# Direct Limit and Tensor Product

Let $I$ be a directed order set. And let $\{ (M_i,\mu_{ij}) | i\leq j\}$ be a directed system of modules with $M = \lim M_i$ the direct limit. With maps $\mu_i :M_i\to M$ satisfying the required properties.

Fix a module $N$. We also have that the system $(M_i\times N),\mu_{ij}\times 1$ is directed and furthermore $\lim (M_i\times N) = M\times N$ with the limiting maps $M_i\times N\to M\times N$ given by $\mu_i\times 1$.

The system $(M_i\otimes N,\mu_{ij}\otimes 1)$ is also directed. Let $P = \lim (M_i\otimes N)$ (do not assume that $P=M\otimes N$, because I am trying to prove this). Passing to the limit we have a map of module $g:M\times N\to P$.

My book says to prove that $g$ is bilinear. This makes no sense. How can it both be linear and bilinear (unless it is trivial map)?

• Um, what's with the algebra-precalculus tag? – Matt Samuel May 8 '15 at 19:14
• Well I didn't learn it until graduate school. Guess that means I'm slow, or something. – Matt Samuel May 8 '15 at 19:17

$g$ is not assumed to be linear. The purpose is to eventually show that $g:M\times N\to P$ is the universal bilinear map characterizing $P$ as the tensor product of $M$ and $N$.
• Let $M_i$ and $N_i$ be modules over the same directed set. If we have a family of maps $g_i:M_i\to N_i$ which makes the diagrams commute then we have a map $g:M\to N$, the direct limit, which is a homomorphism of modules and makes the appropriate diagrams commute. – Nicolas Bourbaki May 8 '15 at 19:23
• What are your maps? Are they homomorphisms of modules or are they bilinear? I've never heard of using a homomorphism of modules $M\times N\to M\otimes N$. – Matt Samuel May 8 '15 at 19:25
• Sorry I don't. But if you have maps $M_i\times N\to M_i\otimes N$, I'd bet a lot of money that they're supposed to be bilinear, not homomorphisms of modules. – Matt Samuel May 8 '15 at 19:29
• Let $(M_i)$ be a directed system of modules. If we multiply through by $N$, we get a directed system $M_i\times N$. The maps within this new directed system are not bilinear, they are linear maps. – Nicolas Bourbaki May 8 '15 at 20:13