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Tag 052U

Lemma 10.107.6. Let $R$ be a ring. The following are equivalent:

1. every $Z \subset \mathop{\rm Spec}(R)$ which is closed and closed under generalizations is also open, and
2. any finite flat $R$-module is finite locally free.

Proof. If any finite flat $R$-module is finite locally free then the support of $R/I$ where $I$ is a pure ideal is open. Hence the implication (2) $\Rightarrow$ (1) follows from Lemma 10.107.3.

For the converse assume that $R$ satisfies (1). Let $M$ be a finite flat $R$-module. The support $Z = \text{Supp}(M)$ of $M$ is closed, see Lemma 10.39.5. On the other hand, if $\mathfrak p \subset \mathfrak p'$, then by Lemma 10.77.4 the module $M_{\mathfrak p'}$ is free, and $M_{\mathfrak p} = M_{\mathfrak p'} \otimes_{R_{\mathfrak p'}} R_{\mathfrak p}$ Hence $\mathfrak p' \in \text{Supp}(M) \Rightarrow \mathfrak p \in \text{Supp}(M)$, in other words, the support is closed under generalization. As $R$ satisfies (1) we see that the support of $M$ is open and closed. Suppose that $M$ is generated by $r$ elements $m_1, \ldots, m_r$. The modules $\wedge^i(M)$, $i = 1, \ldots, r$ are finite flat $R$-modules also, because $\wedge^i(M)_{\mathfrak p} = \wedge^i(M_{\mathfrak p})$ is free over $R_{\mathfrak p}$. Note that $\text{Supp}(\wedge^{i + 1}(M)) \subset \text{Supp}(\wedge^i(M))$. Thus we see that there exists a decomposition $$\mathop{\rm Spec}(R) = U_0 \amalg U_1 \amalg \ldots \amalg U_r$$ by open and closed subsets such that the support of $\wedge^i(M)$ is $U_r \cup \ldots \cup U_i$ for all $i = 0, \ldots, r$. Let $\mathfrak p$ be a prime of $R$, and say $\mathfrak p \in U_i$. Note that $\wedge^i(M) \otimes_R \kappa(\mathfrak p) = \wedge^i(M \otimes_R \kappa(\mathfrak p))$. Hence, after possibly renumbering $m_1, \ldots, m_r$ we may assume that $m_1, \ldots, m_i$ generate $M \otimes_R \kappa(\mathfrak p)$. By Nakayama's Lemma 10.19.1 we get a surjection $$R_f^{\oplus i} \longrightarrow M_f, \quad (a_1, \ldots, a_i) \longmapsto \sum a_im_i$$ for some $f \in R$, $f \not \in \mathfrak p$. We may also assume that $D(f) \subset U_i$. This means that $\wedge^i(M_f) = \wedge^i(M)_f$ is a flat $R_f$ module whose support is all of $\mathop{\rm Spec}(R_f)$. By the above it is generated by a single element, namely $m_1 \wedge \ldots \wedge m_i$. Hence $\wedge^i(M)_f \cong R_f/J$ for some pure ideal $J \subset R_f$ with $V(J) = \mathop{\rm Spec}(R_f)$. Clearly this means that $J = (0)$, see Lemma 10.107.3. Thus $m_1 \wedge \ldots \wedge m_i$ is a basis for $\wedge^i(M_f)$ and it follows that the displayed map is injective as well as surjective. This proves that $M$ is finite locally free as desired. $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 25045–25053 (see updates for more information).

\begin{lemma}
\label{lemma-finite-flat-module-finitely-presented}
Let $R$ be a ring. The following are equivalent:
\begin{enumerate}
\item every $Z \subset \Spec(R)$ which is closed and closed under
generalizations is also open, and
\item any finite flat $R$-module is finite locally free.
\end{enumerate}
\end{lemma}

\begin{proof}
If any finite flat $R$-module is finite locally free then the support
of $R/I$ where $I$ is a pure ideal is open. Hence the implication
(2) $\Rightarrow$ (1) follows from
Lemma \ref{lemma-pure-ideal-determined-by-zero-set}.

\medskip\noindent
For the converse assume that $R$ satisfies (1).
Let $M$ be a finite flat $R$-module.
The support $Z = \text{Supp}(M)$ of $M$ is closed, see
Lemma \ref{lemma-support-closed}.
On the other hand, if $\mathfrak p \subset \mathfrak p'$, then by
Lemma \ref{lemma-finite-flat-local}
the module $M_{\mathfrak p'}$ is free, and
$M_{\mathfrak p} = M_{\mathfrak p'} \otimes_{R_{\mathfrak p'}} R_{\mathfrak p}$
Hence
$\mathfrak p' \in \text{Supp}(M) \Rightarrow \mathfrak p \in \text{Supp}(M)$,
in other words, the support is closed under generalization.
As $R$ satisfies (1) we see that the support of $M$ is open and closed.
Suppose that $M$ is generated by $r$ elements $m_1, \ldots, m_r$.
The modules $\wedge^i(M)$, $i = 1, \ldots, r$ are finite flat $R$-modules
also, because $\wedge^i(M)_{\mathfrak p} = \wedge^i(M_{\mathfrak p})$
is free over $R_{\mathfrak p}$. Note that
$\text{Supp}(\wedge^{i + 1}(M)) \subset \text{Supp}(\wedge^i(M))$.
Thus we see that there exists a decomposition
$$\Spec(R) = U_0 \amalg U_1 \amalg \ldots \amalg U_r$$
by open and closed subsets such that the support of
$\wedge^i(M)$ is $U_r \cup \ldots \cup U_i$ for all $i = 0, \ldots, r$.
Let $\mathfrak p$ be a prime of $R$, and say $\mathfrak p \in U_i$.
Note that
$\wedge^i(M) \otimes_R \kappa(\mathfrak p) = \wedge^i(M \otimes_R \kappa(\mathfrak p))$.
Hence, after possibly renumbering $m_1, \ldots, m_r$ we may assume that
$m_1, \ldots, m_i$ generate $M \otimes_R \kappa(\mathfrak p)$. By
Nakayama's Lemma \ref{lemma-NAK}
we get a surjection
$$R_f^{\oplus i} \longrightarrow M_f, \quad (a_1, \ldots, a_i) \longmapsto \sum a_im_i$$
for some $f \in R$, $f \not \in \mathfrak p$. We may also assume that
$D(f) \subset U_i$. This means that $\wedge^i(M_f) = \wedge^i(M)_f$
is a flat $R_f$ module whose support is all of $\Spec(R_f)$.
By the above it is generated by a single element, namely
$m_1 \wedge \ldots \wedge m_i$. Hence $\wedge^i(M)_f \cong R_f/J$
for some pure ideal $J \subset R_f$ with $V(J) = \Spec(R_f)$.
Clearly this means that $J = (0)$, see
Lemma \ref{lemma-pure-ideal-determined-by-zero-set}.
Thus $m_1 \wedge \ldots \wedge m_i$ is a basis for
$\wedge^i(M_f)$ and it follows that the displayed map is injective
as well as surjective. This proves that $M$ is finite locally free
as desired.
\end{proof}

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