0

I have a little question about of the associativity of the tensor products of two modules.

If $M, N, P$ are $A$-modules, we know that $(M \otimes N) \ P \cong M \otimes N \otimes P$ If i want to prove that, i have to find a bilinear-map from $(M \otimes N) \times P \to M \otimes N \otimes P$, but in Atiyah-Macdonald proposition 2.14, they first induce an homomorphism $f_z: M \otimes N \to M \otimes N \otimes P$.

My question is, why is this step necessary? Why not define the bilinear function $f$ from $ (M \otimes N) \times P \to M \otimes N \otimes P$ and then use the universal property?

I appreciate your help.

Samsara814
  • 1
  • 2
  • The delimiters of MathJax (LaTeX) are not backticks or French accents graves, it's dollarsigns :-) – Dominique Feb 22 '24 at 12:39
  • 2
    Well, how will you define this bilinear function $f$ on the first coordinate? What will be $f(m\otimes n, 0)$ for example? You will need to show that this is well defined, and for that you need the universal property of $M\otimes_R N$. I mean, it makes sense to define $f$ by sending $(m\otimes n, p)\to m\otimes n\otimes p$, but you need to verify this is indeed a well defined map. – Mark Feb 22 '24 at 12:42
  • ¿What is the definition of well defined function in this context? ¿ $f(m \otimes n, 0)$ cannot be equal to $m \otimes n \otimes 0$ – Samsara814 Feb 22 '24 at 12:52
  • I get it. Thank you all. – Samsara814 Feb 22 '24 at 20:15

0 Answers0