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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2
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1answer
91 views

Why is the image of $M\times N\rightarrow M\otimes N, (m,n)\mapsto m\otimes n$ not a submodule?

$M,N$ are $A$-modules. I don't see why the statement is true. Can you explain please?
2
votes
1answer
96 views

Why does $M$ free imply $M\otimes_K N$ is free also?

I don't understand the following implication in some notes I'm reading, either I'm overlooking something simple, or some details are being swept under the rug. Let $K$ be a field, and ...
1
vote
2answers
143 views

Prove that $A^{\oplus n} \otimes_A N \cong N^{\oplus n}$

Let $A,N$ be $A$-modules. I'm trying to prove $A^{\oplus n} \otimes_A N \cong N^{\oplus n}$. First we define $f: F(A,N) \rightarrow N^{\oplus n}$ where $F(A,N)$ is a free module with a basis ...
5
votes
2answers
599 views

Why this element in this tensor product is not zero?

$R=k[[x,y]]/(xy)$, $k$ a field. This ring is local with maximal ideal $m=(x,y)R$. Then the book proves that $x\otimes y\in m\otimes m$ is not zero, but I don't understand what's going on, if the ...
7
votes
1answer
555 views

Irreducible representations of a tensor product

Let $A, B$ be finitely generated (noncommutative) algebras over a field $k$ (say, algebraically closed). Can we get all irreducible representations of $A \otimes_k B$ from tensoring representations of ...
8
votes
1answer
285 views

Dual of a finite dimensional algebra is a coalgebra (ex. from Sweedler)

Let $(A, M, u)$ be a finite dimensional algebra where $M: A\otimes A \rightarrow A$ denotes multiplication and $u: k \rightarrow A$ denotes unit. I want to prove that $(A^*, \Delta, \varepsilon) $ ...
7
votes
1answer
572 views

Condition for a tensor to be decomposable

Let $V$ be a vector space of dimension 3 with basis $e_1,e_2,e_3$. Let $W$ be a vector space of dimension 2 with basis $f_1,f_2$. Is $e_1\otimes f_1+e_2\otimes f_2$ decomposable? What about ...
22
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2answers
2k views

Tensor products commute with direct limits

This is Exercise 2.20 in Atiyah-Macdonald. How can we prove that $\varinjlim (M_i \bigotimes N) \cong (\varinjlim M_i) \bigotimes N$ ? Atiyah gives a suggestion, he says that one should obtain a map ...
1
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0answers
35 views

reducing the dimension of a tensor

I have a tensor $C$ of size $a \times b \times c$, all with real values. I need to compute for a vector $u$ of length $a$ and a vector $w$ of length $c$ the product: $(C \times_{3} w) \times_{1} u$ ...
2
votes
2answers
91 views

How can one see that $\operatorname{tr}(f\otimes g)=\operatorname{tr}f\operatorname{ tr }g$?

Suppose you have two free modules $M$ and $N$ of finite rank over a commutative ring $R$. Let's also take some $f\in\operatorname{End}_R(M)$ and $g\in\operatorname{End}_R(N)$, which gives a ...
9
votes
2answers
186 views

Why is $\mathbb{H}\otimes\mathbb{H}\cong\text{End}_\mathbb{R}\mathbb{H}$?

When I first learned of the quaternion algebra $\mathbb{H}$, the most concrete way to get a grip on the ring of its endomorphisms $\operatorname{End}_\mathbb{R}(\mathbb{H})$ was to view them as ...
9
votes
4answers
277 views

Why does $(A/I)\otimes_R (B/J)\cong(A\otimes_R B)/(I\otimes_R 1+1\otimes_R J)$?

