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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No canonical evaluation map $H^*\hat{\otimes}H \to \mathbb{K}$ in $\mathsf{Hilb}$?

Is it true that there is no canonical continuous evaluation $H^* \hat{\otimes}H \to \mathbb{K}$ on the Hilbert space tensor product coming from the natural pairing $H^* \times H \to \mathbb{K}$? At ...
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41 views

$2\otimes 1$ is non-zero in $2\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$.

I had the following doubt: Show that the element $2\otimes 1$ is $0$ in $\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$ but not a zero in $2\mathbb Z\otimes_{\mathbb Z} \mathbb Z/2\mathbb Z$. ...
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14 views

Why is $\ker(id\otimes \cdot b:R/(a)\otimes_R R \to R/(a)\otimes_R R)=R/(d)$?

Let $R$ be a commutative ring with unit $1_R$, $M$ a $R$-module. Let $a,b\in R\setminus \{0\}$ and $\gcd(a,b)=d$. I want to prove: $$\operatorname{Tor}_1^R(R/(a),R/(b))=R/(d).$$By definition, it is ...
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24 views

Tensor Product of Spaces has Basis of Tensor Products

I am given the following definition of the Tensor Product of spaces Given two vector spaces $V,W$ a vector space S is a tensor product of $V,W$ if there exists a map $M$ $$ M: V \times W ...
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22 views

Intersection of tensor product of vector spaces whose intersection is $\{0\}$ is trivial

Let $V$ and $W$ be subspaces of a finite-dimensional vector space $U$ such that $$V \cap W= \{0 \}.$$ Let $A$ be a second vector space (possibly infinite). Is it true that as subspaces of $A ...
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24 views

If $f : M\otimes_A A/m \to N\otimes_A A/m$ is surjective , so is $f : M \to N$. [on hold]

Let $A$ be a local ring with maximal ideal $m$. Let $f : M \to N$ be a morphism of $A$-modules, where $N$ is finitely generated. Show that if the map $f : M\otimes_A A/m \to N\otimes_A A/m,\quad ...
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1answer
52 views

$\mathbb C$-dimension of vector space $\mathbb C\otimes_{\mathbb R}\mathbb C$

Let $\mathbb R$ be the field of real numbers, $\mathbb C$ be the field of complex numbers. Consider $\mathbb C\otimes_{\mathbb R}\mathbb C$ as a $\mathbb C$-vector space via $a(b\otimes c) := ab ...
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150 views

Example of a Tensor Product of Modules with Non-Decomposable Elements

Given a ring $R$ and $R$-modules $A_R$ and $_{R}B$, we define the tensor product $A \otimes_R B$ as the free abelian group on $A \times B$ modded out by the subgroup generated by the elements of ...
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39 views

Representing a linear operator on $V$ with an element of $V \otimes V^*$

I got interested by the first sentence of this wikipedia subsection. It claims that any linear operator $f:V\to V$ can be represented by an element of $V\otimes V^*$ in a very concrete way: the ...
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1answer
43 views

Tensor of cocomplete categories

Let $C$, $D$ and $E$ be cocomplete categories. Is there a construction $C \otimes D$ such that there is a correspondence between functors $C \otimes D \to E$ preserving colimits and functors $C ...
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1answer
57 views

Basis for Tensor products/bilinear maps

What is the delta in the image? How do I check it is a basis? Why are $i$ and $k$ the same, and $j$ and $l$? Need some clarification.
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1answer
50 views

Tensor product of Hom-module and another ring

Let $A$ be a local noetherian ring, $B$ and $C$ are finitely generated $A$-algebras and $M$ is a finitely generated $B$-module. Is the natural morphism $\mathrm{Hom}_B(M,B) \otimes_A C \to ...
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Express the operator of a tensor product by a form containing an inner product term

Assume $A,X,Y\in \mathbb{R}^{I_1\times\cdots\times I_N}$ are three arbitrary N-th order tensors. How to prove the following equation: $$\langle A,X\rangle Y=(X\otimes Y)(A),$$ where $\langle\cdot ...
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1answer
28 views

How to Prove $V\otimes sl(k)=sl(V)$?

