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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20 views

Counterexample for short exact sequence of tensor product of modules

Let $R$ be a semisimple Artinian ring and $A, B, C$ be right $R$-modules such that $A \subseteq B$ and $B/A \cong C$. I have managed to show that $(B\otimes M)/(A\otimes M) \cong C\otimes M$ for any ...
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Unable to understand a step in an example related to tensor product

This question was told in class notes of our course in commutative algebra and I was unable to comprehend reasoning behind a step. so, I am asking it here. Consider $0 \to \mathbb{Z} \otimes \mathbb{...
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1answer
79 views

Prove $\phi$ to be isomorphism (an exercise in commutative algebra)

The question is from my course exercise of commutative algebra and I am asking here as I was unable to make any significant progress in it. $\DeclareMathOperator{\Hom}{Hom}$ Suppose there exists $f \...
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Simplicial structure on chain complexes

It is known that the homotopy category of chain complexes has the right action by the homotopy category of simplicial sets. For any simplicial sets, $X$, one can define the associated chain complex $...
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Defining an $A$-algebra structure using a basis and “constants of structure”

Let $A$ be a commutative ring and $E$ an $A$-module with a basis $(e_i)_{i\in I}$. Let $(\alpha_{ijk})_{(i,j,k)\in I\times I\times I}$ be a family of elements of $A$ such that, for $i,j\in I$, the ...
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40 views

Sign in definition of Cap product

We define the cap product in the following way : $$a \frown z := \epsilon \otimes 1(a \otimes Ez \circ \Delta_\star(z))$$ Where : $1. \hspace{0.3cm} \Delta_\star$ is the induced map on chain complex ...
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1answer
58 views

Is $R\otimes\mathbb{K}[[t]]$ isomorphic to $R[[t]]$?

Let $K$ be a field of characteristic zero. It's true that if $R$ is a $K$-algebra then $R\otimes_{\mathbb{K}}\mathbb{K}[[t]]\cong R[[t]]$ with the natural inclusion inducted by $x\otimes t^j$$\mapsto$$...
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1answer
54 views

What is the tensor product $\mathbb{Z}_{k} \otimes \mathbb{Z}_{(2)}$?

I am trying to calculate the tensor product $\mathbb{Z}_{k} \otimes \mathbb{Z}_{(2)}$ where $\mathbb{Z}_{(2)} = \{\frac{a}{b} | a \in \mathbb{Z} \text{ and } b \text{ is odd}\}$ and $\mathbb{Z}_{k} = ...
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16 views

Base change to algebraic closure commutes with quotient of polynomial ring by maximal ideal

Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ ...
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2answers
43 views

Tensor product of quadratic number field with itself

If $F=\mathbb{Q}(\sqrt{D})$, is there a nice structure to $F\otimes_{\mathbb{Q}} F$? A spanning set of that tensor product is $1\otimes 1$, $1\otimes \sqrt{D}$, $\sqrt{D}\otimes 1$, and $\sqrt{D}\...
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Meaning of wedge product to calculate volume, and basic concept of tensor [closed]

I heard that wedge product is correct type to calculate volume form from professor and here https://mathworld.wolfram.com/WedgeProduct.html And he said that to calculate surface integral int(FdS)= int(...
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How exactly do tensor products correspond with “tensors” as understood in tensor calculus & mathematical physics?

This is a mathematical construct that I've found rather hard to construct a more intuitive description of than what is usually given - heck, I've sooner with enough digging found what I'd consider a ...
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1answer
28 views

Slice maps on a von Neumann algebra completely positive?

Let $M\subseteq B(H)$ and $N\subseteq B(K)$ be von Neumann algebra and let $\omega$ be a positive (normal) functional on $M$. Consider the von Neumann algebraic tensor product $M \boxtimes N \...
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1answer
9 views

Linear maps defined on pure tensors is well defined.

