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

2
votes
1answer
20 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 ...
0
votes
0answers
13 views

tensor products of $\mathbb Z$-modules over quotient rings

Please I have a question concerning Tensor products. What is $\mathbb Z/a \mathbb Z$ tensor $\mathbb Z /b \mathbb Z$ over $\mathbb Z/c \mathbb Z$ where $a$,$b$ and $c$ are positive integers? Maybe a ...
0
votes
0answers
6 views

Dimension of sum of permutations of tensor products of vector spaces

Sorry for the mouthful of a title! Suppose I have two finite vector spaces $W,V$ with bases $\{w_1\dots w_p\}$ and $\{v_1\dots v_q\}$. Consider some subspace $S$ of $W\otimes V$ of dimension $m$ ...
2
votes
0answers
25 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
0answers
29 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 ...
2
votes
1answer
30 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 ...
0
votes
0answers
7 views

Rewriting a sum of Young Tableaux as Tensors

Is there a straightforward way (perhaps a software) that can write a direct sum of Young Tableaux in terms of tensors? For instance the direct product in $SU(3)$ (taken from this post) ...
8
votes
2answers
141 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
25 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 ...
0
votes
1answer
37 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
14 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 ...
0
votes
0answers
54 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
1answer
31 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 ...
1
vote
0answers
38 views

Tensor products over $\mathbb{Z}$

I am doing some computation using spectrums and I would need to compute the following two tensor products: $\mathcal{O}_K\otimes_{\mathbb{Z}} \mathbb{Q}$ and ...
1
vote
4answers
78 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
25 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
41 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 ...
1
vote
2answers
24 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 ...
3
votes
0answers
24 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 ...
1
vote
1answer
49 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
24 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 ...
2
votes
2answers
44 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
45 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
57 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 ...
2
votes
3answers
26 views

Ideal contained in annihilator problem

Let $M$ be an $A$-module, and $I\subseteq \operatorname{Ann}(M)$ be an ideal. Why do we can endow $M$ in a natural way with the structure of an $A/I$-module, and why do $M\simeq M\otimes_AA/I$?
2
votes
1answer
29 views

Existence of a canonical surjective homomorphism on tensor products

I got stuck with Liu's "Algebraic Geometry and Aritmetic Curves" exercise 1.1.7. Let $B$ be an $A$-algebra, and let $M$, $N$ be $B$-modules. Why do there exists a canonical surjective homomorphism ...
0
votes
1answer
35 views

transform xWz to Wxz using tensor product

The equation I need to solve is $\mathbf{R} = \mathbf{X}^T\mathbf{W}\mathbf{Y}$ where $\mathbf{R}$ is in $\mathbb{R}^{l \times m}$. $\mathbf{X}$, $\mathbf{W}$, and $\mathbf{Y}$ are in ...
0
votes
1answer
39 views

Tensor Products of C*-Algebras

If A, B are C*-algebras, show that there exists a unique $*-isomorphism $ $ ‎\theta‎‎: A ‎\otimes‎_{*}‎‎ B ‎\longrightarrow‎ B ‎\otimes‎_{*}‎‎ A $ such that $\theta( a \otimes‎_{*} b) = b ...
3
votes
2answers
54 views

Tensor product of two finitely generated modules

How can I show that if $M$ and $N$ are finitely generated $A$-modules, then so is $M\otimes_AN$? I understand that I have assumption that there are integers $n,m$ such that there are surjections ...
3
votes
2answers
50 views

Linear maps on tensor products

Short question. Suppose we have vector spaces $V_1,V_2,V_3,V_4$ and a linear map $f: V_1\otimes V_2 \to V_3 \otimes V_4$. Are there always linear maps $f_1: V_1 \to V_3$ and $f_2: V_2 \to V_4$, such ...
1
vote
2answers
51 views

Can we take a tensor product of algebra and module?

I'm trying to learn tensor product and I found that there are at least two different tensor products, tensor product of modules and tensor product of algebras. But can we mix them? Like if $M$ is an ...
0
votes
0answers
45 views

If M is a free R module then $M \otimes S$ is a free S module.

