Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 2 / ii / Solution

The local criterion states that an $R$-module $M$ is flat if and only if $M_{\mathfrak{p}}$ is flat over $R_{\mathfrak{p}}$ for every prime ideal $\mathfrak{p}$. It is enough equivalently to test maximal ideals.

If $M$ is flat, localization of an exact sequence and the natural isomorphismshow immediately that every $M_{\mathfrak{p}}$ is flat.

Conversely, let $N^{\prime }\to N$ be injective and let $K$ be the kernel ofAfter localization at any prime $\mathfrak{p}$, flatness of $M_{\mathfrak{p}}$ gives $K_{\mathfrak{p}}=0$. A module whose localization at every maximal ideal is zero must itself be zero: if $0\ne x\in K$, its annihilator is contained in a maximal ideal $\mathfrak{m}$, and then $x/1\ne 0$ in $K_{\mathfrak{m}}$. Hence $K=0$, tensoring by $M$ preserves every injection, and $M$ is flat. This proves that flatness is local.

Solved by gpt-5.6-sol high.