# Tagged Questions

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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### Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
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### Exterior algebra of a ring

In the book "Cohen-Macaulay rings" by Bruns and Herzog, the quick introduction of tensor algebra and exterior algebra left me a bit bewildered. After referring to the section on tensor algebra ...
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### Balanced Tensor Product of Module Categories

Let $C$ be a $k$-linear ($Vect_k$-enriched) monoidal category and consider the 2-category $Mod_{C}$ of $k$-linear $(C,C)$-bimodule categories in the sense of Ostrik (https://arxiv.org/abs/math/0111139....
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### Let $V$ and $W$ be vector spaces over $\mathbb{C}$. Show that $0\otimes w = v\otimes 0 = 0 \in V \otimes W$.

Algebraically, the vector space $V \otimes W$ is spanned by elements of the form $v \otimes w$, and the following rules are satisfied, for any scalar $c$. The definition is the same no matter which ...
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### Dual space of polynomial algebra

Let $k$ be an infinite field and let's consider the ring $R=k[x_1,\dots,x_n]$. This ring has a structure of $k$-vector space (or a $k$-algebra). I am interested to know about the structure of the ...
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### Tensor product $Z[1/2]\otimes Z/3$

Compute $\mathbb{Z}[1/2]\otimes_{\mathbb{Z}} \mathbb{Z}/3$ My initial instinct was that it is equal to $\mathbb{Z}/3[1/2]$, but I couldn't show this is correct by the universal propert. Hence, I ...
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### how the Kronecker product is a tensor product?

I have seen this question but still don't understand how the Kronecker product is a tensor product. If $T_1 : V_1 \to W_1$ and $T_2 :V_2 \to W_2$ are linear transformations their matrix ...
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### linear decompositions of tensor product vectors

I have a vector space $V^{(1)} \otimes V^{(2)}$, a set of vectors $\{s_i\} \in V^{(1)}$ and $\{s_j\} \in V^{(2)}$, that span the respective spaces such that any one vector, e.g. $s_{i'}$, can be ...
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### Hausdorffness of projective tensor product of locally convex space

I do not see why the following is true: Given two locally convex t.v.s. E and F, if the projective tensor product is Hausdorff then both E and F are Hausdorff. The converse holds and the proof is ...
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### Show that $M\otimes N$ is isomorphic to $N\otimes M$

I want to prove the following: Let $A$ be a ring and $M,N$ be $A$-modules. Show that the tensor products $M\otimes N$ and $N\otimes M$ are isomorphic. I have consulted this question but it did ...
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### How to represent matrix multiplication in tensor algebra?

How can we represent matrix multiplication in tensor algebra? Even if we assume all matrices represent contravariant tensors only, clearly matrix multiplication does not correspond to the ...