tensor products, decomposable tensors, …

I need some help with the following problem: Let $V_1,\ldots,V_m$ be finite dimensional vector spaces over $\mathbb{K}$. Let $\varphi \in L(V_1,\ldots,V_m;U)$ such that $Im(\varphi)=U$. ...
1
vote
1answer
53 views

What is the completed projective tensor product of compactly supported smooth functions on two manifolds?

Let $M$ and $N$ be smooth manifolds (not necessarily closed). It is a lovely fact that $$C^\infty(M \times N) \cong C^\infty(M) \hat{\otimes}_\pi C^\infty(N).$$ See, for the instance, the book ``...
2
votes
1answer
78 views

Computation rules for tensor products and inner products

I'm studying distributions and just came along the formula $$\langle f\otimes g,\phi\otimes\psi\rangle=\langle f,\phi\rangle\langle g,\psi\rangle.$$ I understand what that means in the context of ...
2
votes
0answers
56 views

Reference for closed categories and monoidal categories

I'm looking for a book that: Defines closed categories separately from monoidal categories, and then proves in detail that the structure induced by a left adjoint to the internal hom is closed ...
3
votes
1answer
81 views

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ as a coproduct?

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ is defined grading-wise by $$(A\Rightarrow B)_n=\prod_{i\in \mathbb Z} \text{Hom}_R(A_i, B_{i+n})$$ Intuitively, I would have defined the ...
3
votes
1answer
39 views

Universality of tensor product from monoidal structure

As a follow-up to this previous question of mine, I'm trying to understand how to obtain tensor products from internal homs. I'm having a lot of difficulties and have found myself stuck already in $\...
9
votes
2answers
287 views

Tensor products from internal hom?

Monoidal categories come with tensor products, and sometimes, these categories are biclosed, i.e each restriction of the tensor bifunctor has a right adjoint. If the category happens to be symmetric, ...
1
vote
0answers
57 views

Understanding Tensor product of modules

This is follow up question Understanding the Details of the Construction of the Tensor Product If $M$ and $N$ are 2 A-modules, we define $Z = A^{M\times N}$ as module generated by $M\times N$. Then $...
1
vote
1answer
112 views

Tensoring the exact sequence by a faithfully flat module

I have problems to do the exercise 1.2.13 from Liu's "Algebraic Geometry and Arithmetic curves". First, Liu defines that if $M$ is a flat module over a ring $A$, then $M$ is faithfully flat over $A$ ...
1
vote
0answers
151 views

Isomorphism between tensor product of dual space and space of bilinear maps

I am trying to show $V^* \otimes W^* \simeq L_2(V \times W,\mathbb{R})$ using the definition here. Hence I try to show $\phi:V^* \otimes W^* \to L_2(V \times W,\mathbb{R})$ defined by $\phi(\alpha\...
0
votes
0answers
112 views

Clarification of definition of tensor product

I am reading "Riemannian Geometry" by Gallot. And I am confused with the following definition of tensor product: Let $E$ and $F$ are two finite dimensional vector spaces, a vector space $E\otimes ...
2
votes
2answers
90 views

What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?

differential geometer in training here. With regards to my background, I learned differential and Riemannian geometry from O'Neill and Lee's series. I'm working on my algebra background (which is ...
2
votes
1answer
222 views

Tensor products over $\mathbb{Z}$

I am doing some computation using spectra and I would need to compute the following two tensor products: $\mathcal{O}_K\otimes_{\mathbb{Z}} \mathbb{Q}$ and $\mathcal{O}_K\otimes_{\mathbb{Z}}\mathbb{F}...
2
votes
4answers
97 views

An elementary fact about tensor product of modules

Let $A$ be an $R$-right module, $N$ be a submodule of $R$-left module $M$, $\pi:M\rightarrow M/N $ is the natural epimorphism. What is $\ker\left(\pi\otimes1_{A}\right) $? I appreciate your help.
0
votes
1answer
40 views

Tensor product and localization

This is from Liu, problem 1.2.2. Let $\rho:A\to B$ be a ring homomorphism, $S$ a multiplicative subset of $A$, and $T=\rho (S)$. Show that $T^{-1}B\simeq B\otimes_AS^{-1}A$ as $A$-algebras. I ...
1
vote
1answer
88 views

Tensor product definition?

I am getting a bit confused on the notation used for tensor products, is we have the tensor product space $V\otimes V^*$ if $v\in V$ and $a \in V^*$ then is the following correct? $$v \otimes a=\...
1
vote
2answers
72 views

Rank of a Decomposable Tensor

I'm independently studying Stephen Roman's Advanced Linear Algebra, and I came across a line of reasoning that appears obvious but that I don't understand, and was hoping someone might help me clarify....
3
votes
0answers
78 views

Property of pullback of quasi-coherent sheaves

In Hasrtshorne the pullback $f^{*}\mathcal{F}$ of a sheaf $\mathcal{F}$ on $Y$ via a map $f:X \rightarrow Y$ is defined as $f^{-1}\mathcal{F}\otimes_{f^{-1}\mathcal{O}_Y}\mathcal{O}_X$. It is quite ...
2
votes
1answer
130 views

Abelian categories with tensor product

Is there a standard notion in the literature of abelian category with tensor product? The definition ought to be wide enough to encompass all the usual examples of abelian categories with standard `...
1
vote
1answer
33 views

Canonical homomorphism related to ideal is an isomorphism

I have a problem to do the exercise 1.2.1 b on Liu. Namely, Let $M$ be an $A$-module, $I\subseteq \operatorname{Ann}(M)$ an ideal, $N\ne M$ is an $A$-module such that $I\subseteq \operatorname{Ann}(N)...
4
votes
2answers
122 views

Exterior Product vs Cross Product

I was confused about the relationship between a set of basis vectors in 3D, $ \left\{\hat e_1, \hat e_2, \hat e_3 \right\} $ and their exterior products. In my head, it makes sense that the identity ...
1
vote
2answers
93 views

Help understanding tensor products

I am struggling to understand tensor products. I will first state what I think I understand and then ask questions. Definition of Tensor Product: http://en.m.wikipedia.org/wiki/Tensor_product ...
1
vote
1answer
77 views

The equivalent definitions of linearly disjoint field extensions

I am trying to proove that the following charactrerization of linearly disjontness holds. We have the definition: Given the following extensions, $K\subset L, M \subset N$, $M$ is said to be ...