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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General expression of smooth sections of tensor bundles.

On page 317 of John Lee's Smooth Manifolds it's said that if $(x^i)$ are local coordinates on a smooth manifold $M$, then sections of the tensor bundle $T^kT^*M=\bigsqcup_{p\in M}T^k(T^*_pM)$ over a ...
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122 views
+200

Symmetric and wedge product in algebra and differential geometry

I have been struggling with this issue for a while (and asked a similar question here), but still not found a satisfying answer. The question boils down to: which is the correct identity? $dx \, dy ...
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4answers
76 views

Showing $U\otimes (V\oplus W) \cong U\otimes V \oplus U\otimes W $

I have to show $U\otimes (V\oplus W) \cong U\otimes V \oplus U\otimes W $ where $U,V,W$ are $K-$vector spaces. One way to give a linear map from left to right is: $$u\otimes (v,w)\mapsto (u\otimes v, ...
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0answers
17 views

What is $M\otimes_\mathbf{Z}(N/P)\cong ?$

Let $A=\mathbf{Z}[G]$ where $G$ finite, and let $M,N,P$ be $A$-mods. Is there any particularly nice relationship for $M\otimes_\mathbf{Z}(N/P)$?
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1answer
14 views

Isomorphism of a torsion product and a quotioten of torsion product.

I have the following problem: If $A'$ a submodule of the right $R$-module $A$ and $B'$ a submodule of the left $R$-module $B$, then $A/A' \otimes B/B' \cong (A\otimes B)/C$ where $C$ is the ...
2
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1answer
29 views

Tensor product and its homomorphisms.

Given $f:A_R\rightarrow A'_R$, $g:B_R\rightarrow B'_R$ R-module homomorphism we can define $f\otimes g: A\otimes_R B\rightarrow A'\otimes_R B'$ such that $(f\otimes g)(a\otimes b)=f(a)\otimes g(b)$ ...
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27 views

Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?

so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...
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12 views

Injectivity and uniqueness of some maps from tensor products

Let $V_A$, $V_B$, $W$ be real vector spaces, let $V_A^*$, $V_B^*$, $W^*$ be their dual spaces. Let \begin{align*} &\phi: V_A \times V_B \rightarrow W \\ &\psi: V_A^* \times V_B^* \rightarrow ...
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1answer
19 views

Tensor product of $\mathbb{R}^d$ and $\mathbb{R}^s$ as abelian groups

It is well known (and easy to prove) that $\mathbb{R}^d\otimes_{\mathbb{R}} \mathbb{R}^s$ is isomorphic as a vector space to $\mathbb{R}^{sd}$. Now, I would like to know a simple description of the ...
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79 views

Tensor products over monoids : Element structure

Let $A$ be a (commutative) monoid. Let $M$ be a right $A$-set and let $N$ be a left $A$-set. Then we can construct the tensor product $M \otimes_A N$, which is a set (of even $A$-set when $A$ is ...
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1answer
44 views

How to calculate scalar product of two gradients in indicial notation?

Does someone knows how calculate scalar product of two gradients and put the result in terms of nabla operator? . $(\vec\nabla{\gamma})\cdot(\vec\nabla{\gamma})$ = ?
9
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81 views

Minimal and characteristic polynomials on tensor product spaces

Given two finite-dimensional vector spaces $V$ and $W$ over a common field $k$ as well as linear transformations $\varphi \colon V \to V$ and $\psi \colon W \to W$, what (if anything) can one say in ...
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2answers
21 views

Scalar multiplication on tensor products

I am in the process of getting comfortable with tensor products (it's going very slowly). My question needs the following setup: We have a finite $K$-algebra $A$ and $L\supset K$ fields. We ...
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24 views

Is there a name for this operation: $x_k = \sum_{ij} y_{ij} a_{ik} b_{jk}$?

Is there a name for this operation: $x_k = \sum_{ij} y_{ij} a_{ik} b_{jk}$ ?
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1answer
49 views

Does $V\otimes_k K\cong W\otimes_k K$ imply $V\cong W$?

