The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.162.1. Let $R \to S$ be a ring map. Let $M$ be an $S$-module. Assume that

  1. $R \to S$ is of finite presentation,

  2. $M$ is a finitely presented $S$-module, and

  3. $M$ is flat over $R$.

In this case we have the following:

  1. There exists a finite type $\mathbf{Z}$-algebra $R_0$ and a finite type ring map $R_0 \to S_0$ and a finite $S_0$-module $M_0$ such that $M_0$ is flat over $R_0$, together with a ring maps $R_0 \to R$ and $S_0 \to S$ and an $S_0$-module map $M_0 \to M$ such that $S \cong R \otimes _{R_0} S_0$ and $M = S \otimes _{S_0} M_0$.

  2. If $R = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } R_\lambda $ is written as a directed colimit, then there exists a $\lambda $ and a ring map $R_\lambda \to S_\lambda $ of finite presentation, and an $S_\lambda $-module $M_\lambda $ of finite presentation such that $M_\lambda $ is flat over $R_\lambda $ and such that $S = R \otimes _{R_\lambda } S_\lambda $ and $M = S \otimes _{S_{\lambda }} M_\lambda $.

  3. If

    \[ (R \to S, M) = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } (R_\lambda \to S_\lambda , M_\lambda ) \]

    is written as a directed colimit such that

    1. $R_\mu \otimes _{R_\lambda } S_\lambda \to S_\mu $ and $S_\mu \otimes _{S_\lambda } M_\lambda \to M_\mu $ are isomorphisms for $\mu \geq \lambda $,

    2. $R_\lambda \to S_\lambda $ is of finite presentation,

    3. $M_\lambda $ is a finitely presented $S_\lambda $-module,

    then for all sufficiently large $\lambda $ the module $M_\lambda $ is flat over $R_\lambda $.

Proof. We first write $(R \to S, M)$ as the directed colimit of a system $(R_\lambda \to S_\lambda , M_\lambda )$ as in as in Lemma 10.126.18. Let $\mathfrak q \subset S$ be a prime. Let $\mathfrak p \subset R$, $\mathfrak q_\lambda \subset S_\lambda $, and $\mathfrak p_\lambda \subset R_\lambda $ the corresponding primes. As seen in the proof of Theorem 10.128.4

\[ ((R_\lambda )_{\mathfrak p_\lambda }, (S_\lambda )_{\mathfrak q_\lambda }, (M_\lambda )_{\mathfrak q_{\lambda }}) \]

is a system as in Lemma 10.126.13, and hence by Lemma 10.127.3 we see that for some $\lambda _{\mathfrak q} \in \Lambda $ for all $\lambda \geq \lambda _{\mathfrak q}$ the module $M_\lambda $ is flat over $R_\lambda $ at the prime $\mathfrak q_{\lambda }$.

By Theorem 10.128.4 we get an open subset $U_\lambda \subset \mathop{\mathrm{Spec}}(S_\lambda )$ such that $M_\lambda $ flat over $R_\lambda $ at all the primes of $U_\lambda $. Denote $V_\lambda \subset \mathop{\mathrm{Spec}}(S)$ the inverse image of $U_\lambda $ under the map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(S_\lambda )$. The argument above shows that for every $\mathfrak q \in \mathop{\mathrm{Spec}}(S)$ there exists a $\lambda _{\mathfrak q}$ such that $\mathfrak q \in V_\lambda $ for all $\lambda \geq \lambda _{\mathfrak q}$. Since $\mathop{\mathrm{Spec}}(S)$ is quasi-compact we see this implies there exists a single $\lambda _0 \in \Lambda $ such that $V_{\lambda _0} = \mathop{\mathrm{Spec}}(S)$.

The complement $\mathop{\mathrm{Spec}}(S_{\lambda _0}) \setminus U_{\lambda _0}$ is $V(I)$ for some ideal $I \subset S_{\lambda _0}$. As $V_{\lambda _0} = \mathop{\mathrm{Spec}}(S)$ we see that $IS = S$. Choose $f_1, \ldots , f_ r \in I$ and $s_1, \ldots , s_ n \in S$ such that $\sum f_ i s_ i = 1$. Since $\mathop{\mathrm{colim}}\nolimits S_\lambda = S$, after increasing $\lambda _0$ we may assume there exist $s_{i, \lambda _0} \in S_{\lambda _0}$ such that $\sum f_ i s_{i, \lambda _0} = 1$. Hence for this $\lambda _0$ we have $U_{\lambda _0} = \mathop{\mathrm{Spec}}(S_{\lambda _0})$. This proves (1).

Proof of (2). Let $(R_0 \to S_0, M_0)$ be as in (1) and suppose that $R = \mathop{\mathrm{colim}}\nolimits R_\lambda $. Since $R_0$ is a finite type $\mathbf{Z}$ algebra, there exists a $\lambda $ and a map $R_0 \to R_\lambda $ such that $R_0 \to R_\lambda \to R$ is the given map $R_0 \to R$ (see Lemma 10.126.3). Then, part (2) follows by taking $S_\lambda = R_\lambda \otimes _{R_0} S_0$ and $M_\lambda = S_\lambda \otimes _{S_0} M_0$.

Finally, we come to the proof of (3). Let $(R_\lambda \to S_\lambda , M_\lambda )$ be as in (3). Choose $(R_0 \to S_0, M_0)$ and $R_0 \to R$ as in (1). As in the proof of (2), there exists a $\lambda _0$ and a ring map $R_0 \to R_{\lambda _0}$ such that $R_0 \to R_{\lambda _0} \to R$ is the given map $R_0 \to R$. Since $S_0$ is of finite presentation over $R_0$ and since $S = \mathop{\mathrm{colim}}\nolimits S_\lambda $ we see that for some $\lambda _1 \geq \lambda _0$ we get an $R_0$-algebra map $S_0 \to S_{\lambda _1}$ such that the composition $S_0 \to S_{\lambda _1} \to S$ is the given map $S_0 \to S$ (see Lemma 10.126.3). For all $\lambda \geq \lambda _1$ this gives maps

\[ \Psi _{\lambda } : R_\lambda \otimes _{R_0} S_0 \longrightarrow R_\lambda \otimes _{R_{\lambda _1}} S_{\lambda _1} \cong S_\lambda \]

the last isomorphism by assumption. By construction $\mathop{\mathrm{colim}}\nolimits _\lambda \Psi _\lambda $ is an isomorphism. Hence $\Psi _\lambda $ is an isomorphism for all $\lambda $ large enough by Lemma 10.126.8. In the same vein, there exists a $\lambda _2 \geq \lambda _1$ and an $S_0$-module map $M_0 \to M_{\lambda _2}$ such that $M_0 \to M_{\lambda _2} \to M$ is the given map $M_0 \to M$ (see Lemma 10.126.5). For $\lambda \geq \lambda _2$ there is an induced map

\[ S_\lambda \otimes _{S_0} M_0 \longrightarrow S_\lambda \otimes _{S_{\lambda _2}} M_{\lambda _2} \cong M_\lambda \]

and for $\lambda $ large enough this map is an isomorphism by Lemma 10.126.6. This implies (3) because $M_0$ is flat over $R_0$. $\square$


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