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Tag 08BQ

Chapter 20: Cohomology of Sheaves > Section 20.27: Flat resolutions

Tor measures the deviation of flatness.

Lemma 20.27.15. Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. The following are equivalent

  1. $\mathcal{F}$ is a flat $\mathcal{O}_X$-module, and
  2. $\text{Tor}_1^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) = 0$ for every $\mathcal{O}_X$-module $\mathcal{G}$.

Proof. If $\mathcal{F}$ is flat, then $\mathcal{F} \otimes_{\mathcal{O}_X} -$ is an exact functor and the satellites vanish. Conversely assume (2) holds. Then if $\mathcal{G} \to \mathcal{H}$ is injective with cokernel $\mathcal{Q}$, the long exact sequence of $\text{Tor}$ shows that the kernel of $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{H}$ is a quotient of $\text{Tor}_1^{\mathcal{O}_X}(\mathcal{F}, \mathcal{Q})$ which is zero by assumption. Hence $\mathcal{F}$ is flat. $\square$

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 5959–5972 (see updates for more information).

    \begin{lemma}
    \label{lemma-flat-tor-zero}
    \begin{slogan}
    Tor measures the deviation of flatness.
    \end{slogan}
    Let $(X, \mathcal{O}_X)$ be a ringed space.
    Let $\mathcal{F}$ be an $\mathcal{O}_X$-module.
    The following are equivalent
    \begin{enumerate}
    \item $\mathcal{F}$ is a flat $\mathcal{O}_X$-module, and
    \item $\text{Tor}_1^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) = 0$
    for every $\mathcal{O}_X$-module $\mathcal{G}$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    If $\mathcal{F}$ is flat, then $\mathcal{F} \otimes_{\mathcal{O}_X} -$
    is an exact functor and the satellites vanish. Conversely assume (2)
    holds. Then if $\mathcal{G} \to \mathcal{H}$ is injective with cokernel
    $\mathcal{Q}$, the long exact sequence of $\text{Tor}$ shows that
    the kernel of
    $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G} \to
    \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{H}$
    is a quotient of
    $\text{Tor}_1^{\mathcal{O}_X}(\mathcal{F}, \mathcal{Q})$
    which is zero by assumption. Hence $\mathcal{F}$ is flat.
    \end{proof}

    Comments (1)

    Comment #2593 by Rogier Brussee on June 4, 2017 a 7:23 pm UTC

    Suggested slogan: Tor measures the deviation of flatness

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