Let V be a vector space over a field $k$. Let $sl(n)$ be the set of all matrices elements from $k$ with trace zero. Is it true that $V\otimes _k sl(n)=sl(V)= \text{set of all $n\times$ n matrices ...
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1answer
36 views

Tensor Product on Hilbert Spaces (well definedness)

Let $H_1$, $H_2$,...,$H_n$ be $n$ Hilbert Spaces. For each $\phi_i \in H_i$, Let $$ \phi_1 \otimes \phi_2 \otimes... \otimes \phi_n:= \text{Conjugate multilinear form which acts on $H_1 \times H_2.. ...
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1answer
32 views

how to prove an element is non-zero in a tensor-product

I was studying the following example from Atiyah & MacDonald's Introduction to Commutative Algebra: let $x$ be the non-zero element in $N := \mathbf{Z}/ 2\mathbf{Z}$, $M := \mathbf{Z}$, and $M' := ...
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1answer
33 views

Associativity of tensor product over various rings

From Atiyah-MacDonald: Exercise 2.15. Let $A$, $B$ be rings, let $M$ be an $A$-module, $P$ a $B$-module and $N$ an $(A,B)$-bimodule (that is, $N$ is simultaneously an $A$-module and a $B$-module ...
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28 views

tensor identity for cross product

I've read somewhere the following identity for a tensor rank 2 $ \nabla \times \nabla v =0 $ where $v$ is a vector of "j" components and $\nabla = \frac{\partial}{\partial x_i}$, such that $ \nabla ...
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25 views

$R[S]\cong \mathbb{Z}[S]\otimes_{\mathbb{Z}} R$, $R\otimes_{\mathbb{Z}} \mathbb{Z}\cong R?$

I don't know much about tensor products in general but for for some lectures I need basics about tensor products of rings, modules and abelian groups. I have the following questions: 1)Let $R$ be a ...
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1answer
63 views

different approaches of defining tensors

This Wikipedia article says that tensor can be defined as miltilinear maps or be defined using tensor products. Could anybody explain with a simple example why these two approaches give the same ...
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1answer
27 views

Universal Linearizer of Alternating Multi-$F[x]$-Linear Maps is Same as that of Multi-$F$-Linear Maps.

Let $V$ be a an $n$-dimensional vector space over a field $F$. Let $M=F[x]\otimes_F V$. We can consider $M$ as an $F[x]$-module by extending scalars using the inclusion $F\to F[x]$. Fact 1. There is ...
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1answer
30 views

Dual of a module and base change [closed]

Let $\phi:A \to B$ be a ring homomorphism and $M$ an $A$-module (not necessarily free). Is it true that $$\mathrm{Hom}_A(M,A) \otimes_A B \cong \mathrm{Hom}_B(M \otimes_A B,B)?$$ If necessary assume ...
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1answer
34 views

reduced density matrix for the given composite system

Given the composite system of two qubits $$ |\psi^{AB}\rangle=\frac{1}{\sqrt{2}}(|0^{A}\rangle \otimes|0^{B}\rangle+|1^{A}\rangle\otimes|1^{B}\rangle) $$ with the density matrix of the composite ...
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1answer
32 views

$V(1)$ generates the tensor category of representations of $sl_2(\mathbb{C})$ - what exactly does this mean?

