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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Given ring $A$, ideal $I$, and $A$-module $M$, show that $A/I \otimes_A M$ is isomorphic to $M/IM$.

The question is stated as in the title; the hint I am given is to "tensor the exact sequence" $0 \rightarrow I \rightarrow A \rightarrow A/I \rightarrow 0$, which I take to mean using that sequence ...
3
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212 views

Tensor product of affine space and algebra

When reading about Quantum Mechanics, I always feel a bit disappointed when physicists consider that (for example) the 3-dimensional position of a particle must be decomposed into 3 coordinates $x, y, ...
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20 views

How to define the tensor product via an initial morphism? [duplicate]

I am just getting to know some category theory. My understanding of the "universal property" (based on this Wikipedia article) is that it is characterized in terms of an "initial morphism" or ...
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1answer
243 views

Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$?

I have a couple of questions about tensor products: Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? Why is an element of $V^{*\otimes m}\otimes V^{\otimes n}$ the same thing as a ...
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155 views

Tensor Product = Multilinear Maps on Algebraic Duals?

Problem: I'm wondering what could be a realization for algebraic tensor product of infinite dimensional vector spaces? Any ideas are welcome, of course! Attempts: My first guess would be the space ...
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47 views

Calculating a certain tensor

Consider the $\mathbb{Z}$ module $\mathbb{Z}/n\mathbb{Z}$. What is $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb Z} \mathbb{Z}/n\mathbb{Z}$?
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101 views

Prove $A_\mathfrak{p} \otimes_A B_\mathfrak{q} = B_\mathfrak{q}$, where $\mathfrak{q}$ prime in $B$

$\require{AMScd}$ Hi, I think I have the answer for this question, but I'm not sure if it's correct. So I would be very glad if someone could have a quick look through it. Let $A$, $B$ be ...
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4answers
77 views

$\{u_{i} \otimes w_{j} \}_{i , j}$ forms a basis for $U \otimes W$

Suppose $U$ and $W$ are $k$-vector spaces with bases $\{u_{i}\}_{i=1}^{n}$ and $\{w_{j}\}_{j=1}^{m}$. How to prove that $\{u_{i} \otimes w_{j} \}_{i , j}$ forms a basis for $U \otimes W$ ?
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67 views

Faithfully flat checkable on finitely generated modules

A left $R$-module $_RM$ is said to be faithfully flat if it is flat and, for any $N_R$, $N \otimes_R M = 0$ implies $N = 0$. I would like to show that $M$ is faithfully flat if it is flat and, for any ...
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64 views

dimension of tensor products over a submodule

Let $k$ be a field, $A$ be a finite dimensional $k$-algebra (say of dimension $n_A$) and let $B\subset A$ be a sub-algebra. What can be said about the $k$-dimension of $A \otimes_B A$ ? The easiest ...
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96 views

Monoidal categories and tensor products

Does the multiplication $-\square-$ biendofunctor in a Monoidal category, $\mathfrak{C}$ necessarily commute with coproducts? This is true in some familiar categories, such as $_RMod$, $Grp$ $CRings$ ...
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60 views

Unramification stable under change base

I want to show that if $f:X\to Y$ is an unramified scheme morphism (ie $m_y\mathcal{O}_{X,x}=m_x\mathcal{O}_{X,x}$ and $k(x)\leftarrow k(y)$ finite and separable) then any base change $X\times_Y Z\to ...
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404 views

Examples proving that the tensor product does not commute with direct products

Examples proving why the tensor product does not distribute over direct products? In fact the canonical map is not surjective; can you give me a simple example?
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53 views

Derivative on a function of tensor products

Assume I have defined an operator $A \otimes B$ on a $H \otimes L^2(\mathbb R^d)$ where $H$ is a Hilbert space as in Reed/Simon p. 299. $A$ is an operator on $H$ and $B$ is an operator on $L^2(\mathbb ...
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3answers
202 views

Tensor products and monomorphisms of $A$-modules.

This problem addresses the same question that has been asked in $A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$, which is exercise 2.11 of Atiyah and Macdonald's Introduction to ...
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240 views

Commutating operators and tensor products

I have this lecture slides about commutators and tensor products, but there is one part that I don't understand: The operators and are commuting operators on the tensor product and their sum has ...
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2answers
126 views

If for $A$ a commutative nonzero ring $A^m ≅ A^n$ as $A$-modules, then $m = n$

This is the problem I need to solve: Let $A$ be a nonzero ring. Show that if $A^m ≅ A^n$, then $m = n$. The book I got this problem from suggests using the following method to solve it: Let ...
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735 views

What, Exactly, Is a Tensor?

