# 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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### 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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### 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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### 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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### 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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### 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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### 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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