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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1answer
32 views

Showing that $A \otimes_B A \to A$ is a surjective homomorphism.

Let us define a homomorphism $\phi: A \otimes_B A \to A$ by $a \otimes a' \to aa'$ where $A$ is a $B$-algebra, and both $A$ and $B$ are commutative rings. I want to show that this is a surjective ...
0
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0answers
35 views

Prime spectrum of tensor product of two R-algebras [on hold]

Let $R$ be a commutative ring and $A_1$ and $A_2$ two commutative unital $R$-algebras. Is there any characterization for $\mathrm{Spec}(A_1\otimes_R A_2)$? Or how can we deduce that $ \mathrm{Spec}(...
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0answers
8 views

Irreducible tensor for fundamental representation of SU(N=3)

I am trying to calculate the singlets of the tensor product $N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^* \otimes N_c \otimes N_c^*$. I know that $N_c \otimes N_c^*=1\oplus (...
1
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2answers
64 views

If $R\otimes_\mathbb R\mathbb C$ is finitely generated $\mathbb C$ - algebra then $R$ is a finitely generated $\mathbb R$ - algebra?

Let $R$ be an $\mathbb R$ - algebra. Suppose $A=R\otimes_\mathbb R\mathbb C$ is a finitely generated $\mathbb C$ - algebra then is $R$ a finitely generated $\mathbb R$ - algebra? I thought along the ...
3
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1answer
49 views

Symmetric decompositions of $SU(2)$ representations.

Let us consider the representation theory of $SU(2)$. There is a unique irreducible representation of dimension $n$ for each $n \ge 1$, which we will denote $\mathbf{n}$, with the defining $2$-...
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0answers
56 views

Dimension of polynomial rings and tensor products of residue fields

In Matsumura textbook to show that $\dim A[x] = \dim A + 1$, first it states that $A[x] \otimes k(\mathfrak{p}) = k(\mathfrak{p})[x]$ which is one dimensional. Then it uses the theorem 15.1.(ii) since ...
6
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2answers
170 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
0
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1answer
80 views

Tensor products and Residue fields

Given a ring homomorphism between two Noetherian rings, $f:A \to B$. Let $P$ be a prime ideal in $B$ and let $\mathfrak{p}$ be an ideal in $A$ such that $f^{-1}(P) = \mathfrak{p}$. How can we prove ...
0
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1answer
37 views

Proof of the exactness of the tensor product

In Atiyah and MacDonald, Prop 2.18 establishes that for any exact sequence $$M'\xrightarrow{f}M\xrightarrow{g}M''\xrightarrow{}0\tag{1}$$ of $A$-modules and homomorphisms, and for any $A$-module $N$, $...
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1answer
34 views

Tensor product space dense in $H_0^1$?

Given $C_c^{\infty}((a_i,b_i))$ for $i \in \{1,2\}.$ Is it then true that $$C_c^{\infty}(a_1,b_1) \otimes C_c^{\infty} (a_2,b_2)$$ is dense in $H_0^1((a_1,b_1) \times (a_2,b_2))$? Clearly, by ...
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0answers
44 views

How can we visualize a tensor?

I would greatly appreciate it if someone could explain to me how to visualize a tensor in the analogous way that we visualize a vector. Thanks!
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0answers
9 views

How to tell if a contracted tensor operation yields an invertible matrix?

Say I have a tensor T whose components are $T_{i j}^k$ and a vector v with components $v^i ;$ $i, j, k \in [1,4]$ When I perform the tensor contraction operation $T_{i j}^k v^i = M^k_j $, a matrix ...
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0answers
24 views

Notation - Tensor product of dirac deltas

In a paper I'm reading I found some notation I've never seen before. For some functions $f,g$ and $h$ they write $$\delta_f\otimes\delta_g\otimes\delta_h.$$ What does this expression mean? I guess it'...
0
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1answer
66 views

Is $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\cong 0$?

Since each 'generator' of $m\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}$ has the form $km\otimes_{\mathbb{Z}}\bar{a}=k\otimes_{\mathbb{Z}}m\bar{a}=k\otimes_{\mathbb{Z}}0=0$.
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0answers
40 views

How do I get from the universal product of the tensor product to other definitions.

I was wondering how you can "derive" the common (or "classical") definition of the tensor product before the universal property was established (I think there is no need to repeat it here), i.e. a ...
3
votes
1answer
87 views

Show that $\Omega^*(M)\otimes \Omega^*(N)$ is isomorphic to $\Omega^*(M\times N)$

Let $M, N$ be compact manifolds and $\Omega^*$ its algebra exterior. How to prove that $\Omega^*(M)\otimes \Omega^*(N)$ is isomorphic to $\Omega^*(M\times N)$? I thought about the function $f(\omega,...
1
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1answer
51 views

Calculus of rank three tensor

Let $A(\alpha)$ be a matrix that depends to vector parameter $\alpha$. I want to approximate $A(\alpha+\Delta\alpha)$ using Taylor expansion. My work: $$ A(\alpha+\Delta\alpha) \approx A(\alpha)+\...
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0answers
27 views

A embedding of tensor product over semisimple algebras

Let $R$ be a semisimple Artinian algebra over the complex number field $\mathbb{C}$, that is, $R$ is isomorphic a finite direct product of matrix rings over $\mathbb{C}$. Let $S$ be a ideal of $R$, ...
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0answers
21 views

when can i move a sum through this tensor product?

If I have a vector space $V^{(1)}\otimes V^{(2)}$ and I have some ray $\sum\limits_k x_k s_k\otimes s'_k = s\otimes \sum\limits_k x_k s'_k$, is the only solution that $s_k = s$ $\forall$ $k$? All $x_k$...
1
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2answers
45 views

If $M_1$ and $M_2$ are distinct maximal ideals of, then $R/M_1 \otimes _R R/M_2=0$ [duplicate]

$R$ is a commutative ring with $1$ and $M_1,M_2$ are distinct maximal ideals of $R$. Then $R/M_1 \otimes _R R/M_2=0.$ Consider $f:R/M_1 \times R/M_2\to R/M_1 \otimes _R R/M_2 $ defined by $f(r+M_1,...
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0answers
30 views

Is tensor product of weighted Sobolev spaces dense?

Let $W^{k}_{2,w_1}(\mathbb{R})$ be a weighted sobolev space with positive continuous weight function $w_1$ for the integrals of the function and its derivatives. Let $W^{k}_{2,w_{1,1}}(\mathbb{R}^2)$ ...
0
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0answers
29 views

Extension of Scalars of Complex Numbers

If we consider the complex numbers as a $\mathbb{R}$-module (vector space in this case), then its natural extension of scalars to $\mathbb{C}$ seems to be the complex numbers themselves, with the ...
2
votes
0answers
38 views

How do I compute the gradient of a tensor?

From this paper, we have three matrices $U\in \mathbb{R}^{n\times d_U}$, $M\in \mathbb{R}^{m\times d_m}$, $C\in \mathbb{R}^{c\times d_C}$ and a tensor $S\in \mathbb{R}^{d_U \times d_M \times d_C}$, ...
1
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1answer
34 views

Understanding the notation $\nabla u \otimes \nabla u$

On a Riemannian manifold $(M,g)$ let $u = u(t,x)$ the solution to the heat equation $\partial_t u = \frac 12 \Delta u$. The Laplace-Beltrami operator etc. are taken with respect to the metric $g$. I'...
0
votes
1answer
63 views

Showing $K = I \otimes_R K$, where $K$ is the field of fractions of $R$.

A question in a previous Commutative Algebra Exam reads: Problem. Let $R$ denote an integral domain, $K$ its field of fractions. Let $I$ denote a non-zero ideal of $R$. Show that $K=I \...
0
votes
1answer
18 views

Tensor product space: dual of the space of bilinear functionals on the Cartesian product

My reading (link provided for completeness only, clicking is not necessary) defines the tensor product space as follows: Let $V$ and $W$ be vector spaces. The symbol $v\otimes w$ is defined to be the ...
3
votes
1answer
42 views

Question about the tensor algebra

If $V$ is a vector space of finite dimension over $\mathbb{F}$, we define the tensor algebra: $T(V)= \oplus_{k=0}^\infty (\otimes^k V)$, where by convention $\otimes^0 V= \mathbb{F}$. My question is: ...
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0answers
17 views

A “contraction” on tensor product spaces

Let $X$ be a topological vector space. Let $X'$ denote its continuous dual. Consider the (algebraic) tensor product $\mathbb{X} := X \otimes X'$. For simple tensors $x \otimes x' \in \mathbb{X}$ set $...
0
votes
1answer
22 views

Algebra linearly isomorphic to a tensor product is a free module

I have an algebra $\mathcal P$ that is isomorphic as a vector space to $\mathcal H \otimes \mathcal J$ for $\mathcal H\subset \mathcal P$ and $\mathcal J\subset \mathcal P$. It follows that $\mathcal ...
0
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1answer
47 views

Localization of $\mathbb{Z}/p^k\mathbb{Z}$ at $S=\begin{Bmatrix}b^n : n\in \mathbb{N}\end{Bmatrix}$

Could I say that $\left(\mathbb{Z}/p^k\mathbb{Z}\right)_{b}$, namely the localization at $S=\begin{Bmatrix}b^n : n\in \mathbb{N}\end{Bmatrix}$ when $(b,p)=1$, is equal to $\mathbb{Z}/p^k\mathbb{Z}$ ...
0
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0answers
34 views

If $A$ is a local ring, is it always true that $(A/Q)_{\mathfrak{m}}\cong A/Q$?

$A$ is a local ring with $\mathfrak{m}$ its maximal ideal and $Q\subset A$ is an ideal of $A$. I thought that $$(A/Q)_{\mathfrak{m}}\cong A/Q\otimes_{A} A_{\mathfrak{m}}\cong A/Q\otimes_{A} A\cong A/Q,...
1
vote
1answer
36 views

Tensor Product and Flat Module

While doing some exercises about flat modules I got to this particular one: Let $I$ be an ideal of $A$ (commutative ring with 1), and $M$ an $A$-module. Show that $I \otimes_{A} M \simeq IM$ if $M$...
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0answers
19 views

Tensor Product is an abelian group? Confused.

In Wikipedia, it is written that $M\otimes_R N$ is an abelian group together with a balanced product $\otimes$. I am a bit confused about the abelian group part. Just to confirm, does that mean if $x,...
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0answers
15 views

The general method to prove tensor product isomorphisms?

For questions that require proving that $A\otimes B\cong C$, what is the easiest method to do so? (We may assume $A,B$ are modules) What I roughly know is that the recommended method is to use ...
1
vote
1answer
41 views

Show $\Bbb{Z}_m\otimes_{\Bbb{Z}}\Bbb{Z}_n=0$ if and only if $m,n$ are coprime. [duplicate]

I'm trying to do this problem out of Atiyah-Macdonald: Show $\Bbb{Z}_m\otimes_{\Bbb{Z}}\Bbb{Z}_n=0$ if and only if $m,n$ are coprime. First, suppose $m,n$ are coprime. Then there exist $s,t$ ...
1
vote
1answer
22 views

Why this map implies the decomposition of the tensor product space?

Let $V$ be a vector space and consider the tensor product of $V$ with itself, that is $V\otimes V$. Define $\alpha' : V\times V\to V\otimes V$ by $$\alpha'(v,w)=w\otimes v.$$ In that case, $\alpha'$ ...
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0answers
29 views

Induced Representations from Tensor Product of Modules

In Chapter 7 of J P Serre's book on Representation Theory, he defines a $\mathbb{C}[G]$-module as $W'= \mathbb{C}[G]\otimes_{\mathbb{C}[H]}W$ and claims it to be the $\mathbb{C}[G]$-module obtained ...
1
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1answer
30 views

Separability and reduced tensor fields

Let $L/K$ be a finite field extension. I need to prove that if $L/K$ is separable, then $E\otimes_KL$ is reduced for every algebraic field extension $E/K$. I've readed that I need to use the ...
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0answers
47 views

What is extension of scalars used for in algebraic geometry?

Given a ring homomorphism $f:A \rightarrow B$ and an $A$-module $M$, one can construct and $A$-module with the tensor product: $M_B=B \otimes_A M$ which has a $B$-module structure. This is said to be ...
1
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1answer
18 views

Unitization of Suspension

Let $A$ a C*-algebra (unital or not). Its suspension is defined to be: $$ S(A) \equiv A\otimes C_0((0,1);\,\mathbb{C}) $$where $C_0$ denotes all continuous functions which vanish at infinity. We ...
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0answers
14 views

Text introducing $T^{i,j}$-tensor algebra

I'm reading a lecture note here : http://www.cis.upenn.edu/~cis610/diffgeom7.pdf It introduces $T^{•,•}(M)$ the tensor algebra and says that this is a necessary tool in differential geometry. Well, ...
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0answers
18 views

Is the map $K\otimes_F Hom_F(V, U)\to Hom_K(K\otimes_F V,K\otimes_F U)$ surjective?

Given $ U, V$ vectorial space over F and $K/F$ a field extension ( even of infinite degree) the map $K\otimes_F Hom_F(V, U)\to Hom_K(K\otimes_F V,K\otimes_F U)$ is a monomorphism of K vectorial space....
3
votes
0answers
34 views

Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V):=B(V,V)$ are linear bounded endomorphisms with operator norm? If so, ...
1
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1answer
29 views

Free modules and their tensor product

can tensor product of two free non zero module over commutative ring with unity be zero? And can tensor product of two non zero vector spaces be zero space?
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0answers
35 views

Levi-Civita symbol (permutation tensor)

I was going over a past exam and the following two questions came up: Show that the Levi-Civita symbol $\varepsilon_{ijk}$ is a tensor. Evauluate the following: $\varepsilon_{ijk}\varepsilon_{ipq}\...
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0answers
29 views

Linear algebra textbook for quantum computing?

I'm looking for an recommendation for a linear algebra textbook specifically to give me the background for learning about quantum computing, and quantum mechanics more generally. In particular, none ...
2
votes
1answer
25 views

General procedure to prove something is a tensor product of modules

I'm trying to understand some proofs of statements of the form: Show that some module is the tensor product of two other modules. When I'm looking at these proofs I always see that they start ...
0
votes
0answers
56 views

Express sum using tensor notation

So I have this expression $\sum_{l,m} c_{lmn} a_l a_m$. Where c is rank 3 tensor, and a is a rank 1 tensor. I would like to express this without specific reference to the tensor indices. Is there a ...
0
votes
1answer
27 views

Tensor derivative

What is the result of $$ \frac{\partial^2 \left(A^{ij}y^ix^j+B^{ij}x^iy^j\right)}{\partial \bf x\partial \bf y} $$ where $i,j$ obey Einstein summation convention, $A,B$ are constant, ${\bf x}=[x^1,x^2,...
0
votes
1answer
31 views

Writing $e_1\otimes e_1+e_2\otimes e_2$ in another way [duplicate]

Let $V=\mathbb C^2$ with standard basis $\{e_1,e_2\}$. Are there $v,w\in V$ s.t. $e_1\otimes e_1+e_2\otimes e_2\in V\otimes V$ can be written as $v\otimes w$ Is the answer no ? for example if $v=...