# 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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### What is the empty tensor product of vector spaces?

The tensor product of a space with itself once is $V^{\otimes1}$, but what is $V^{\otimes0}$? Since it is an empty tensor product, it is - a fortiori - an empty product. So I'm looking for a "$1$&...
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### What's the definition of dual number at perspect of exterior algebra?

In Dual Number it said that "It may also be defined as the exterior algebra of a one-dimensional vector space with $\varepsilon$ as its basis element." But I can't find the detailed rigorous ...
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### Confusion Over Distributive Property in Tensor and External Tensor Products

I've been delving into the properties of tensor ($\otimes$) and external tensor products ($\boxtimes$) within the context of coalgebra, particularly examining how the coproduct $\Delta$ applies to ...
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### Module of type $FP_n$

I'm trying to understand the converse of theorem 1.3 in Robert's Bieri Homological dimension of discrete groups which says that a $\Lambda$-module $A$ is of type $FP_n$ if and only if for every direct ...
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### Comultiplication on the tensor algebra

Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
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### What are the names of the conventions for defining the double dot product?

The double dot product of two matrices $A : B$ can be defined as either: $A : B = Tr(AB^T) = A_{ij}B_{ij}$ $A : B = Tr(AB) = A_{ij}B_{ji}$. I've seen the first convention called Frobenius or ...
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### Induced change of basis on a (p,q) tensor

I'm struggling to simplify the last step of a $(p,q)$ tensor and how its components change with a linear change of basis on the associated vector space. So far I have: Given a vector space $V$ over ...
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### Proving a criterion for flatness of modules

I am following Qing Liu's textbook "Algebraic Geometry and Arithmetic Curves," and have come upon the following statement (the truth of which is well-known): Theorem: Let $M$ be an $A$-...
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### $\sigma$-weak continuity of $x \mapsto x \otimes 1$ from $B(H)$ to $B(H \otimes H)$

Let $H$ be a separable Hilbert space. Write $B(H)$ for the set of linear bounded operator on $H$ and $H \otimes H$ the tensor product of Hilbert space. For every $A,B \in B(H)$ we can define an ...
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### Is the infinite tensor product of flat modules still flat? [closed]

Suppose $(M_i)_{i \in I}$ is a collection of flat $A$-modules ($A$ is a commutative ring with $1$). Is the tensor product $\bigotimes_i M_i$ still flat? This is obviously true by induction when $I$ is ...
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### The projective tensor norm on tensor product of Banach spaces implies the inner product on tensor product of Hilbert spaces?

As presented in the answer of this post, the projective tensor norm on the algebraic tensor product of two Banach spaces $X$ and $Y$ is given by \[ \Vert \omega\Vert_{\pi} = \inf\left\{\sum \lVert x_{...
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### Double dual and tensor product for infinite dimensional spaces

It is well known that finite dimensional vector spaces (over any field $K$) canonically satisfy $V''\otimes W''\simeq (V\otimes W)''$. My question is, roughly, if this canonical map can somehow be ...
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