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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Rank of a locally free $\mathbb Z[G]$- module

Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$. Here $\mathbb Z_p[G]=\mathbb Z_p\otimes_\mathbb ...
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What are the main differences between Tensor Products of Vector Space vs Modules vs Algebras

What I understand so far: I understand the definitions of vector space, modules, algebras. I am also acquainted with basic properties of Tensor Product of Vector spaces, especially the "universal ...
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Query on Tensor Product of Quaternion Algebras

In this proof of isomorphism of tensor product of quaternion algebras I have two queries: 1) Why does one need to "check $xy=yx$ for all $x\in C$ and $y\in D$? 2) How is the product $CD$ defined? ...
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How to make $S\otimes_R M$ into a left $S[G]$-module

I have a basic question on tensor products which might have been asked before-sorry if this is the case. Let $G$ be a finite group and $R\subset S$ any (unital) ring extension. Let $M$ be a finitely ...
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34 views

When is tensor product isomorphic to product?

Let $A$ and $B$ be algebras. When do we have $A\otimes B\cong AB$, where $$AB=\{\sum a_ib_i\mid a_i\in A, b_i\in B\}$$ Is commutativity $ab=ba$ for $a\in A$, $b\in B$ a sufficient condition? Thanks ...
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Is $l^2 \cong l^2 \otimes l^2 \cong l^2 \oplus l^2$?

I'm trying to learn about tensor products of Hilbert spaces and started to wonder if $l^2 \cong l^2 \otimes l^2 \cong l^2 \oplus l^2$? If $(e_n)$ denotes the standard basis, in the first case, it ...
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Tensor Product of Modules over R and a subring S

If we have a commutative Ring R with a subring S and two R-Modules (or S-Modules). What can we say about the correlations between $$ M \otimes_R N \quad and \quad M \otimes_S N \quad ?$$ Wikipedia ...
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26 views

vanishing tensor product, geometric meaning

We can derive the following property directly. " $\mathbb{Z}_m \otimes_{\mathbb{Z}} \mathbb{Z}_n = O$. where $m,n$ are relatively prime. But, this is just a simple computation. I want to know the ...
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37 views

A question about tensor product and multiplication?

Let $k$ be a commutative ring, $A$ be an algebra over $k$. The tensor product $A\otimes A$ is over $k$. If $\sum_{i} a_i \otimes_k b_i=\sum_j c_j\otimes_k d_j$, where $a_i,b_i,c_j,d_j\in A$, I wonder ...
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11 views

Suppose $N \cong R^n $ then for every non-zero $R$ module $M$ show that $M \otimes N$ can be written uniquely by $\sum m_i \otimes e_i $

Suppose $N \cong R^n $ be free $R$ module of rank $n$ with basis $\{e_1,...,e_n\}$ and $R$ is commutative then for every non-zero $R$ module $M$ show that $M \otimes N$ can be written uniquely by ...
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If $G$ be a abelian Then $G\otimes_{\mathbb{Z}}\mathbb{Z}_m\simeq G/mG$ [on hold]

Let $G$ be a abelian group. Then $G\otimes_{\mathbb{Z}}\mathbb{Z}_m\simeq G/mG$ suggestion please.
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Proof of a vector identity for adjoint heat transfer

I have a domain $\Omega$ and the expression below. S, T and $\alpha$ are scalar fields. $$\int_\Omega \frac{\partial}{\partial \alpha}T\nabla\cdot(\kappa(\alpha)\nabla S)\;d\Omega$$ I have read in a ...
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25 views

How do outer products differ from tensor products?

From my naive understanding, an outer product appears to be a particular case of a tensor product, but applied to the "simple" case of tensor products of vectors. However, in quantum mechanics there ...
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63 views

Fibers of extension of scalars to algebraic closure of an affine variety is of dimension 0.

Let $X$ be an affine variety over $k$ and $\overline{k}$ be the algebraic closure of $k$. Then the projection morphism $X_\overline{k} = X \times _k \overline{k} \to X$ has dimension 0 fibers. ...
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53 views

How does extension of restriction of $M$ relate to $M$?

Let $A,B$ be rings, $f:B\to A$ be a ring homomorphism, and $M$ be an $A$-module. We can view $M$ as a $B$-module via restriction, and we may then extend the restriction of $M$ to an $A$-module by ...
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Is $p\in B(\mathbb{C}^4)$ a s.o.t-limit of a sequence $(a_n\otimes b_n)_{n\in\mathbb{N}}\subseteq B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)$?

Let $L(H)$ the bounded linear operators on a hilbert space $H$. I proved that the inclusion $$i:B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)\hookrightarrow B(\mathbb{C}^4)$$ is not surjective: take ...
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22 views

Is this tensor contraction correct?

