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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Shuffle the Kronecker Product

Suppose $D$ is a orthogonal matrix of size $N\times N$, and we have \begin{equation} S = D F D^{T}, \end{equation} where $D^T$ means the transformation of $D$. Using Kronecker product, we can rewrite ...
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47 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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55 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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25 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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26 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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40 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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43 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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58 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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38 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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28 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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47 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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60 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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70 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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31 views

Necessary and sufficient condition which tensor product of two central simple algebras is a division ring.

I think this is an interesting and important question that when a tensor product of two central simple $K$-algebra is a division ring. If we know the answer, we can make some counterexamples in ...
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37 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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47 views

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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60 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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162 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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109 views

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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63 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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103 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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66 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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82 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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Universal property of free spaces

In class we had developed the notion of a free space in preparation for our study of tensor product. There is one statement in my notes about the universal property of free spaces that has been ...
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53 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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71 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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39 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 ...
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110 views

Prove that $M_{n}(F)\otimes _{F}M_{m}(F)\simeq M_{nm}(F)$ .

Suppose $F$ is a field. Then prove that $$M_{n}(F)\otimes _{F}M_{m}(F)\simeq M_{nm}(F)$$ as $F$-algebras. I know that I should take $$\alpha :M_{n}(F)\otimes _{F}M_{m}(F)\rightarrow M_{nm}(F)$$ ...
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21 views

If $A$ and $B$ are $F$-algebras, then $A\otimes _{F}B $ is $F$-algebra. [duplicate]

Suppose $A$ and $B$ are two $F$-algebras ($F$ is a field). Prove that $A\otimes _{F}B $ is an $F$-algebra with the multiplication: $$(a\otimes b )({a}'\otimes {b}')=(a{a}')\otimes (b{b}').$$ With ...
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88 views

Tensor product of a module and a localized ring

Let $A$ be a commutative ring with unity. Let $S$ be a multiplicative subset of $A$. Let $M$ be an $A$-module. Let $x \in M$. Suppose $x\otimes 1 = 0$ in $M\otimes_A S^{-1}A$. Then there exists $s \in ...
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73 views

Question about tensor product of modules, when does $c \otimes r = 0$?

Let $R$ be an integral domain, and let $K$ be its field of fractions. Let $C$ be an $R$-module. Then is $C \cong C \otimes_R R$ injectively contained in $C \otimes_R K$? Define $\phi: C \otimes_R ...
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50 views

Given a group homomorphism $R \otimes_{\Bbb{Z}} M \rightarrow M$, I need to show that this makes $M$ into a left $R$-module.

Given a group homomorphism $R \otimes_{\Bbb{Z}} M \rightarrow M$, I need to show that this makes $M$ into a left $R$-module. We can see that $\bullet$ $r(m+m') = r \otimes (m+m') = r \otimes m ...
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58 views

Gelfand triple in Tensor product spaces

Let's define $A^k={\rm span} \{\phi_1,\dots,\phi_k\}$ and $B^k={\rm span} \{\psi_1,\dots,\psi_k\}$ where $\phi \in H^1(X)$ and $\psi \in L^2(X)$ simple functions and $X$ is closed subset of ...
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81 views

Prove this isomorphism of $K$-algebras

Seen this is a lot of literature, usually without proof. Was just wondering how is it: $A$ is a $K$-algebra, where $K$ is a field. Then $A \otimes_K A^{op} \simeq M_r(K)$ where $r=\dim(A)$. I ...
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17 views

Is there any vanishing criterion for elements of a tensor product of algebras?

Let $k \subset L$ be a field extension, $\mathrm{char}(k) = p$. I have some polynomial $f(X_1, \ldots, X_n) \in k^{1/p}[X_i], f^p \in k[X_i]$, with at least one coefficient not in $k$, such that ...
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49 views

Show that we have an algebra homomrohpsim

I need to show that we have an algebra homomorphism $\phi: M_n(K)\otimes_KA \simeq M_n(A)$ Where A is a K-algebra and K is some field. I suspect it's really easy but I don't know what to do. Is ...
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1answer
43 views

$\ker(f\otimes g)=\langle\{a\otimes b:(a\in \ker f \text{ or }(b\in\ker g)\}\rangle$

There's a theorem that if $f:A\to B$ and $g:A'\to B'$ are epimorphisms, then their tensor product's kernel is given as $$\ker(f\otimes g)=\langle\{a\otimes b:(a\in \ker f )\ or\ (b\in\ker ...
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57 views

Tensor Product problem.

Let $V$ be a vector space over $\Bbb F$, and let $x\not=0,y\not=0 $ be two elements in $V$. I want to show that $x\otimes_{_F} y=y\otimes_{_F} x$ iff $x=ay$ where $a\in \Bbb F$. I know the second ...
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1answer
49 views

Norm of the dual of the Tensor product of Hilbert spaces

Let $V$ and $W$ be Hilbert spaces, we can define inner product and induced norm on Tensor product of these spaces as: Let $v_1,v_2 \in V$,and $w_1,w_2 \in W$. then $(v_1 \otimes w_1, v_2 \otimes ...
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1answer
39 views

When is a simple tensor equivalent to natural multiplication?

Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules. Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold? $$x_1 \otimes x_2 = x_1x_2$$ The textbook I'm ...
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22 views

Sum of creation operators as one creation operator

Let $A = a^\dagger \otimes 1 + 1 \otimes a^\dagger$ act on $H \otimes H$ where $a^\dagger$ is the usual creation operator. I.e. we have $a^\dagger |n \rangle = \sqrt{n+1}|n+1\rangle \\ N |n \rangle = ...
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2answers
84 views

Tensor product of a field with itself.

I am proving the fact that if $A$ and $B$ are two central $k$-algebras where $k$ is a field (so then $Z(A) = Z(B) = k$), then $A \otimes B$ is also central. I made almost everything except this: I ...
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2answers
53 views

Arbitrary element in a tensor product is a finite sum.

Many books on algebra say that any arbitrary element in a tensor product is a finite sum of pure tensors, however, I cannot clearly see why this is true. Actually, if both of the spaces $U$ and $V$ ...
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46 views

Extending the universal property of tensor product

Suppose that we defined a tensor product of vector spaces $U$ and $V$ as a quotient of a vector space with basis $V \times W$ by the vector space spanned by $-(u_1+u_2,v)+(u_1,v)+(u_2,v), ...
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3answers
75 views

$0$-th exterior power, empty product of modules and their tensor product

$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object in the category of multilinear maps from $\finiteprod En$ where ...
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1answer
64 views

Tensor product equivalent definitions

I'm studying tensor products right now and I've came across multiple definitions. The one I'm confused with is when we have vector spaces $V$ and $W$ and we define the tensor product as the quotient ...
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66 views

Tensor product, Artin-Rees lemma and Krull intersection theorem

I asked another question about tensor product, but can't conclude from the answer, so here is another more concrete question. Let $(A,m)$ be a local ring then by Artin-Rees Lemma $m^k \bigcap I ...
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1answer
52 views

Question about tensor product of modules and ideals

Trying to prove some properties of tensor product with a given module, I came up with questions some of them I can't prove. Maybe it is also because Im not very used to work with tensor products and I ...
3
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2answers
58 views

$\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup of $A$

Let $A$ be a finite abelian group of order $n$ and let $p^k$ be the largest power of the prime $p$ dividing $n$. Then $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup ...