Questions tagged [tensor-products]

For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.

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Tensor product calculation: verification

Let $f \in C^2(\mathbb R^3)$ and $p=(x,y,z) \in \mathbb R^3$. Here $\;\text{Id}$ stands for the $3\times 3$ identity matrix. Is the following estimation correct? $\begin{align*} -(\text{Id}-p\otimes p)...
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Homological Algebra - Prove the question [closed]

Let $M$ be a right $R$-module and consider a left exact sequence or left $R$-modules $$0\longrightarrow A\stackrel{f}\longrightarrow B\stackrel{g}\longrightarrow C.$$ Show that in general $\ker(g\...
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Onto function between $M \otimes_{A} L$ and $N \otimes_{A} L$

I'm trying to prove the following statement, but I'm having so much trouble with it. Let $L, M$ y $N$ modules over a ring $A$. Assume that exist an onto application of $A$-module between $M$ and $N$. ...
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Tensor product, direct, product, and direct sum?

My question is whehter or not I'm using the direct sum $\oplus$ and direct product $\otimes$ symbols correctly, or if I need a tensor product. I have no formal training with these symbols and if I am ...
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Relation between symmetric outer product decomposition and symmetric multilinear decomposition

Suppose tensor $\mathcal{A}$ is a symmetric real tensor of order $k$. Then, symmetric outer product decomposition of $\mathcal{A}$ is $$ \mathcal{A} = \sum_{i=1}^p \lambda_i v_i^{\bigotimes k}, $$ ...
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Coherence in closed monoidal categories

Let $(M, \otimes, I)$ be a left-closed (non-symmetric) monoidal category with left-internal hom $\underline{\operatorname{hom}}(-,-)$. Denote by $\sigma_{A,B,C}: M(A\otimes B, C) \xrightarrow{\sim} M(...
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Tensor Product, proof of isomorphism

Let V and W be two vector spaces and $$\Phi: Hom_K(\land^rV,W)\rightarrow Alt_K^r(V,W)\\ f \mapsto [(v_1,....,v_r)\mapsto f(v_1\land \cdot \cdot \cdot \land v_r)]$$ I need to show that $\Phi$ is an ...
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Tensor product of non-abelian groups

My algebra professor says that tensor product can be defined for non-abelian groups.$$$$how is this defined?
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Linear Algebra - Tensor Products

Let $V$ be a finite-dimensional vector space over $\Bbb Q$ and $\bigwedge^kV$ be the $k$-th antisymmetric power. Let $\{ v_1, \dots, v_n \}$ be a basis of V. Define $$\pi: V^{\otimes k} \to \bigwedge^...
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Why is $\sigma \in S_n$ acting covariantly on $n$-covector?

This is from Tu's book in smooth manifold. Let $V$ be a vector space, and let us define $S_n$ acting on $\Lambda^n(V^*)$ as follows: For $\sigma \in S_n$ and $f \in \Lambda^n(V^*)$, $\sigma f(v_1, \...
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Eigenvalues of non-diagonalizable representations of a group acting on $V^\otimes V$, $\text{Sym}^k V$, $\bigwedge^k V$, etc. (Fulton & Harris)

In Section 2.1 of Fulton & Harris, it states that if the action of a group element $g$ has eigenvalues $\{\lambda_i\}$ when acting on a vector space $V$, and eigenvalues $\{\mu_j\}$ when acting on ...
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Find decomposition of a kronecker product

I know that $\underbrace{A}_{mn \times mn} = \underbrace{B}_{n \times n} \otimes \underbrace{C}_{m \times m}$. Both A and B are know symmetric and positive definite matrices (they are covariance ...
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1 answer
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Is $\mathrm{Hom}_S(M,E) \otimes_R X \cong \mathrm{Hom}_S(\mathrm{Hom}_R(X,M),E)$ with $R$ a Noetherian ring?

Given a left Noetherian ring $R$, a ring $S$, a $R$-$S$-bimodule $M$, an injective cogenerator $E$ of right $S$-module, is there an natural isomorphism $$ \mathrm{Hom}_S(M,E) \otimes_R X \cong \mathrm{...
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3 votes
2 answers
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Why is $\mathbb{R} \otimes \mathbb{S}^{1} $ infinite?

Intuitively I see why this is infinite, but there is no obvious way to apply the tensor relations to reduce every element into a finite set.I don't really have an idea on how to approach this problem (...
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Decomposition of order 3 tensor symmetric along two dimensions

I have a 3rd order tensor $\mathbf{A}$ consisting of symmetric covariance matrices (with dimensions of space by space) stacked in time. I would like to compute the leading spatial features that ...
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Scalar extension in tensor product.

Let $V$ be a vector space over $\Bbb Q$ a) Show that $\Bbb C$ takes the structure of a $\Bbb Q$ vector space. b) For elements $\sum'\lambda_i \otimes v_i$ in $\Bbb C \otimes_{\Bbb Q} V$ and $\lambda \...
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Tensor product of modules and vector spaces

Let $K$ be a field, $K[t]$ be a polynomial ring and be $K(t)$ be the fraction field of $K[t]$. We take the $K[t]$-module $K[t]/(t) \otimes_{K} K(t)$ with the action $r(t)\left(\overline{p(t)} \otimes \...
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Example of invertible modules?

I am trying to find an example of a non-trivial invertible module (let's say over $\mathbb Z$). This seems to be very simple, but after trying and searching around, I do not find any examples. (Many ...
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Is $G\otimes\mathbb Z^X$ isomorphic to $G^X$?

Let $G$ be an abelian group and let $X$ be a finite set. For a given group $T$, let $T^X$ be the group of functions $X\to T$ where the operation is defined pointwise. Is it true that $G\otimes\mathbb ...
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Tensor Product $\mathbb{Q} / \mathbb{Z} \otimes \mathbb{Z} / 2\mathbb{Z}$ [duplicate]

I believe that the tensor product $\mathbb{Q} / \mathbb{Z} \otimes \mathbb{Z} / 2\mathbb{Z}$ is trivial, i.e. any element is 0, but apparently there are in fact 2 elements. Why is this?
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Linearly distributive categories: Principles of excluded middle and contradiction

1. Context Let $(\mathscr{C}, \otimes, \top, \oplus, \bot, \delta ^l, \delta^r)$ be a linearly distributive category. Let $(S,S', \alpha, \beta, \alpha‘, \beta‘)$ be a negation on $\mathscr{C}$. This ...
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How to make precise the meaning of naturality for a specific natural isomorphism?

What I know: (1) If $V$ is a finite dimensional vector space then there is a natural isomorphism from $V$ to its double dual $V^{**}$. (2) There is no natural isomorphism from $V$ to its dual $V^*$. (...
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7 votes
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Show that $S^{\otimes T}=\underset{I \,is\, a\, countable\, subset\, of\, T}{\cup}\,\,\,\pi_1^{-1}(S^{\otimes I})$

Let $S$ be a measurable space and $T$ an uncountable index set. For a subset $I\in T$, we write $\pi_1:S^T\to S^I$ for the natural projection. Show that $S^{\otimes T}=\underset{I \,is\, a\, countable\...
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Equivalence of kronecker product of matrices and tensor product of vectors.

I know how kronecker product of matrices $A\otimes B$ where $A,B\in M_n(\mathbb{R})$ and tensor product of vectors $x\otimes y=xy^T$ are defined. I am perplexed about the equivalence of both the ...
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Give an example of a symmetric matrix $A$ such that $A\in S_4^+$ but $A\neq \sum_{k=1}^NX_k\otimes Y_k$ where both $X_k,Y_k\in S_2^+$.

