# 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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### Coalgebras & Coideals: Why does $ker(\pi \otimes id_C ) = I\otimes C$ hold?

In a proof on comodules and coideals I found the following passage: “Let $C$ be a coalgebra, and $I \subset C$ a vector subspace. Let $\pi$: $C \rightarrow C/I$ be the canonical projection. Consider ...
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### Tensor product of two $n\times n$ orthogonal matrices with determinant $+1$

If two square matrices A and B are two $n\times n$ orthogonal matrices with determinant unity i.e., $\det A=\det B=+1$ and $A^TA=B^TB=I$, will the tensor product $C\equiv A\otimes B$ also be ...
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### Proof that the exterior power fulfils an universal property using tensor product

before I ask my question, I briefly review two definition which I use: (1) Tensor product of vector spaces: Let $V_{1},\dots,V_{n}$ be vector spaces over the same field $\mathbb{F}$. Then the ''...
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### Does quadruple product identity holds for any metric tensor?

$(A\times B)\cdot(C\times D)=(A\cdot C)(B\cdot D)-(A\cdot D)(B\cdot C)$ If I take any metric tensor $g_{ij}$ instead of $\delta_{ij}$, will this identity still holds? I found as if $g$ is a symmetric ...
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### How are the following two rings isomorphic?

The paper here in Construction 4.16 makes the following claim that I'm unable to unpack (though my question should be self-contained here): Let $R$ be a commutative ring, and let $r\in R$ be an ...
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### About two definitions of tensor product. (in James R. Munkres's book and Ichiro Satake's book)

I am reading "Analysis on Manifolds" by James R. Munkres. Definition: Let $f$ be a $k$-tensor on $V$ and let $g$ be an $l$-tensor on $V$. We define a $k+l$ tensor $f \otimes g$ on $V$ by the ...
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### Tensor of s.e.s. of projective modules [closed]

Let $f:R \rightarrow S$ be a ring homomorphism. All modules are right R-modules.Let $\otimes_R S:\mathbb{P}(R) \rightarrow \mathbb{P}(S)$ be the functor sending a projective $R$-module $P$ to the ...
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### Finitely generated module over PID with tensor with itself is zero

Hello I have the next doubt about this problem: Show that if $A$ is a finitely generated module over a PID and $A\otimes_{\Lambda}A=0$, then $A=0$. I have done the next thing, I consider the next ...
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### Existence of the tensor product

Let $E_1$ and $E_2$ be $\mathbb{K}$-vector spaces. Lets also consider the set $F$ the functions $\xi:E_1\times E_2\to \mathbb{K}$ such that \begin{equation} |\{(\mathbf{u},\mathbf{v})\in E_1\times E_2:...
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### $\mathbb{Z}[T]$-module and extension

I consider the structure of $\mathbb{Z}[T]$-module on $\mathbb{Z}$ given by the multiplication $P \times a := P(0) \times a$. Now i consider a morphism of ring $\phi : \mathbb{Z}[T] \to R$, that ...
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### Isomorphism concerning tensor product

While reading algebraic number theory, I came across the following fact which I don't know how to prove. Let $A$ be an integral domain with field of fraction $K$ and $L/K$ is a finite separable ...
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### Derivative of Frobenius norm of tensor product in component-free notation

I need to take a derivate with respect to $w$ of the following function: $$f(x,w)=\left\lVert x \otimes w\right\rVert _{F},$$ where $x \in \mathbb{R^{n}}$, $w \in \mathbb{R^{n}}$, $\otimes$ is a ...
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### $\mathbb{Z}[G^{n+1}] \otimes_{\mathbb{Z}[G]} \mathbb{Z} \cong \mathbb{Z}[G^n]$ as $\mathbb{Z}$-module

Let $G$ be a group. If $M$ is any right $G$-module, then we can consider $M$ as left $G$-module also under the action $g.m:= mg^{-1}$, where $m \in M$ and $g \in G$. Consider $\mathbb{Z}[G^{n+1}]$ as ...
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### What is the submodule of $M \otimes_R N$ generated by $1 \otimes_R n$?

For some context, I am trying to prove that extension and restriction of scalars are an adjoint pair. Formally, given a commutative ring map $f: A \rightarrow B$ and $M$ an $A$-module, $N$ a $B$-...
Consider two vectors such as $$v_1=(a,b,c)~~~,~~~v_2=(d,e,f)$$ I am wondering how many different useful ways to take a tensor product of these two vectors exist such that a matrix is produced? For ...