While reading, there is an isomorphism that I'm having trouble fulling seeing. If you have two algebras $A$ and $B$ over a commutative ring $R$, with $I$ and $J$ two sided ideals in $A$ and $B$, ...
3
votes
1answer
426 views

Tensor products

I'm trying to get my head round tensor products of vector spaces (I'm happy to see arguments in a more general setting, though). I am concerned principally with two statements: i) If $U,V,W$ are ...
4
votes
1answer
189 views

Corollary 2.13 of Atiyah - Macdonald

I just started learning about tensor products and I have some trouble understanding this corollary in Atiyah - Macdonald. All modules are assumed to be $A$ - modules for $A$ a commutative ring. ...
0
votes
1answer
75 views

What does it mean for tensoring to be exact?

I read a phrase saying that tensoring $\mathbb{Q}$ over $\mathbb{Z}$ is exact. What does it mean for tensoring to be exact?
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2answers
97 views

Why is this homomorphism on an infinite product well defined?

Browsing over some questions, I found that the natural homomorphism from $(\prod M_i)\otimes N\to \prod(M_i\otimes N)$ is given by $(\prod m_i)\otimes n\mapsto \prod(m_i\otimes n)$. This of course ...
11
votes
2answers
627 views

Examples proving why the tensor product does not distribute over direct products?

I recently read about the result that the tensor product distributes over direct sums. I was curious if it also distributes over direct products, but google tells me it doesn't. What are some simple ...
2
votes
1answer
128 views

Antisymmetric functions in higher dimensions

For an antisymmetric function $f:\mathbb R\rightarrow \mathbb R$ (i.e. f(x)=-f(-x)) we have: necessary condition for the differential of f of order $r$ to not vanish at $0$ is that $r$ is odd. My ...
16
votes
1answer
437 views

Modules with $m \otimes n = n \otimes m$

Let $R$ be a commutative ring. Which $R$-modules $M$ have the property that the symmetry map $$M \otimes_R M \to M \otimes_R M, ~m \otimes n \mapsto n \otimes m$$ equals the identity? In other ...
3
votes
1answer
94 views

Why is $M_A\otimes_A N\cong M\otimes_R N$?

I've been doing some tensoring, but am having a hard time understanding the following isomorphism. Suppose $A$ is a commutative $R$-algebra, and for any $R$-module $M$, denote by $M_A=A\otimes_R M$. ...
2
votes
1answer
59 views

Finding a $\mathbb{C}$-balanced map from $\mathbb{C}^2\to\mathbb{R}^2$?

I was trying to show $\mathbb{C}\otimes_\mathbb{C}\mathbb{C}\cong\mathbb{R}^2$ as $\mathbb{R}$-modules. At one point I would like to prove the existence of an $\mathbb{R}$-module homomorphism from ...
6
votes
3answers
352 views

For $M\otimes_R N$, why is it so imperative that $M$ be a right $R$-module and $N$ a left $R$-module?

I was reading about the construction of a tensor product of any right $R$-module $M$ and left $R$-module $N$ over a ring $R$. Why is it required that $M$ and $N$ be right and left modules, ...
2
votes
0answers
112 views

induced representation of tensors of irreducibles

Let $V_{\lambda}$ and $V_\mu$ be representations of the symmetric groups $\mathfrak{S}_d$ and $\mathfrak{S}_m$ respectively where $\lambda$ is a partition of $d$ and $\mu$ is a partition of $m$. It is ...
2
votes
0answers
92 views

Isomorphism modulo torsion is stable under tensor products

Let $\phi : A \to B$ be a homomorphism of grades $k$-algebras and $N$ be a graded $A$-right module. Assume that $\mathrm{ker}(\phi)$ and $\mathrm{coker}(\phi)$ are torsion $A$-right modules, i.e. ...
16
votes
2answers
1k views

Geometric intuition of tensor product

Let $V$ and $W$ be two algebraic structures, $v\in V$, $w\in W$ be two arbitrary elements. Then, what is the geometric intuition of $v\otimes w$, and more complex $V\otimes W$ ? Please explain for me ...
3
votes
1answer
186 views