Suppose $V_1, \ldots, V_k, W$ are vector spaces, and suppose we define a map $f:V_1\otimes \cdots \otimes V_k \to W$ by sending pure tensor elements $v_1 \otimes \cdots \otimes v_k$ to $f(v_1 \otimes \...
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14 views

rank of a matrix formed from a tensor product

I know that rank of a matrix expressed in term of product of 2 matrices, satisfies the following equation rank(AB) ≤ min(rank(A), rank(B)). When we write a matrix as a product of tensors(of tensor ...
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1answer
17 views

Tensor Product of Matrix and Vector

I am studying tensor networks and tensors. A commonly described operation is the tensor product (denoted by $\otimes$) which is a generalization of the outer product (as I understand it). It makes ...
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16 views

$\mathbb{R}^{\Omega}\otimes\mathbb{R}^{\Omega}\cong\mathbb{R}^{\Omega\times\Omega}$ and beyond

Notations : $V^*$ is the dual of $V$, $\otimes$ is tensor product of vector spaces/modules, $\mathbb{R}^{\Omega}$ is the set of functions from $\Omega\subset\mathbb{R}$ to $\mathbb{R}$. I found in Lee'...
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Matrix times Vector where the elements are vectors

Whats the correct operation to calculate the "product" of matrix $A$ of the size $M \times L$ $$A= \begin{bmatrix} \vec{A}_{1,1} & \vec{A}_{1,2} \\ \vec{A}_{2,1} & \vec{A}_{2,2} \\ ...
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Module structure of tensor product when changing basis

I'm trying to understand a part of a proof in an algebra course. Setting: $A,B$ commutative, unital rings and $B$ is an $A$-Algebra, where $f:A\to B$ is a ring-homomorphism. Furthermore $M$ is an $A$-...
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1answer
35 views

Rewriting a tensor product as a quotient of a polynomial ring

Good evening. While trying to show that the base change of étale morphisms is étale, I have ended up stuck on a commutative algebra question I can't seem to be able to solve. Suppose you are given two ...
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1answer
54 views

Alexander Whitney map is a chain map

Define $a_p : \Delta_p \longrightarrow \Delta_n$ and $b_q : \Delta_q \longrightarrow \Delta_n$ to be $\begin{cases}a_p(e_i) = e_i & p \leq n \\ b_q(e_i) = e_{n-q+i} & q \leq n \end{cases}$. ...
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1answer
71 views

Are $(1, 0)$ tensors always vectors? (resolved)

An $(r, s)$ tensor $T$ is defined to be an element of the tensor product of a vector space and its dual: $$T \in T^r_sV := V^{\otimes r}\otimes V^{* \otimes s}.$$ However, when $V$ is finite ...
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1answer
50 views

Tensor algebra $T(A \oplus B)$ is a coproduct of $T(A)$ and $T(B)$ in the category of unitary associative $(R, R)$-algebras.

Let $A$ and $B$ be unitary $(R, R)$-bimodules. Let $f_1 \colon T(A) \to C$ and $f_2\colon T(B) \to C$ be identity preserving $(R, R$)-algebra homomorphisms. I need to define morphisms $\iota_1 \colon ...
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Find the correct sign so that the equation is satisfied

Context Let $A=\bigoplus_{i,k} A^i_k$ be a bigraded module and let us say we have a linear map $\circ :A\otimes A\to A$ of bidegree $(0,0)$. This operation satisfies certain associativity relation ...
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1answer
62 views

$Hom(K,R) \otimes Hom(C,R) \simeq Hom(K \otimes C,R)$

Is the following statement true when $K$ is a chain complex of finitely generated free module over $R$ PID and $C$ is a chain complex over $R$ ? $$Hom(K,R) \otimes Hom(C,R) \simeq Hom(K \otimes C,R)$$ ...
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0answers
41 views

Showing that spectrum of tensor product of linear operators having respectively $\{μ_i\}$ and $\{λ_j\}$ as eigenvalues is $\{μ_iλ_j\}$.

I'm hoping to get a clue to show the following problem, for which I only have an intuition: Considering a finite-dimensional vector space $V$ and operators $φ:V→V$ and $ψ:U→U$, I would like to show ...
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1answer
46 views

Confusion regarding the use of horizontal padding of $(1,1)$-tensor indices: which is the “correct” interpretation?

Fix finite-dimensional vector spaces $V,W$. My whole life I'm been content with viewing linear maps $T:V\to W$ as $(1,1)$-tensors, i.e. elements of $W\otimes V^*$: such a $T$ yields a bilinear map $B:...
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1answer
23 views

Well-definedness of Wedge Product

Quick preface - this is problem 14-3 in Lee's Intro to Manifolds and a past homework problem I never really finished. Let $\Re=$ space generated by all $w^1\otimes\cdots\otimes w^k|w^i=w^j$ for some $...
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1answer
39 views

Proving that a wedge product is closed/exact [closed]

I am trying to prove the following and don't quite know where to begin. Let $A \subseteq \mathbb R^n$ be an open set. Suppose $\omega$ is a close k-form on $A$, i.e. $d\omega =0$, and $\eta$ is an ...
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1answer
66 views

$0 \longrightarrow C(A)\otimes C(Y) + C(X)\otimes C(B) \longrightarrow C(X)\otimes C(Y) \longrightarrow C(X,A)\otimes C(Y,B) \longrightarrow 0$

In the relative version of Eilenberg-Zielber Theorem the following exact short sequence is taken $$0 \longrightarrow C(A)\otimes C(Y) + C(X)\otimes C(B) \longrightarrow C(X)\otimes C(Y) \...
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95 views

Dense set and the tensor-product

Let $F, G, X, Y$ be Hilbert spaces, where $G \subset F$ is dense in $F$ and $Y \subset X$ is dense in $X$ with respect to the corresponding norms. Furthermore, the estimates $$\forall g \in G: \quad \|...
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1answer
32 views

$Tor(A,B) = 0$ if $A,B$ is torsion free

I don't understand the proof given in Hatcher p.265 of $Tor(A,B) = 0$ if $A,B$ is torsion free. The proof is the following : The line I don't get is "This means [...] can be reduced to $0$ by a ...
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0answers
32 views

Tensor product of adjoint operators

I'm working over $G = SU(2)$. $g$ represents an element of $G$, and $E,F$ are elements of the Lie algebra $\mathfrak{su}(2)$. Since these are $2\times 2$ matrices, the natural tensor product space is $...
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1answer
23 views

The trace of $A\otimes A$ as a self-map of the symmetric square $S^2V\leq V\otimes V$

Let $V$ be a finite-dimensional complex vector space and $A\in \text{End}(V) $. Then we may define $A\otimes A\in \text{End}(V\otimes V)$ via $$ (A\otimes A)(v\otimes w)=(Av)\otimes(Aw)$$ and ...
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3answers
274 views

A tensor product vs the tensor product.

I have been reading about tensor products in different books and struggle with the definition of tensor products in the following sense. There is a common construction of a tensor product as a ...
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1answer
61 views

Clarification about $C \otimes D$ with $C,D$ contractible chain complex

The question is strictly related to Eilenberg-Zielber theorem, since the ones I'm about to write are the Propositions needed in order to prove the Theorem. Given $(C,d_c),(D,d_d)$ chain complex and ...
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1answer
37 views

$Hom(C,R) \otimes G \simeq Hom(C,G)$

Given $C,G$ free $R-$modules, I was interesed in proving that $Hom(C,R) \otimes G \simeq Hom(C,G)$ if $G$ is finitely generated. I thought $\psi : \varphi \otimes g \longmapsto \varphi \cdot g$ could ...
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1answer
80 views

When a power series in two variables is a finite sum of products of series

Consider the infinite sum $ \sum_{k \geq 1} p_k(x,y), $ where $$ \begin{aligned} p_1(x,y) &= 1, \\ p_2(x,y) &= y + xy^2, \\ p_3(x,y) &= y^2 + xy^3 + x^2 y^4, \\ \vdots \\ p_n(x,y) &= y^...
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1answer
40 views

If $V$ is simple then is $V\otimes W$ is simple given $W$ is simple?

