Localization commutes with tensor products
= Localization commutes with tensor products
{title2=$T^{-1}(M\otimes_AN)\cong T^{-1}M\otimes_{T^{-1}A}T^{-1}N$}
For a multiplicatively closed set $T$ in a commutative ring $A$, the map sends $(m\otimes n)/t$ to $(m/t)\otimes(n/1)$. An inverse sends $(m/s)\otimes(n/t)$ to $(m\otimes n)/(st)$. The fraction relations and the <tensor product of modules> balancing relation make these maps well-defined and mutually inverse.