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Tag 01U5

Chapter 28: Morphisms of Schemes > Section 28.24: Flat morphisms

Lemma 28.24.3. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. The morphism $f$ is flat.
  2. For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is flat.
  3. There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is flat.
  4. There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is flat, for all $j\in J, i\in I_j$.

Moreover, if $f$ is flat then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is flat.

Proof. This is a special case of Lemma 28.24.2 above. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4251–4272 (see updates for more information).

    \begin{lemma}
    \label{lemma-flat-characterize}
    Let $f : X \to S$ be a morphism of schemes.
    The following are equivalent
    \begin{enumerate}
    \item The morphism $f$ is flat.
    \item For every affine opens $U \subset X$, $V \subset S$
    with $f(U) \subset V$ the ring map
    $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is flat.
    \item There exists an open covering $S = \bigcup_{j \in J} V_j$
    and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such
    that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$
    is flat.
    \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$
    and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such
    that $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is flat, for all
    $j\in J, i\in I_j$.
    \end{enumerate}
    Moreover, if $f$ is flat then for
    any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$
    the restriction $f|_U : U \to V$ is flat.
    \end{lemma}
    
    \begin{proof}
    This is a special case of Lemma \ref{lemma-flat-module-characterize}
    above.
    \end{proof}

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