Let $V$ and $W$ be two $k$-vector spaces of the same dimension and $K/k$ any field extension. If $V\otimes_k K\cong W\otimes_k K$ as $K$-vector spaces then are $V$ and $W$ already isomorphic over $k$? ...
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1answer
107 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
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1answer
36 views

On isomorphisms of tensors of certain type

I've got a question form Gille and Szamuely's "Central Simple Algebras' and it's about vector spaces equipped with tensors of certain types. Let $V$ be a $k$-vector space. For a field extension ...
4
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0answers
86 views

Consistency of different ways to define the tensor product

It seems my trouble with understanding tensors stems from the following statement: More specifically, the statement: Namely, given $B: V \times W \to U$ and $\xi: U \to \mathbb{R}$, $\xi ...
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1answer
48 views

Decomposition of Tensor into a Product of Tensors

I was working on a text that made use of the assumption that, for some rotation matrix $a_{ij}$, and some tensor $U$, the tensor under rotation is represented by $U'_{\alpha}=a_{\alpha i}U_i$. It made ...
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56 views

Tensor product of free modules over free algebra

Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$. Question: Under what ...
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22 views

efficient least squares A = BX+CXD (solving for matrix X)

I am interested in solving a least-squares solution of the form $$ \operatorname{argmin}_X \| A - BX - CXD \|_F^2 $$ for large (rank in hundreds to thousands) matrices $A,B,C,D,X$ I know this is ...
3
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1answer
94 views

Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
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74 views

Tensor product of arbitrary categories

I would like to consider a definition of tensor product in the category of (small, finite, whatever is needed) categories, analogous to the tensor product of vector spaces. I will first rewrite ...
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43 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
2
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1answer
54 views

Tensor products and isomorphic algebras

I found that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus\mathbb{C}$ and that $ \mathbb{H} \otimes_{ \mathbb{R}} \mathbb{C} \simeq M_2( \mathbb{C})$. Could anybody hint me how ...
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0answers
12 views

Expanding a product of matrices with tensor product and transpose

I'm trying to expand the following product of $\pm1$ matrices $H_1, H_2, K_1, K_2$: $(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 (H_1-H_2)\otimes K_2^T)(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 ...
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0answers
36 views

Tensor product with an L on top

Looking at the definition of the Kunneth morphism in SGA4, XVII, 5.4.1.4, there is the notation $$Rf_*K \overset{\mathbb{L}}{\boxtimes}_{\mathcal{A}_0} Rg_* L \rightarrow Rh_*(K ...
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0answers
60 views

Surjectivity implies injectivity of finitely generated modules, localization?

The following problem is canonical: Suppose $A$ is a commutative unitary ring, and $M$ is a finitely generated module over $A$. If an endomorphism $f\colon M\to M$ is surjective, then it's also ...
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1answer
25 views

Ring structure on tensor product of two $A$-modules

Let $A, B, C$ be a commutative rings. Suppose I have two ring homomorphisms, $\alpha : A \rightarrow B$ and $\beta : A \rightarrow C$. I am trying to show that $B \otimes_A C$ has a ring structure ...
3
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1answer
60 views

Tensor product of duals

Can we show $V^* \otimes W^* \simeq (V \otimes W)^*$, when $V$ or $W$ is finite-dimensional without referring to the basis? I can inject $V^* \otimes W^*$ into $(V \otimes W)^*$ using the obvious ...
2
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1answer
53 views

Why are these dimensions equal?

For a finite $K$-algebra $A$ and $L\supset K$ fields, why do we have$$\dim_K A=\dim_L(A\otimes_KL)?$$ I ran across this a couple of times and it's always assumed to be quite obvious, which it isn't to ...
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2answers
51 views

On a basic tensor product question

I am trying to show that $$ \mathbb{Z} / (10) \otimes \mathbb{Z} / (12) \cong \mathbb{Z}/(2) $$ by defining a map $$ h([a]_{10} \otimes [b]_{12}) = [ab]_2 $$ and extend it linearly. I am having ...
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2answers
57 views

Why is tensor product of linear maps defined as $(S\otimes T)(v\otimes w)=S(v)\otimes T(w)$?