Does anyone know what it means for a $V(1)$ to generate the tensor category of (finite dimensional) representations of $sl_2(\mathbb{C})$. Here $V(1)$ is the 2 dimensional irreducible Lie algebra ...
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1answer
23 views

Elements of tensor product

Let $R, S$ be rings such that $R$ is a subring of $S$ and $1_R = 1_S$. Let $N$ be a left $R$-module. Let the free $\mathbb{Z}$-module on $S \times N$ be $F_\mathbb{Z}(S \times N)$. Let $H$ be the ...
2
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0answers
68 views

What are the non-vanishing directions of $A_{a(b} B_{cd)} - B_{a(b} A_{cd)}$? ($A$ and $B$ symmetric tensors)

Question Consider two symmetric tensors $A_{ab}=A_{ba},\, B_{ab}=B_{ba}$ which are generally of full rank (non-degenerate, all eigenvalues non-zero but of general sign) and their tensor product under ...
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2answers
23 views

Tensor product with irreducible representation has no $G$-invariant submodules

Let $\rho: G \to GL(V)$ be a finite dimensional irreducible representation of a group $G$ over an algebraically closed field $\mathbb{F}$ of characteristic $0$, and let $R$ be a commutative ring with ...
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34 views

What kind of operation is available here?

What kind of operation $\clubsuit$ is available here? $$ \underbrace{\left(v^{\mathrm T}A_{n\times n\times n}\right)}_{\in\mathbb R^{n\times n}}\underbrace{\left(v^{\mathrm T}B_{n\times n\times ...
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0answers
20 views

Books to learn tensor product on hilbert spaces

I have just started to work on Quantum Computing. I have began to read a paper which deals with tensor product on hilbert spaces. I have a had a course in functional analysis. So I don't have an ...
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2answers
80 views

If $M \otimes N \cong M' \otimes N$, is it true that $M \cong M'$? [duplicate]

I tried using the universal property of tensor products to show that there are mutually inverse maps from $M \times N$ to $M' \times N$, and use this to show that $M \cong M'$, but I didn't get far. I ...
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3answers
51 views

Showing that a ring homomorphism induces a correspondence that maps free modules to free modules

Let $f: \Lambda \rightarrow \Lambda'$ be a ring homomorphism. This induces a mapping $M \mapsto \Lambda' \otimes_{\Lambda} M$, where $M$ is a $\Lambda$-module and $\Lambda' \otimes_{\Lambda} M$ is a ...
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1answer
30 views

Cotangent bundle tensor product tangent bundle

What is the meaning of Cotangent bundle tensor product tangent bundle: $T^*M\otimes TM$? what will an element of this space be?
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2answers
47 views

Understanding of an example of “extending scalars”

The following is an example in the Abstract Algebra by Dummit and Foote: I don't understand in this example why $\iota$ is an isomorphism. By Theorem 8, I can get $$ id_N=\Phi\circ\iota $$ which ...
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1answer
87 views

$A\otimes_{\mathbb C}B$ is finitely generated as a $\mathbb C$-algebra. Does this imply that $A$ and $B$ are finitely generated?

Consider $A$ and $B$ two $\mathbb C$-algebras such that $A\otimes_{\mathbb C}B$ is finitely generated as a $\mathbb C$-algebra. Does this imply that $A$ and $B$ are finitely generated? I know that ...
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2answers
28 views

Checking if an element of tensor product is zero

I am trying to understand tensor product. Let $R$ be a ring and $R_0$ its subring. Then $R$ is a right $R_0$-module. Let $M$ be a left $R_0$-module. Is the element $1\otimes m$ of $R\otimes_{R_0} ...
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47 views

a property of the tensor product of modules

The following theorem is from the Abstract Algebra by Dummit and Foote (in the section 10.4 tensor products of modules): Would anybody illustrate how Theorem 8 is used to get $$ \textrm{ker ...
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1answer
30 views

Multiplication of tensor and vector

How to do the multiplication of the multidimensional array $A_{n\times n\times n}$ and the vector $v_n$ (indices denote dimensions)? Can you kindly give suggestions or references? Thanks in advance.
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1answer
31 views

Functor right adjoint to $.\otimes_BA$

Given a ring morphism from $B$ to $A$, we can regard an $A$-module $M$ as a $B$-module. Then how can I prove the functor $._B:\operatorname{Mod}_A \to \operatorname{Mod}_B$ is right adjoint to ...
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1answer
46 views

Does a basis for $V \otimes_{\mathbb{F}} W$ always consist of pure tensors?