Let $W$ be a simple, finite dimensional, $\mathbb{C}G$-module and $G$ is a finite group. Suppose $V$ is also a simple, finite dimensional, $\mathbb{C}G$-module, would it be the case that $V\otimes W$ ...
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1answer
18 views

Base of tensor product

Given the tensor product $V_1 \otimes V_2$ (so we have a bilinear map $\otimes : V_1 \times V_2 \to V_1 \otimes V_2$) and bases $B_i$ of $V_i$ (not need to be finite) how to prove using only the ...
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0answers
41 views

Diagonalisation of tensor product

Say I have two diagonalisable matrices $A$ and $B$, diagonalisable in different basis'. \begin{equation} A=\sum_ia_i|a_i\rangle\langle a_i|\qquad\text{and}\qquad B=\sum_ib_i|b_i\rangle\langle b_i|\;. \...
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1answer
68 views

How to calculate a tensor product

I’m used to the abstract definition of tensor product as an universal solution, but I’m getting trouble in how to calculate one: Take an example, I think $m\mathbb{Z}\otimes \mathbb{Z}/(m)=0$, since $...
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1answer
48 views

Why doesn't right exactness of a functor (say tensor product) imply exact?

Let's work in $R$-mod for the remainder of this question. I know I'm probably missing something super basic, but here goes. Let $F$ be the functor $- \otimes_R M$ for some $R$-module $M$. Let's use ...
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2answers
34 views

Prove that there is an $R$-module isomorphism between $Q \otimes_R N \cong N$, Q is a quotient field of $R$.

Let $R$ be an integral domain with quotient field Q and $N$ be a unitary, divisible, torsion-free left $R$-module. Show that there is an $R$-module isomorphism so that $Q \otimes_R N \cong N$. Here ...
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1answer
22 views

How to make a multidimensional SVD?

Is it possible to define a tensor Singular Value Decomposition (SVD)? For example for the 3 tensor $$\left[\begin{array}{rr}1&-1\\1&1\\\hline 1&-1\\-1&-1\end{array}\right]$$ Can be ...
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1answer
37 views

unique factorization of product of linear functionals

Let $V$ be a vector space over the complex numbers. Let $f_{1\leq i \leq n}$ and $g_{1\leq i \leq n}$ be two sets of nonzero linear functionals on $V$. Suppose we have $$f_1(v) f_2 (v)\ldots f_n (v) = ...
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1answer
71 views

Is that a exact sequence?

Let $A$ be an Algebra, $S$ a subalgebra ans $W$ a subspace s.t $SW \subset W$, $WS \subset W$ and $S,W$ generate $A$. Show that for any $A-$module $M$ the follow sequence is exact. $$A \otimes_S W \...
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2answers
35 views

Unique definition of $T(V) = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$

We know that $V \otimes(W \otimes U) \cong (V \otimes W) \otimes U \cong V \otimes W \otimes U$ but as far as I know there is not a canonical isomorphism. Now consider $T(V) = \bigoplus_{n \in \...
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1answer
33 views

Trace of tensor product identity

Let $A: V\to V$ and $B: W\to W$ be linear operators on vector spaces $V$ and $W$. I know how to prove $$\operatorname{tr}(A\otimes B) = \operatorname{tr}(A)\operatorname{tr}(B)$$ by appealing to a ...
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0answers
17 views

Tensor products of group algebras

We know that if $G$ and $H$ are finite groups and $F_p$ is a field of characteristics $p$, then $$F_pG\otimes_{\mathbb{F}_p} F_pH\cong F_p(G\times H).$$ Here $\otimes_{\mathbb{F}_p} $ denotes the ...

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