In my understanding, the definition of tensor product of linear maps cannot be directly derived from the definition of tensor product of vector spaces (or modules), since it's not clear what is the ...
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0answers
41 views

Exercise 7.10 Atiyah, $M[x] $ is a noetherian $A[x] $-module [duplicate]

The exercise is: Let $M$ be a noetherian $A$-module. Then $M[x] $ is a noetherian $A[x] $ module. The action of $A[x] $ on $M[x] $ is the obvious one. In a previous exercise it was shown that ...
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30 views

Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
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0answers
36 views

can quaternions be expressed in terms of tensor products?

QUESTIONS does this arithmetic check out? if so, is there a geometric interpretation? note: my aim was to try to find a very simple but non-trivial example which might help me begin to understand ...
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1answer
24 views

natural isomorphism of polynomial functions on $V$ and $S(V^*)$

In Humphreys, reflection groups and coxeter group book, Humphreys denotes $S(V^*)$ as the ring of polynomial function on the finite dim vector space $V$. Why we are considering $S(V^*)$ rather than ...
3
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2answers
78 views

Is the tensor product of two representations a representation?

I am a little bit uncertain about an argumentation showing that a given map of a topological group is somehow obviously continuous. In the following I will rely on the book of Anthony W. Knapp „Lie ...
2
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1answer
47 views

Monoidal product is coproduct in category of commutative monoids

If $V$ is a symmetric monoidal category, the category $\text{CMon}(V)$ of commutative monoids in $V$ has binary coproducts given by $\otimes$, the monoidal product of $V$. See for example Johnstone’s ...
2
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1answer
37 views

how do I view the tensor product $X^*\otimes Y$ as a subspace of $\mathcal{L}(X,Y)$?

Background. According to Raymond A. Ryan, in his book Introduction to Tensor Products of Banach Spaces, a Banach ideal $\mathcal{J}$ is an assignment to each pair of Banach spaces $X$ and $Y$ a ...
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25 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) ...
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2answers
125 views

Proving that tensoring a projective module with a flat module gives a projective module?

If $P$ is a projective module and $M$ is a flat module, both over some commutative ring $R$, then how do you prove that $M\otimes_R P$ is flat? I tried using the fact that $P$ is the direct ...
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1answer
26 views

Tensor product and free $R$-module construction

I am trying to understand an exposition on Tensor products by Keith Conrad. In the proof of Theorem 3.2 on page 7 it considers the free $R$-module on the set $M \times N$: $$F_R(M \times N) = ...
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1answer
61 views

What is the wedge product of multilinear forms?

The construction of $V^* \otimes V^*$ involves creating formal symbols and then adding in relations such as bilinearity by quotienting out. A bilinear form $V\times V\to F$ can be thought of as a ...
2
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2answers
48 views

Relation between $\operatorname{Coker}(f)$ and $\operatorname{Coker}(f \otimes 1_P)$

Let $M,N,P$ be $R$-modules ($R$ commutative ring with $1$) and let $f:M\to N$ be a $R$-module homormorphism. Let tensor the homomorphism to get $ f \otimes 1_P : M \otimes P \to N \otimes P $. I ...
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1answer
66 views

Completion of a torsion-free module

Let $R$ be a Dedekind domain, $K$ its field of fractions, $P$ a non-zero prime ideal of $R$. Let $\hat R_P$ be the completion of $R$ wrt to the valuation $v_P$ induced by $P$ and let $L$ be a ...
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47 views

Product between the Kronecker product of three matrices and a vector

I need to perform the product between a matrix defined as the kronecker between three matrices and a vector : $$v = (A \otimes B \otimes C)u.$$ I know there is an identity for such a thing with two ...
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1answer
32 views

Operator Tensor Product

Let $S$ and $T$ be bounded operators over a Hilbert space $\mathcal{H}$. Define their tensor product $S\otimes T$ as acting on $\mathcal{H}\otimes\mathcal{H}$ by $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
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1answer
44 views

Localization of a torsion-free module

I have a Dedekind domain $R$ with field of fractions $K$ and a non-zero prime ideal of $P$ of $R$. Let $L$ be a torsion-free $R$-module. How can I show that $R_P\otimes_R L$ is torsion-free as an ...
2
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1answer
40 views

When is $M \times N$ contained in$ M \otimes_{R} N$.

Let $R$ be a ring with 1. $M$ is a right R-module and $N$ is a left R-module. A tensor product of M and N comes with a map from $M \times N \to M\otimes N$ which is actually a composition of maps ...