I have trouble in finding an example of a symmetric matrix $A\in M_4(\mathbb{R})$ such that $A$ is positive semidefinite matrix but $A$ cannot be written as sum of tensor product of matrices $X_k,...
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Tensor products of finitely generated algebras is a finitely generated algebra (proving a particular case using universal property)

Related to this question, I am trying to prove a simpler case: Let $R$ be a commutative ring and $I \leq R[x_1, \ldots, x_n]$ an ideal. Let $S:=R[x_1, \ldots, x_n]/I$. I want to see that: $$ S \...
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2 votes
1 answer
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Isomorphism between $K\otimes_{\mathbb{Q}}\mathbb{R}$ and $K_{\mathbb{R}}$, Minkowski theory

This is a basic question from Neukirch’s book ‘Algebraic Number theory’, chapter 1, $\S$5. Let $K$ be a number field, $K_{\mathbb{C}}$ the $\mathbb{C}$-vector space $\prod_{\tau}{\mathbb{C}}$ where $\...
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Injectivity of morphism in the conormal sequence with the Kähler differentials module

Consider the conormal sequence proposition from Eisenbud's Commutative Algebra: If $\pi:S\to T$ is a surjective morphism of $R$-algebras, and $I=\ker(\pi)$ we have an exact sequence $$I/I^2\to T\...
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2 votes
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Tensor product of division algebra with $\mathbb R[x]$ or $\mathbb R(x)$

I learned that the tensor product of two division algebras may not be a division algebra. Thus, I am curious if there is some case in which this is true. To be precise, given a division algebra $A$ ...
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A "trivial" question on tensors that I'm not sure about

(Notation $L^k$ is the space of $k$-covariant tensors) A friend posed this question to me - he's been studying about tensors and this question is an exercise there, claimed to be "trivial". ...
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1 vote
1 answer
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Tensor Product "commutes" with ring homomorphism

Suppose $R,S$ are (unital and commutative) rings and $\phi:R\to S$ is a homomorphism of rings so we can view $S$ as an $R$-module. Let $A,B$ be $S$-modules, so using $\phi$ we can also view them as $R$...
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1 answer
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Tensor product of finite Galois extensions

Let $K_1$ and $K_2$ be finite Galois extensions of $k$, set $G_k = \text{Gal}(K_1 K_2/k)$, and $H= \text{Gal}(K_1 K_2/K_1 \cap K_2)$. I want to prove that as $k$ algebras, $$K_1 \otimes_k K_2 \cong \...
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1 vote
1 answer
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Prove that $Hom_{R}(M,N)\cong M^*\otimes_{R} N$ as groups

So the statement of the problem is thus: Let $R$ be a ring with identity such that $1_{R} \ne 0_{R}$, $M$ be a free left $R$-module of finite rank, and $N$ be a left $R$-module. Then $M^*=Hom_{R}(M,R)...
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Associated prime of certain tensor product

Let $A$ be a Noetherian ring and $p\subset A$ a prime ideal. Is $p\otimes_A A/p$ a torsion-free $A/p$-module? The case which I'm interested in is $A:=k[X,Y,Z]/(X^2-YZ)$ and $p:=(x,y)$, where $k$ is a ...
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1 answer
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How to define this function in terms of tensor products

This problem was left as an exercise in algebra class of mine but I am not able to make any progress. I have been following book by atiyah and macdonald and I must say I had difficulty in ...
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What does it mean for a matrix to wedge product itself? (Cartan's second structural equation)

In page-440 of Tristan Needham's Visual Differential geometry, the following equation is given: $$ d \left[ \omega \right] = \left[ \omega \right] \wedge \left[\omega \right]$$ Where $\left[ \omega \...
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How do I simplify $\delta_{ij} \delta^{jk}$?

How do I simplify $\delta_{ij} \delta^{jk}$? I know that $\delta_{ij} \delta_{jk}=\delta_{ik}$, but what do I do if the there's a Kronecker Delta symbol with upper indices and one with lower indices?
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Checking my understanding of Einstein summation convention

I have a limited understanding of the convention so I'd like to check. Given a (0,2) tensor or bilinear map $(e_{i}\otimes e_{j})=\zeta_{ij}$ with $i,j=1,2,...,n$, and two covectors $\Omega^{1}=\sum^{...
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Why differ the definition of the tensor product in algebraic and analysis books?

