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I found it on page 31 of Atiyah's Commutative Algebra. It says:

If $M_k$, $N_k$ are vector spaces over a field, then $M_k \otimes N_k = 0$ implies $M_k=0$ or $N_k=0$.

Where $A$ is a local ring, $k$ is the residue field $A/\frak m$, $M,N$ are two $A$-modules, $M_k=k\otimes M,N_k=k\otimes N$.

I did not learn much linear algebra, and Atiyah did not give any further explanation.. So I'm stuck here until I found this: Tensor product of two vector spaces .

It seems to solve my problem, but I'm starting to have more questions about ternsor product:

  1. What is tensor product? I just know it is a multi-linear mapping of some modules, is there any geometric explanation?

  2. In linear algebra, i.e., module over field, how to understand tensor product? and the example above ($M_k \otimes N_k = 0 \implies M_k=0$ or $N_k=0$)?

Youxing
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  • Your questions needs more focus. If you want to know what a tensor product is, read the definition, e.g., in wikipedia. – Dietrich Burde Jun 03 '22 at 13:18
  • @DietrichBurde I know what the definition is, but I want a geometric view. I know in differential geometry, they use tensor to describe something, so I hope someone can give a different view about it, not just the definition. – Youxing Jun 03 '22 at 13:27
  • For the geometric intuition see for example this post and many other posts here. – Dietrich Burde Jun 03 '22 at 15:25
  • @DietrichBurde Thank you for your comment! I found this https://math.stackexchange.com/a/126936/1048243 in the link you just gave. Does that mean I can imagine the tensor as equivalence of a kind of 'area'? – Youxing Jun 03 '22 at 15:48

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