Given a field $\mathbb{F}$ and two $F$-vector spaces $V$ and $W$, it's true that if $\{v_i\}$ and $\{w_j\}$ are bases for $V$ and $W$, respectively, then the set $\{v_i \otimes w_j\}$ is a basis for ...
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2answers
26 views

Notation for equal via isomorphism

I am working with tensor products and there are a lot of identities, etc. that are true after appropriate identifications (between tensor products) are made. For example, if $V$ is a vector space, ...
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2answers
58 views

Show that $a \otimes 1$ will vanish in $A\otimes_{\Bbb{Z}}\Bbb{Q}$ only when $a=0$.

Let $A$ be a $\mathbb{Z}$-module. I want to see that if $a \otimes 1=0$ in $A\otimes_{\Bbb{Z}}\Bbb{Q}$, then $a=0$. I'll assume $A$ is torsion free. Can I use that there exists an injective ...
2
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0answers
67 views

about the the tensor product

Because I'm a beginner in those stuffs I find some difficulties in the following (Using the universal property of the tensor product) prove that $\mathbb {Q}\otimes \mathbb{Q} \cong \mathbb{Q}$ ...
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1answer
35 views

What is considered to be the natural (injective) homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$?

Let $R$ be a ring and $I,J,L \unlhd R$ such that $J \subseteq I$. What is considered to be the natural homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$ ? Remark: It must be ...
2
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1answer
60 views

Induction functor from $\mathbb{K}$-mod to ($\mathbb{K}\times\mathbb{K}$)-mod

Let $\mathbb{K}$ a field. Given a $\mathbb{K}$-mod (a vector space) there is an induction functor on $\mathbb{K} \times \mathbb{K}$-mod that is, as usual, $- \otimes_\mathbb{K} (\mathbb{K} \times ...
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1answer
78 views

When considering vector spaces $A$ and $B$, what is the difference between $A \times B,$ $A \otimes B$ and $A \wedge B?$

I have looked at this resource http://hitoshi.berkeley.edu/221a/tensorproduct.pdf to instinctively differentiate between the tensor product and the direct sum of two vector spaces. I am currently ...
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0answers
19 views

Product of a matrix and a tensor

I need to know how to compute the following product: $M(x)\frac{\partial M(x)}{\partial x}M(x)$ $\quad$ where $x \in R^{n}$. Assuming the dimensions of the matrices are compatible,how do we take ...
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0answers
27 views

tensor product of a number field with $ \mathbb R $

Let $K$ be a number field, i.e. a finite extension of the field of rational numbers $ \mathbb Q .$ And consider the tensor product: $$ K \otimes_{\mathbb Q } \mathbb R $$ I have the questions: ...
2
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1answer
38 views

Confusion regarding notation of a dual transformation

I'm reading Spivak's Calculus on Manifolds and in Chapter 4 he defines the dual transformation (although he doesn't call it that) as follows: If $f:V \rightarrow W$ is a linear transformation, a ...
0
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1answer
28 views

Does the tensor product distribute over a direct sum exactly if all the involved modules are bimodules?

Given a non-commutative ring $R$, a right module $M$, and a left module $N$, we can define the tensor product $M \otimes_R N$. I suspect that $$ \left(\bigoplus_{i=1}^n M_i\right) \otimes_R N \simeq ...
0
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1answer
44 views

Short exact sequences of quasi-coherent sheaves and closed subschemes

I am confused by this exercise in Ravi's notes: 16.3F (paraphrasing) Suppose $i : p \to A_k^1$ is the inclusion of the origin. Consider the associated short exact sequence of quasi-coherent modules: ...