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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202 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 ...
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4answers
96 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.
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1answer
38 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 ...
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1answer
87 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 ...
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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 ...
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77 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
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1answer
127 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 ...
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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 ...
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2answers
114 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 ...
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2answers
87 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 ...
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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 ...
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3answers
51 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 $M\simeq M\otimes_AA/I$?
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1answer
70 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 ...
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1answer
46 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 ...
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1answer
63 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 ...
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2answers
361 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
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2answers
62 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 ...
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65 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 ...
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1answer
99 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 $V_1, \cdots, V_m$, ...
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2answers
37 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 ...
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1answer
89 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 ...
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1answer
464 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 ...
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0answers
70 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 ...
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1answer
35 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
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1answer
78 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\\ ...
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1answer
91 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 ...
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2answers
52 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 $\text{Ann}(M)^e=\text{Ann}(M_B)$? ...
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1answer
68 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 ...
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2answers
104 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 ...
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91 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 ...
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71 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
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1answer
214 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
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3answers
167 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 ...
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2answers
349 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 $\mathsf{Mod}$-$R$?
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1answer
64 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 ...
2
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2answers
66 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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2answers
185 views

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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0answers
30 views

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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1answer
132 views

Tensor Product: Vector Spaces

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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231 views

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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4answers
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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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2answers
88 views

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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1answer
91 views

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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1answer
43 views

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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2answers
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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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3answers
116 views

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 ...
2
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1answer
110 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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1answer
62 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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1answer
36 views

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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1answer
89 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) ...