If I start off with the $\left(p,q\right)$-tensor given by $$T_{i_1,\dots,i_p}^{j_1,\dots,j_q}e_{i_1}\otimes\dots\otimes e_{i_p}\otimes\varepsilon^{j_1}\otimes\dots\otimes\varepsilon^{j_q}$$ and I ...
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34 views

Understanding isomorphism of Hom's

During some reading I came across the statement $$Hom_R(M,N) \otimes R/I \cong Hom_{R/I}(M/IM,N/IN) $$ where $M,N$ are $R$-modules and $I$ an ideal of the commutative ring $R$. This is proved, under ...
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Geometrical Interpretation of Tensors (intuition)

How would one describe to a non-mathematician (an undergraduate physicist actually- so please do use mathematics-i am not even sure if they can be described without mathematics!) what do tensors ...
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1answer
17 views

Does this symmetric rank-3 tensor vanish?

Suppose we have a rank-3 tensor $T$ on some vector space $\mathbb{V}$. We can view $T$ as a map: $$T: \mathbb{V} \times \mathbb{V} \times \mathbb{V} \to \mathbb{R},$$ which maps triples of vectors ...
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45 views

Tensor product of coordinate rings corresponds to pullback

Here in Milne's notes on algebraic geometry, he proves that if $k$ is an algebraically closed field, and $A$ and $B$ are reduced finitely generated $k$ algebras, then $A \otimes_k B$ is reduced. (This ...
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Tensor products in separable Hilbert spaces

In a variety of scientific publications, I have come across the use of tensor products of random Hilbert-Schmidt operators defined on separable Hilbert spaces. Let us introduce some notations. Define ...
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21 views

Commutation of Tensor Products as operators

Suppose I have unitary operators $$A: \mathbb{C}^{2^k} \rightarrow \mathbb{C}^{2^k}$$ $$B: \mathbb{C}^{2^j} \rightarrow \mathbb{C}^{2^j}$$ For some $k,j \in \mathbb{Z}, j,k \ge 0$. How do show that ...
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23 views

$\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert space

I want to describe $\mathbb{C}^n\otimes \mathbb{C}^m$ as tensor product of Hilbert spaces; $\mathbb{C}^n\otimes \mathbb{C}^m$ is endowed with the scalar product $\langle x\otimes y, x'\otimes ...
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23 views

Tensor Product of Algebras Commutes with Products?

It seems to me that if $R$ is commutative and $A,B,C$ are (associative, unital) $R$ algebras then $(A\times B)\otimes_R C\approx A\otimes_RC\times B\otimes_RC$. On the level of $R$ modules, we can use ...
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13 views

How to write clebsch gordan decomposition using tensor notation

Let be $G$ a Lie Group and $\textbf{N}$ its complex representation. It is well know that any state $|\ ab\ \rangle\in \textbf{N}\otimes\textbf{N} = \oplus_I\textbf{r}_I$ can be decomposed through the ...
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48 views

Linear algebra prerequisites for abstract algebraic geometry

I'm interested in what linear/multilinear algebra does one need to study algebraic geometry(following EGA and Harthshorne). Texts I have in mind are like "Foundations of algebraic geometry" by Ravi ...
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eigenvalues of sum of tensor product of matrices

Lets say $A, B$ are rank 1 hermitian real matrices. It follows immediately that $A\otimes A$ and $B\otimes B$ individually are rank 1. My question is how does the largest eigenvalue of $A+B$ relate ...
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Prove the pullback of the wedge product is the wedge product of the pullbacks.

Let $F:V \rightarrow W$ be a linear map. Show that $F^{\ast}(\omega \wedge \eta)=(F^{\ast}\omega) \wedge (F^{\ast}\eta)$ for all $\omega \in \Lambda^{p}(W) , \eta \in \Lambda^{q}(W)$. Where ...
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Correspondence between tensors and multivectors

In one of its definitions the Clifford algebra is defined as a quotient space of a (infinite dimensional) Tensor algebra. The question is: Given the metric signature and the tensor $T$ and its ...
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Question about two ways to induce an inner product on $S^2V$

$\newcommand{\til}{\tilde}$ Let $(V,g)$ be an $n$-dimensional inner product space, and let $S^2V^*$ be the symmetric algebra. I am familiar with a natural way to endow $S^2V^*$ with an inner product ...
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Equivalent conditions for a short pure exact sequence

Let $L$, $N$, and $M$ be right $R$-modules and let $\widehat{L}=\mathrm{Hom}_{\mathbb{Z}}\left(L,\mathbb{Q}/\mathbb{Z} \right)$ ($\widehat{N}$ and $\widehat{M}$ are defined analogously). Show that ...
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Definition of Outer Product of (abstract) vectors

I was reading an article from the American Mathematical Monthly on the Caratheodory derivative for functions of several variables, and in one of the proofs the authors construct a linear ...
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Ordering of basis elements of a Lie-group representations tensor product

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \bigoplus_J \textbf{r}_J $$ where $\textbf{r}_J$ ...
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29 views

Hadamard product involving operators

If we have two matrices $A=(a_{i,j})_{i,j}$, $B=(b_{i,j})_{i,j}$ representing linear and continuous operators from $\ell^2$ to $\ell^2$, it is known that the Hadamard product of them, $A\ast ...
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Can this matrix equation be solved?

