Freeness of stalk Implies locally free

Question

Let $ A $ be a Noetherian ring, and $ M $ a finitely generated $ A $ module. Suppose that $ \mathfrak { p } \in M $ such that $ M_{\mathfrak{p}} $ is free. Show that there is a $ f \in A \setminus \mathfrak{p} $ such that $ M_{f} $ is free over $ A_{f} $.

P.S. Some related questions are 1) Flatness and Local Freeness 2) Locally free sheaves, though, both of these don't answer the specific question that I have above, in that I am just looking around one prime (so my module is not projective etc). I have seen this in Vakil but I can't find it at the moment. I will post my proof of the fact above below but I would like to see what are some other ways to do it.

Answer 1

Suppose that $M $ is generated by $ b_{1}, \ldots, b_{k} $ over $ A $, and $ M_{\mathfrak{p}} $ has a basis given by $ \beta_{1}, \ldots, \beta _{ n} \in M_{ \mathfrak{p}} $. Let $ \beta_{i} = m_{i} / s_{i} $ for $ m_{i} \in M $, $ s_{i} \in A \setminus \mathfrak{p} $. There exist $ a_{ij} \in A $, $ t_{ij} \in A \setminus \mathfrak{p} $ for $ 1 \leq i \leq n $, $ 1 \leq j \leq k $ such that $$ b_{j} = \sum_{ i = 1 } ^ { n } \frac{a_{ij} } { t_{ij} } \frac { m_{i} } { s_{i } } $$ because $ \beta_{i} $ form a basis. Let $ g = \prod _{ i, j } t_{ij} \cdot \prod _ { i } s_{i} $ Consider the sequence $$ 0 \to I \to A _{g} ^ { n } \xrightarrow { \varphi } M_{g} \to 0 $$ where the map $ A_{g}^{n} \to M_{g} $ sends the the $ e_{i} $ to $ m_{i} $. Localizing this sequence at $ \mathfrak { p } $ kills $ I $, and since $I $ is finitely generated, one can localize at an element $ h $ such that $ I_{h} = 0 $. The element $ f = gh $ then works.

Answer 2

$\newcommand{\gp}{{\mathfrak p}}$Let $\left\{ \frac{m_1}{s_1}, \cdots, \frac{m_n}{s_n} \right\}$ be an $A_\gp$-basis of $M_\gp$, where $m_i \in M$ and $s_i \in A-\gp$. Observe that $\{m_1, \cdots, m_n\}$ is still a $A_\gp$-basis of $M_\gp$.

This gives an map $A^n \to M$ that localizes to an isomorphism $A_\gp^n \to M_\gp$ at $\gp$. Let $K$ and $Q$ be the kernel and cokernel respectively of the map $A^n \to M$. This gives an exact sequence: $$0 \to K \to A^n \to M \to Q \to 0$$

Localizing is exact, so we have an exact sequence: $$0 \to K_\gp \to A_\gp^n \to M_\gp \to Q_\gp \to 0$$

But we know that the map $A_\gp^n \to M_\gp$ is an isomorphism, so we must have $K_\gp = 0$ and $Q_\gp = 0$.

$A$ is noetherian, and $K$ is a submodule of $A^n$, so $K$ is finitely generated, so the fact that $K_\gp = 0$ implies that there is $s \in A-\gp$ such that $K_s = 0$; $M$ is finitely generated, and $Q$ is a quotient of $M$, so $Q$ is finitely generated, so the fact that $Q_\gp = 0$ implies that there is $t \in A-\gp$ such that $Q_t = 0$.

Now take $f = st \in A-\gp$. Localize the exact sequence at $f$ to get: $$0 \to K_f \to A_f^n \to M_f \to Q_f \to 0$$

But $K_f = (K_s)_t = 0_t = 0$ and $Q_f = (Q_t)_s = 0_s = 0$, so the exact sequence becomes: $$0 \to 0 \to A_f^n \to M_f \to 0 \to 0$$

So $M_f$ is free over $A_f$ as required.

Answer 3

$\def\p{\mathfrak{p}} \def\coker{\operatorname{Coker}} \def\ker{\operatorname{Ker}} \def\supp{\operatorname{Supp}} \def\Spec{\operatorname{Spec}}$We don't need Noetherian hypotheses. Instead, we just require the module to be finitely presented (over a Noetherian ring this is equivalent to being finitely generated).

Lemma. Let $A$ be a ring (not necessarily Noetherian). Suppose $M$ is a finitely presented $A$-module such that $M_\mathfrak{p}$ is $A_\mathfrak{p}$-free with basis $\mathcal{B}\subset M_\mathfrak{p}$. Then there is $f\in A\setminus\mathfrak{p}$ such that $\mathcal{B}$ lifts to an $A[f^{-1}]$-basis of $M[f^{-1}]$.

Proof. Pick an isomorphism \begin{align*} R_\p^n&\to M_\p\\ e_i&\mapsto x_i/s_i \end{align*} Without loss of generality, all $s_i=1$ (replace $A$ and $M$ respectively by $A[s^{-1}]$ and $M[s^{-1}]$, where $s=s_1\cdots s_n$). Define a module homomorphism \begin{align*} \varphi:A^n&\mapsto M\\ e_i&\mapsto x_i \end{align*} Then $C=\coker\varphi$ is $A$-finite (it is a quotient of an $A$-finite module). Thus $\supp C$ is closed in $X=\Spec A$ [00L2]. Since $\p\not\in\supp C$, there is $g\in A$ with $\p\in D(g)\subset X\setminus\supp C$. Replacing $\varphi$ by $\varphi[g^{-1}]$, we may assume $\varphi$ is onto. Hence, from this result we obtain that $K=\ker\varphi$ is $A$-finite [00L2]. Since $\supp K\subset X$ is closed and $\p\not\in\supp K$, there is $h\in A$ with $\p\in D(h)\subset X\setminus\supp K$. Thus $\varphi[h^{-1}]$ has zero kernel and zero cokernel, so it is an iso.

Taking all subsequent localizations into account, $f=sgh$ satisfies the claim from the statement. $\square$