I've repeatedly read things that reference tensors, and despite reading the wiki page and other answers here on stackexchange I still don't know what a tensor is. I'm fine with hearing things in the ...
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1answer
54 views

Relations in tensor product of vector space

From the definition of tensor product, we have $V\otimes W=F(V\times W)/R$. Now, I understand $(v_1+v_2)\otimes w=v_1\otimes w+v_2\otimes w$, because the left-side is $(v_1+v_2,w)+R$, and the ...
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2answers
75 views

Tensor product of $\mathbb{R}^m$ and $\mathbb{R}^n$

I read this definition of a tensor product on Wikipedia. I'm quite confused.. so the elements of $F(V\times W)$ are the pairs $(v,w)$, where $v\in V,w\in W$. Now, if we take ...
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118 views

group of automorphisms of vector spaces

let α is an automorphism of G as a vector space and β is an automorphism of H as a vector space as well, then the tensor product of G and H is an automorphism of G tensor H? Let suppose that G and H ...
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47 views

Free Module over a tensor product

Suppose that $A$ and $B$ are $\mathbb{K}$-algebras for some field $\mathbb{K}$. Is $B$ free over $A \otimes_{\mathbb{K}} B$ ?
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97 views

Question about tensor product of homomorphisms

I've come to think about this problem when reading a proof in Commutative Algebra by N. Bourbaki. Say, let $R$ be a commutative ring, given 3 $R-$modules $A$, $B$, $C$, and the $R$-homomorphism $f:B ...
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100 views

tensor product of a vector space and finite field

I want to know how we can interpret and define the tensor product of V as a vector space with a F as a finite field?
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115 views

Operation on a Tensor Product of Modules

Suppose that $M$ and $M'$ are $\mathbb{K}$-algebras where $\mathbb{K}$ is a field. Now suppose that $N$ is an $M$-module and $N'$ is an $M'$-module, then I can view both of them as ...
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1answer
62 views

A Property of Tensor products

So I'm new to tensor products and there's something that's been confusing me ... Suppose that $A$ and $B$ are $R$-modules, I know that $A \otimes_R B$ is an abelian group; and what I understood is ...
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61 views

$Tor_1^\mathbb{Z}(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/2\mathbb{Z})$

How do I find $Tor_1^\mathbb{Z}(\mathbb{Z}/2\mathbb{Z}, \mathbb{Z}/2\mathbb{Z})$? and is it free or at least projective? I tried using the obvious short exact sequence then tensoring with ...
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1answer
29 views

Is a one-form a derivation on $C^ \infty$?

I know that a vector field is a derivation on $C^ \infty$, meaning that it is R-linear and Leibnizian. Is it the same case for one-forms?
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55 views

Injective norm on tensor algebra of a finite-dimensional Banach space

Suppose that $V$ is a finite n-dimensional Banach space, and suppose that $T(V)$ is the tensor algebra on it. Furthermore, suppose that $T^{(n)}(V)$ is the "abridged" tensor algebra obtained by taking ...
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59 views

A problem in division rings and Brauer group

Suppose that $D, E$ are two division rings in the Brauer group of $F$ ($Br(F)$), where $F$ is local field. Show that $D\otimes_FE$ is a division ring iff $([D:F],[E:F])=1$.
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Origin of the modern definition of the tensor product

Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
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62 views

On scalar extension of module and annihilator

Let $A, B$ be commutative rings with identity, $f: A \longrightarrow B$ a ring morphism, $M$ an $A$-module. Given $b\in B, x\in M$, does the following statement hold? $b\otimes x=0$ in $B ...
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329 views

Prime ideals in tensor products of algebras and their pullbacks

Suppose $\mathfrak{p}$ is a prime ideal in $B\otimes_CA$, and $\mathfrak{p}_A,\mathfrak{p}_B,\mathfrak{p}_C$ are its pullbacks in $A,B,C$. Does it hold: $(B\otimes_CA)_{\mathfrak{p}}\cong ...
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75 views

Homomorphism of modules and Tensor Product.