$\mathrm{Tor}$ functor not left exact

Is there an example which shows that the functor $B\otimes_R(-)$ is not left-exact, given a ring $R$ and a right $R$-module $B$?
3
votes
2answers
276 views

$1\otimes \arccos \frac{1}{3}$ is not zero in $\mathbb{R}{\otimes}_{\mathbb{Q}}(\mathbb{R}/\mathbb{Q})$

How to prove that the element $1\otimes \arccos\frac{1}{3}\in\mathbb{R}{\otimes}_{\mathbb{Q}}(\mathbb{R}/\mathbb{Q})$ isn't equal to zero? I know why $$\arccos\frac{1}{3}\neq \frac{m}{n}\pi,$$ where ...
7
votes
2answers
317 views

Tensor Decomposition

Consider a tensor product $$ V^{\otimes n} = \underbrace{V\otimes\cdots\otimes V}_{n} $$ where $V$ is a vector space over $\mathbb R$, $\dim V = m$ , hence $\dim V^{\otimes n} = m^n$ . So every $A ...
1
vote
1answer
386 views

Question about perfect pairings

Suppose we had a bilinear map $V \otimes W \rightarrow \mathbb{C}$. What is meant by the fact that this map is a perfect pairing? How does one go about and show that something is a perfect pairing?
3
votes
1answer
52 views

$\hom_{k}\left(V_{p,q},V_{r,s}\right)\simeq V_{q+r,p+s}$

Define $V_{p,q}=\underset{p}{\underbrace{V\otimes\cdots\otimes V}}\otimes\underset{q}{\underbrace{V^{*}\otimes\cdots\otimes V^{*}}}$. In a previous question here I was shown that ...
4
votes
1answer
489 views

Endomorphisms of $V$ and the dual space

I was told that $V\otimes V^{*}\simeq\mbox{End}\left(V\right)$. I can't find the isomorphism itself though. Can anyone tell me what it is with a proof? Thanks!
6
votes
1answer
173 views

Modules over a tensor product

Let $k$ be a field. Suppose $A$ and $B$ are two commutative $k$-algebras. Let $M$ be a finite $A\otimes_k B$-module. Can one find a finite $A$-module $N$ and a finite $B$-module $L$ such that $M ...
4
votes
1answer
387 views

Tensor products: proving that $I \otimes_R M \cong IM$

Assume it if it´s neccesarly that the ring has an 1 or is commutative ( I´m not sure if it´s needed) Given a ring $R$ an ideal $I$ of $R$, and a $R$ module $M$ , prove that: $ I \otimes _R M \cong ...
2
votes
1answer
181 views

Linearly disjoint field extensions and the tensor product

The problem: Let $K$ and $L$ be subfields of a field $\Omega$, and let $k\subset K\cap L$ be a common subfield. (a) Show that there exists a unique ring homomorphism $f_{K,L}:K\otimes_k L\to ...
0
votes
0answers
149 views

Induced homomorphisms of tensor products?

I'm trying to verify for myself some isomorphisms of tensor products. If $M$, $N$, and $P$ are $A$-modules ($A$ commutative, unital), I would like to see why $$(M\otimes N)\otimes ...
6
votes
1answer
243 views

A characterization of finite purely inseparable extensions of fields

Let $K/k$ be a finite extension of fields. Let $A=K \otimes_k K$. An exercise: Show that $K/k$ is purely inseparable $\Leftrightarrow A/J(A) \cong k$, where $J(A)$ is the Jacobson radical of $A$. It ...
3
votes
2answers
702 views

What is the categorical diagram for the tensor product?