Question is to prove that : If M is a free R module then $M \otimes S$ is a free S module, where S is an R-algebra via $\phi$. What i have done so far is : If M is a free and finitely generated R ...
1
vote
1answer
27 views

The use of universal properties to prove the existence of isomorphism

I just start self learning tensor and I find the universal property is difficult to use. I think I understand the basic concept of the universal property. The tensor product of $V_1, \cdots, V_m$, ...
0
votes
2answers
17 views

dual map to the exterior multiplication

I came across with a concept problem which ask me to describe the dual map to the exterior multiplication $$m: \bigwedge^iV\otimes\bigwedge^jV\to\bigwedge^{i+j}V$$ by the formula independent of the ...
1
vote
1answer
56 views

Vectors and Tensors

I am currently studying Reliability engineering and hence need to deal with material properties like elasticity modulus and poisson's ratio. I am basically an electrical engineer and hence never had ...
7
votes
1answer
167 views

Eigenvalues of Kronecker Product

Maybe it's simple but I can't see the solution of this problem (Russell Merris, Multilinear Algebra, CRC Press, 1997, chapter 6, p.202, exercise 4): Let $\lambda_1,\ldots,\lambda_p$ be the ...
0
votes
0answers
43 views

Local expression of a differential form

During the course, we have defined differential forms as maps $$ \omega : D(M) \times \cdots \times D(M) \rightarrow \mathcal{C}^\infty(M), $$ where $D(M)$ are the global derivates of the manifold ...
0
votes
0answers
9 views

Transition functions of sheaf tensor product

Suppose $\mathcal{L},\mathcal{M}$ are invertible sheafs on a scheme $X$. I've seen an abstract construction of $\mathcal{L}\otimes_X \mathcal{M}$, but I'm having trouble connecting this with a more ...
3
votes
1answer
38 views

jordan canonical form with direct product?

I met some problems when solving Jordan canonical forms. Here are two problems: Let $f: K^3\to K^3$ be a map in JCF having the matrix: $$\begin{pmatrix} -1 & 1 & 0\\ 0 &-1&1\\ ...
1
vote
1answer
42 views

Tensor product over Lie group isomorphic to that over its Lie algebra

Hi: I have a question about the tensor product of representation of Lie groups as follows: Let $G$ be a connected, complex Lie group with Lie algebra $\mathfrak{g}$. Let $V$ and $W$ are ...
-4
votes
1answer
72 views

Example: $ u\in A \otimes_R B$, but $u \neq a \otimes b $ for any $ a \in A, b \in B.$ [closed]

Give an example to show the following may actually occur for suitable ring $R$ and modules $A,B$: $ u\in A \otimes_R B$, but $u \neq a \otimes b $ for any $ a \in A, b \in B.$
0
votes
2answers
30 views

Annihilator of extension of scalars vs. the extension the annihilatar

Let $A,B$ be commutative rings with 1, $f:A\to B$ a morphism of rings, $M$ an $A$-module, and $M_B=B\otimes_AM$ the extension of scalars. Then is it the case that $\text{Ann}(M)^e=\text{Ann}(M_B)$? ...
0
votes
1answer
60 views

acrobatics with $2$-form in $\mathbb{R}^{2n}$ [closed]

In the space $V = \mathbb{R}^{2n}$ with coordinates $(x_1, \dots, x_n, y_1, \dots, y_n)$ consider the $2$-form $\omega = \sum_{i=1}^n x_i \wedge y_i$. Let $A$ be a $n \times n$ matrix. Consider a ...
1
vote
2answers
53 views

Is there a more intuitive way to define tensor products other than using free vector spaces?

Tensor products come up a lot in some literature I am reading. but every time I go to Wikipedia, it says a prerequisite for understanding tensor products is understanding free vector spaces. Having ...
5
votes
0answers
68 views

Decomposition of order-$n$ tensors

If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$. The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...
1
vote
0answers
37 views

Can the gradient be expressed with contravariant components?

I read that the gradient is an example of a quantity that transforms covariantly since in the below expression for the gradient $$\frac{\partial x^j}{\partial x'^i}$$ appears instead of ...
2
votes
0answers
54 views

Components of vector in dual basis transform covariantly

I am trying to understand how components of a vector in the dual basis transform covariantly as mentioned in this quote. If you seek to define a quantity (such as vector A) that remains ...
5
votes
3answers
72 views

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 ...
1
vote
2answers
91 views

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 $\mathsf{Mod}$-$R$?
0
votes
1answer
13 views

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 ...