In algebraic books, Tensor product is defined using quotient space and in analysis books, tensor product is defined using the bilinear map and linear functional. Why is it defined in two ways? Thanks ...
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Checking Metric Tensor relations

I need to check some simple tensor relations for continuing my calculations. Please write me if you think anyone is incorrect. P.S.: I know this is very simple question, but i need to be assured about ...
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Isomorphism in the space of section of a trivial vector bundle

In the answer of this post Fundamental result on the projective tensor product of sections of a vector bundle we have $\Gamma(M, V\times M) \cong C^\infty(M) \otimes V$ Where $\Gamma(M, V\times M)$ ...
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2 votes
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Bounding norm of a vector

The problem Consider two $d$-dimensional POVMs, $ \mathcal{E}$ and $\mathcal{F}$, given by POVM elements $\left\{E_{1}, \ldots E_{n}\right\}$ and $\left\{F_{1}, \ldots F_{n}\right\}$, respectively. (...
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How to compute this tensor product?

I need to compute this tensor product $$ \mathbb{R}[t]_{t}\otimes_{\mathbb{R}[\frac{t^2+1}{t}]} \mathbb{R}(\frac{t^2+1}{t})=\mathbb{R}[t]_{t}\otimes_{\mathbb{R}[\frac{t^2+1}{t}]} \mathbb{R}[\frac{t^2+...
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tensor with the separable closure

Let $l/k$ be a finite field extension and $K$ the separable closure of $k$. The finite $K$-algebra $l \otimes _k K$ is a product of finitely many separable fields $L_i$ over $l$. Are these fields $L_i$...
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Let $L/K$ be a field extension. When $x\otimes1=1\otimes{x}$?

Let $L/K$ be a field extension. Let's consider $R=L\otimes_K{L}$. In $R$, when $x\otimes1=1\otimes{x}$ ? I understand when $x∈K$, $x\otimes1-1\otimes{x}=1\otimes{x}-1\otimes{x}=0$, but I don't ...
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Laws of a tensor product of algebras

Given a commutative field $k$ and two $k$-algebras $A$ and $B$, the tensor product $A\otimes_k B$ is also an algebra. What are its laws ? I mean, the composition law is given by $(a\otimes b) \circ (c\...
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Show that $\ker(\phi\otimes\phi)\subseteq \ker\phi\otimes A+A\otimes\ker\phi$

Context: Let $\phi:A\to B$ be a morphism of Hopf algebras. To show that $\ker\phi$ is a Hopf ideal of $A$, we must verify that $\Delta(\ker\phi)\subseteq \ker\phi\otimes A+A\otimes\ker\phi$, where $\...
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Clarification on definition of representation of $\mathrm{Hom}(V,W)$

From Representation Theory by W. Fulton and J. Harris: Let $V$ be a finite dimensional vector space, and $G$ a finite group. Let $\rho: G \to \mathrm{GL}(V)$ be a representation of $V$. The dual of ...
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Justification for $\operatorname{dim} \wedge^k(V)$ is $n\choose k$ the alternating $k$-linear form.

I was told that: $\operatorname{dim} \wedge^k(V)$ is $n\choose k.$ and I am trying to justify this by computing it for different dimensions of $V.$ If $\operatorname{dim}(V) = 1,$ then $v_1 \wedge v_1 ...
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Conditions for Weighted Norm

I'm constructing a weighted norm of the form $||N||_A = \sqrt{tr(N^T A N)}=\sqrt{N_{ij}A_{jk}N_{kl}\delta_{il}}$, where $\{A,N\} \in \mathbb{R}^{3x3}$. In order to keep the argument of the square root ...
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