I have a matrix equation of the form [Z][C][Z] - [Z][D] = [A][Z] - [B] and I need to solve for [Z]. Is it possible? If so, what is the solution? The matrices are all general, complex, with no ...
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35 views

Covariant Contravariant approach for Tensors

I'm reading a book on Geometry from the '70s and when speaking about Tensors it defines them starting from the covariant and contravariant commutation rule. I know this definition was quite widespread ...
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36 views

Tensor algebra with semidirect product

I am having trouble parsing the following problem: Let $F$ be a field, $A$ an $F$-algebra with 1 (not necessarily commutative) and let $M$ be an $F$-vector space which is also an $A$-bimodule. Let ...
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44 views

Differing definitions of a connection on a vector bundle

My lecture notes define a connection on a vector bundle $\pi:E\rightarrow{M}$ to be an $\mathbb{R}$-linear map: \begin{equation} \nabla:\Gamma(E)\rightarrow\Gamma(T^*M\otimes{E}) \end{equation} ...
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Construction of Tensor Product on $\mathbb R^2$

I am trying to construct a tensor product on $ \mathbb R^2$. I have defined a bilinear map by $\phi:\mathbb R \times \mathbb R\to \mathbb R $, as $\phi(x,y) = xy$ Which clearly is a bilinear map. ...
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Understanding the construction of Exterior Algebra

Background The tensor space of type $(r,s)$ associated with $V$ is the vector space $$\underbrace{V\otimes \ldots \otimes V}_{\text{r copies}} \otimes \underbrace{V^* \otimes \ldots \otimes ...
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Using Universal Mapping Property of Tensors to show that $\dim V \otimes W=(\dim V)(\dim W)$

Background Let $V,W$ be finite dimensional vector spaces, $V^*$ the dual space of $V$ and $\mathrm{Hom} \ (V,W)$ the vector space of all linear transformations from $V$ to $W$. The universal mapping ...
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Why does this tensor product has exactly 2 elements?

I am trying to prove that $(\mathbb{Z}/10\mathbb{Z})\otimes (\mathbb{Z}/12\mathbb{Z}) \cong (\mathbb{Z}/2\mathbb{Z})$. I understand now that every element in $(\mathbb{Z}/10\mathbb{Z})\otimes ...
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25 views

What are the submodules of $M⊗K$?

Let $k$ be a field and $K$ is a extension of this field. And let $A$ be a finite type $k$-algebra, and $M$ be a finitely generated module over $A$. Then, is the form of submodules of $M⊗_kK$ always ...
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Help with a statement in P.286 of Hatcher

In P.286 of Hatcher's 'Algebraic Topology', it is stated that $\alpha_{p^i}$ is primitive in $ \bigotimes_{i\geq 0} \mathbb{Z}_p[\alpha_{p^i}]/(\alpha_{p^i}^p)$, where 'primitive' means the coproduct ...
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51 views

Is there a relation between Cartesian and tensor product of function spaces and function factorizability

H1 and H2 are two Hilbert spaces represented by a function space, say f1(x1) and f2(x2) are its vectors. If H3 is tensor product of H1 and H2 I assume one can say that f(x1,x2) now represents vectors ...
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42 views

When (and why) tensor product? When dot product?

This looks a lot like physics, but it is actually a math question! I will be omitting unnecessary constants for simplicity so the units might be off. I want to reduce the equation $-i\omega ...
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Is the tensor product over $B$ of two flat $A$-modules flat over $A$?

Given a morphism of commutative rings $A\to B$ such that $B$ is a flat $A$-module and given $M$, $N$ two $B$-modules flat as $A$-modules, is the tensor product $M\otimes_B N$ flat over $A$?? The ...
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14 views

Tensor product on a hilbert space

If I have the expression: $\langle\phi|\hat{A}$ $ \otimes \hat{I}|\phi\rangle$ $ $ $ $$ $(*) where $\hat{A}$ is a linear operator $\hat{I}$ is the identity operator and $| \phi \rangle ...
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25 views

Tensor prodct on Hilbert Space

How do I show that $\langle \phi|I \otimes I|\phi\rangle=1$ ? where: $I$ is the identity operator and $\phi \in \mathbb{C^2}\otimes\mathbb{C}^2$