# Proof of Prop 1.5.d. in Qing Liu Algebraic Geometry and Arithmetic Curves

I am reading through this book and the following proof for the distributivity of the tensor product has me confused.

Let $$A$$ be a ring and let $$M_i,N$$ be $$A$$-modules. Then we have a canonical isomorphism $$\left( \oplus_{i \in I} M_i \right) \otimes_A N \simeq \oplus_{i \in I} \left( M_i \otimes_A N \right)$$

The proof:

Let $$\phi : \left( \oplus_{i \in I} M_i \right) \times N \to \oplus_{i \in I} \left( M_i \otimes_A N \right)$$ be the map defined by $$\phi : \left( \sum_i x_i,y\right) \mapsto \sum_i \left( x_i \otimes y \right)$$.

Let $$f : \left( \oplus_{i \in I} M_i \right) \times N \to L$$ be a bilinear map. For every $$i \in I$$, $$f$$ induces a bilinear map $$f_i : M_i \times N \to L$$ which factors through $$\tilde f_i : M_i \otimes_A N \to L$$. One verifies that $$f$$ factors uniquely as $$f = \tilde f \circ \psi$$ with $$\psi : \left( \oplus_{i \in I} M_i \right) \times N \to \left( \oplus_{i \in I} M_i \right) \otimes_A N$$ the canonical map and $$\tilde f = \oplus_i \tilde f_i$$.

Hence $$\oplus_{i \in I} \left( M_i \otimes_A N \right)$$ is the tensor product of $$\left( \oplus_{i \in I} M_i \right)$$ with $$N$$.

I am confused since the map $$\phi$$ defined at the start of the proof doesn't occur anywhere else. I also don't see how the Hence follows.

Please note I have seen a couple proofs of the statement before and am convinced of its validity. I am only interested in understanding Qing Liu's proof.

Can anyone help?

One verifies that $$f$$ factors uniquely as $$f = \tilde f \circ \psi$$ with $$\psi : \left( \oplus_{i \in I} M_i \right) \times N \to \left( \oplus_{i \in I} M_i \right) \otimes_A N$$ the canonical map and $$\tilde f = \oplus_i \tilde f_i$$.
One verifies that $$f$$ factors uniquely as $$f = \tilde f \circ \phi$$ with $$\tilde f = \oplus_i \tilde f_i$$. This shows that the pair $$\left( \bigoplus_{i\in I} \left( M_i \otimes N \right) , \phi \right)$$ satisfies the universal property of the tensor product of $$\bigoplus_{i \in I} M_i$$ and $$N$$. But there is only one pair (up to isomorphism) satisfying this property.
• Thanks. This is what I also suspected. It also makes sense since the $\tilde f_i$ come from $M_i \otimes N$, so the $\oplus \tilde f_i$ should come from $\oplus (M_i \otimes N)$. In Qing Liu, his $\psi$ doesn't seem to map here, so the whole thing becomes ugly. – Ruben Sep 27 '19 at 16:41