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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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
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1answer
185 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
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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
315 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
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1answer
374 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
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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
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1answer
470 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!
7
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1answer
169 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
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1answer
384 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
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1answer
179 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
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0answers
147 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
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1answer
242 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
682 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
467 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
334 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
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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
154 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]\ \ \ ...
1
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2answers
365 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
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2answers
277 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
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2answers
254 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
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1answer
734 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", ...
13
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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
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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
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2answers
236 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
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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 ...
17
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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 ...
3
votes
3answers
197 views

Do tensor products support cancellation?

If $A\otimes B\simeq A\otimes C$, is it true that $B\simeq C$? If not, under what conditions might this be true? EDIT: More specifically, I'm working on a problem where I have a field $K$ and an ...
5
votes
2answers
294 views

Using the notation of wedge product to solve a linear system of equations

I am trying to solve a problem that seems like a standard idea from linear algebra but with a the notion of wedge product and exterior algebra added it gets more complicated for someone who isn't ...
1
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1answer
103 views

Alternative definition of rank involving tensor products

I have come across a definition of rank in some lecture notes I am using to prepare for a linear algebra qualifying exam and the notes define the rank of a linear map in a somewhat different manner ...
3
votes
2answers
210 views

what is the tensor product $\mathbb{H\otimes_{R}H}$

I'm looking for a simpler way of thinking about the tensor product: $\mathbb{H\otimes_{R}H}$, i.e a more known algbera which is isomorphic to it. I have built the algebra and played with it for a ...
3
votes
1answer
225 views

Tensor products and linear transformations

If $E$ and $F$ are linear transformations. How does one prove that $rank(A \otimes B)=rank(A) rank(B)$ where $\otimes$ is the tensor product. This is a question I do not know how to approach. Can I ...
3
votes
2answers
172 views

a simple tensor product question

I have just started to learn about this today, and though i understand this is probably a very simple question, i'm still quite not sure about it: is ...
2
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2answers
613 views

Tensor product as a colimit

I've been dealing with category theory for three weeks now and we just started covering limits and colimits, meanwhile in my geometry class we defined the tensor product of vector spaces. Then I ...
7
votes
2answers
355 views

Why does $M \mathbin{\otimes_R} N \cong M_\mathfrak{p} \mathbin{\otimes_{R_\mathfrak{p}}} N$?

Let $R$ be a commutative ring, $\mathfrak{p}$ a prime ideal of $R$, $M$ a $R$-module, and $N$ a $R_\mathfrak{p}$-module. Why do we have this isomorphism? $$M \mathbin{\otimes_R} N \cong M_\mathfrak{p} ...
2
votes
2answers
185 views

Why does a tensor algebra not contain any zero divisors?

The following question is related to this post What elements in a tensor algebra are invertible? and mainly I am confused if the answer in the above post applies to showing that the tensor algebra of ...
4
votes
1answer
248 views

What elements in a tensor algebra are invertible?

A question was brought up to me about if it is possible to come up with a module that has no non trivial invertible elements in its respective tensor algebra. I am not sure if this is trivial based ...
2
votes
1answer
69 views

Showing that $\wedge^2 \operatorname{Frac}(I) = 0$ where $I$ is an integral domain

The following is based on a follow up to some of the comments from this post Exterior powers of a module contained in a field of fractions Let $I$ be an integral domain and let ...
2
votes
1answer
241 views

Why is the tensor algebra of a vector space non-commutative?

I was just curious in describing the notion that the tensor algebra of a vector space (That is the direct sum of all spaces containing k-tensors for each k) need not be commutative because I am having ...
1
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0answers
136 views

Why is there no isomorphism between exterior powers of a free module?

The following question is related to this post Showing an isomorphism between exterior products of a free module about the existence of a canonical isomorphism of the exterior product of free modules ...
0
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0answers
102 views

Questions about a factorization theorem using tensor products

I think this problem is a simple generalization of a standard factorization theorem over the integers formulated in the language of abstract algebra but I have not quite been able to generalize the ...
1
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1answer
164 views

Showing an isomorphism between exterior products of a free module

The following is a question that comes from a counterexample to a problem about exterior algebras of modules and the notes mentions somehow this all relates to the fact that symplectic geometry is ...
1
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2answers
231 views

How do we describe standard matrix multiplication using tensor products?

Let $V$ be a finite dimensional vector space over a field $F$. Consider the bilinear map $End(V) \times End(V) \rightarrow End(V)$ given by $(u,v) \rightarrow u \circ v$ and the map associated linear ...
2
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1answer
103 views

Exterior powers of a module contained in a field of fractions

I am working some problems that can potentially be on a qualifying exam about tensor algebras and have come across some questions about field of fractions which is something I have not seen for a ...
2
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1answer
81 views

Equivalent conditions for existence of an invertible element in an exterior algebra

I am working on old qualifying problems involving tensor products. I am stuck on a statement about invertible elements in an exterior algebra and was wondering if this was a well known fact in a book ...
8
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1answer
464 views

A question about the tensor product of $\mathbb{Q}$

I'm reading this blog post about $\mathbb{Q} \otimes_\mathbb{Z} \mathbb{Q}$ and I have two questions about it: Is a simple tensor a tensor that cannot be written as a sum of tensors? On the first ...
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5answers
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Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$ Can you tell ...
3
votes
2answers
409 views

Two definitions of tensor product: when are they equivalent? [duplicate]

Possible Duplicate: understanding of the “tensor product of vector spaces” Take vector spaces $V, W$ over the field $\mathbb{K}$. I've come across two different definitions of ...
2
votes
1answer
188 views

Isomorphism involving tensor products of homomorphism groups

This post is related to the following question: A counterexample to the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$. I have been trying to isolate what hypothesis can be eliminated in these ...
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1answer
233 views

Solution space of Kronecker product of variables

i want to solve the following equation. $\begin{bmatrix} \mathbf{A_1} & \mathbf{A_2} & \mathbf{A_3} & \mathbf{A_4} \\ \mathbf{A_5} & \mathbf{A_6} & \mathbf{A_7} & \mathbf{A_8} ...
5
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1answer
479 views

Understanding the Details of the Construction of the Tensor Product

This is essentially a follow-up on this question here. Specifically, I'm studying the construction of the tensor product of two vector spaces as indicated by the OP's 2nd definition. Namely, that the ...