The title says it all: what is the diagram that defines a tensor product? (I'm using the term diagram here in the technical sense it has in category theory.) Edit: This question was motivated by the ...
3
votes
1answer
481 views

Complexification of Tangent Bundle

I am currently reading a book where the author says that the tangent and cotangent bundles $TM$ and $T^*M$ of a manifold $M$ are complexified. I am not familiar with Complex Manifolds so looked it ...
3
votes
1answer
337 views

Spatial tensor product of algebras and normed spaces

Let $A_1$ and $A_2$ be two $C^*$-algebras considered as closed $*$-subspaces of some $\mathcal{B}(H_1)$ and $\mathcal{B}(H_2)$. More preciesly there exist faithfull $*$-representations ...
1
vote
1answer
73 views

Action of $L$ on $End(V)$.

I'm reading Introduction ot Lie Algebras and Representation Theory from James Humphreys and I do not understand the statement made at the top of page 27. Given a vectorspace $V$ (finite dimensional) ...
2
votes
1answer
155 views

Extending an ideal of a polynomial ring to a polynomial ring with more indeterminates. Is it a tensor product?

Let $\mathbb{k}$ be a field, let $S'=\mathbb{k}[x_1,x_2,\dots,x_m]$, and let $I'\subseteq S'$ be an ideal. For some $n>m$, let $$S=\mathbb{k}[x_1,x_2,\dots,x_n]\ \ \ ...
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vote
2answers
373 views

What is the difference between tensors and tensor products?

The tensor product $S\otimes_R T$ of $S$ and $N$ over $R$ is a module. A multilinear form $L:V^r \to R$ is called an $r$-tensor on $V$. On the one hand a tensor is a function sending elements of ...
5
votes
2answers
283 views

Unique symmetric covariant $k$-tensor satisfying $(\operatorname{Sym} T)(A,…,A)=T(A,…,A)$ for all $A \in V$

Let $T$ be a covariant $k$-tensor on a finite dimensional vector space $V$. I want to prove that the symmetrization of $T$ is the unique symmetric $k$-tensor satisfying the following condition: ...
5
votes
2answers
257 views

$e_1\otimes e_2 \otimes e_3$ cannot be written as a sum of an alternating tensor and a symmetric tensor

Let $(e_1,e_2,e_3)$ be the standard dual basis for $(\mathbb{R}^3)^\ast$. How can I show that $e_1\otimes e_2 \otimes e_3$ cannot be written as a sum of an alternating (or antisymmetric) tensor and a ...
2
votes
1answer
756 views

Matrix Multiplication for n-dimensional arrays

Background Within the context of my research, I have been working with a vector-based model that treats entities of a function-like language as vectors: Some of these entities are "objects", ...
14
votes
2answers
1k views

What does a zero tensor product imply?

I'm trying to prove that for two finitely generated $A$-modules $M,N$ ($A$ being any cmmutative ring), the tensor product $M\otimes_A N$ is zero iff $\operatorname{Ann}(M)+\operatorname{Ann}(N)=A$. ...
4
votes
2answers
117 views

Prove a linear combination of tensor product can not be written as a single tensor product

Prove that in $\mathbb{C}^{3}\otimes\mathbb{C}^{3}$, the state vector $$\mathbf{h}=\frac{1}{\sqrt{8}}=e_{1}\otimes e_{1}+e_{2}\otimes e_{2}+e_{1}\otimes e_{2}+e_{2}\otimes e_{1}+e_{1}\otimes ...
6
votes
2answers
241 views

Example of a non-algebraic $\ell^2$-function in two variables

Let's call an $\ell^2$-function $\mathbb{N} \times \mathbb{N} \to \mathbb{C}$ algebraic if it is in the image of the natural algebra homomorphism $\ell^2(\mathbb{N}) \otimes \ell^2(\mathbb{N}) \to ...
1
vote
1answer
74 views

Relation between free algebras and polynomials

Given a vector space $V=\mathbb{F}^{d}$, the free algebra, or tensor algebra, of $V$ is $T\left(V\right)=\oplus_{n\geq0}V^{\otimes n}$. Now, it is stated everywhere, that this is exactly the algebra ...
18
votes
2answers
2k views

Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that ...