Let $\phi: A \rightarrow B$ be a ring homomorphism. Let $M$ be an $A$-module. We can think $B$ as $A$-module via the map $\phi$ defined by $\phi:A\times B \rightarrow B$, $(a,b)\mapsto\phi(a)\cdot ...
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1answer
80 views

Vector by Matrix derivitive

According to wikipedia, there is no widely accepted definition of a Vector by Matrix derivative. I have a need of such a notion. For matrix w, and vector h. $$\mathbf{y=w \;h} $$ $$ ...
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1answer
84 views

Tensor Product definition (help with certain step)

I'm going over some notes I took from the blackboard, and reached a slight hitch. I thought that maybe someone could help. Let $E,F$ be vector spaces over a field $\mathbb K$. A tensor product of $E$ ...
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128 views

Tensor product and valence of a tensor?

Given a vector space $V$, its dual vector space $V^{*}$ and a tensor $\mathbf{T}$: $\mathbf{T} \in \underbrace{V \otimes\dots\otimes V}_{n\text{ copies}}\otimes \underbrace{V^{*}\otimes\dots\otimes ...
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1answer
87 views

A criterion for an extension to be Galois

This is an exercise given during my Commutative Algebra course. I reached to solve just the "if" arrow, but not the "only if". The question is: Let $F\subseteq L$ be a finite degree extension of ...
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1answer
274 views

Logarithm and tensor products

We define Von Neumann Entropy for a density matrix $\rho$ (hermitian, positively defined, with trace 1) as : $S(\rho)=-tr(\rho \ln(\rho))$ Considering $\rho = \rho_1 \bigotimes \rho_2$, I want to ...
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Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
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1answer
110 views

Tensor product of modules preserve injectiveness and surjectiveness or not?

Let $R$ be a commutative ring with identity and $M$ an $R$-module. If $N_1\longrightarrow N_2$ is injective (resp. surjective), is the induced map $M\otimes_R N_1\longrightarrow M\otimes N_2$ ...
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448 views

Exactness of the Tensor Functor

This might turn out to be a very stupid question, so I apologize in advance, but it is confusing me a little bit. I know in general that if $$M'\rightarrow M\rightarrow M''\rightarrow 0$$ is an exact ...
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to find the graphs having vertices with same eccentricity

I was reading a paper http://www.discuss.wmie.uz.zgora.pl/php/discuss3.php?ip=&url=plik&nIdA=11134&sTyp=HTML&nIdSesji=-1 There is a formula to calculate eccentricity in the section ...
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194 views

Transitivity of representation induction

Let $K\subset H\subset G$ be some triple of finite groups and $T: K\longrightarrow \mathrm{GL}(V)$ - some representation f $K$. We are to prove the transitivity of induction: $Ind_K^G(V)\simeq ...
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142 views

Gelfand triple for tensor product of Hilbert spaces

Is there any dense embeding $\to$ that makes $H^1_0(D) \otimes L^2(\Gamma) \to L^2(D) \otimes L^2(\Gamma) \to (H^1_0(D) \otimes L^2(\Gamma))^{*}$ a Gelfand tripe? In fact we may only answere to the ...
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1answer
127 views

Ideals and exactness of projective tensor product of Banach spaces / algebras

Thanks for suggesting this question: Image of the tensor product of strict maps of Banach spaces I read the reference and realize that for a short exact sequence of Banach algebra: $0 \to J \to A \to ...
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1answer
378 views

Trace of tensor product

I am currently struggeling with the following problem: Imagine that you have 4 vectors $v_1,v_2,v_3,v_4$ in $\mathbb{R}^2$ and then you want to calculate $\hbox{trace}(v_4 \otimes v_3 \cdot ...
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68 views

Does there exist a $\nabla$-notation variant of the product rule applied to $\nabla[\mathbf{f}(\mathbf{x})\otimes\mathbf{g}(\mathbf{x})]$?

This is a vector-calculus notation question; as a disclaimer, I am working in rectilinear space! For vector functions $\mathbf{f},\mathbf{g}:\mathbb{R}^n\rightarrow\mathbb{R}^n$, the chain rule for ...
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82 views

Prove that $M_{nm}(A \otimes B) \cong M_{n}(A) \otimes M_m(B)$

Prove that $M_{nm}(A \otimes B) \cong M_{n}(A) \otimes M_m(B)$ There's a similar question floating around but I was merely wondering if the result holds in the same way when we let A and B be fields, ...
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54 views

Tensor power of a quotient space

Let $E$ and $F$ be vector spaces, $F$ a subspace of $E$. Is there any canonical isomorphism between $E/F \otimes E/F$ and a quotient of the form $E \otimes E/G$, where $G$ is a subspace of $E \otimes ...