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Tag 0B44

36.43. Projective schemes

This section is the analogue of Section 36.42 for projective morphisms.

Lemma 36.43.1. Let $S$ be a scheme which has an ample invertible sheaf. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. $X \to S$ is projective,
  2. $X \to S$ is H-projective,
  3. $X \to S$ is quasi-projective and proper,
  4. $X \to S$ is H-quasi-projective and proper,
  5. $X \to S$ is proper and $X$ has an ample invertible sheaf,
  6. $X \to S$ is proper and there exists an $f$-ample invertible sheaf,
  7. $X \to S$ is proper and there exists an $f$-very ample invertible sheaf,
  8. there is a quasi-coherent graded $\mathcal{O}_S$-algebra $\mathcal{A}$ generated by $\mathcal{A}_1$ over $\mathcal{A}_0$ with $\mathcal{A}_1$ a finite type $\mathcal{O}_S$-module such that $X = \underline{\text{Proj}}_S(\mathcal{A})$.

Proof. Observe first that in each case the morphism $f$ is proper, see Morphisms, Lemmas 28.41.3 and 28.41.5. Hence it suffices to prove the equivalence of the notions in case $f$ is a proper morphism. We will use this without further mention in the following.

The equivalences (1) $\Leftrightarrow$ (3) and (2) $\Leftrightarrow$ (4) are Morphisms, Lemma 28.41.13.

The implication (2) $\Rightarrow$ (1) is Morphisms, Lemma 28.41.3.

The implications (1) $\Rightarrow$ (2) and (3) $\Rightarrow$ (4) are Morphisms, Lemma 28.41.16.

The implication (1) $\Rightarrow$ (7) is immediate from Morphisms, Definitions 28.41.1 and 28.36.1.

The conditions (3) and (6) are equivalent by Morphisms, Definition 28.38.1.

Thus (1) – (4), (6) are equivalent and imply (7). By Lemma 36.42.1 conditions (3), (5), and (7) are equivalent. Thus we see that (1) – (7) are equivalent.

By Divisors, Lemma 30.28.5 we see that (8) implies (1). Conversely, if (2) holds, then we can choose a closed immersion $$ i : X \longrightarrow \mathbf{P}^n_S = \underline{\text{Proj}}_S(\mathcal{O}_S[T_0, \ldots, T_n]). $$ See Constructions, Lemma 26.21.4 for the equality. By Divisors, Lemma 30.29.1 we see that $X$ is the relative Proj of a quasi-coherent graded quotient algebra $\mathcal{A}$ of $\mathcal{O}_S[T_0, \ldots, T_n]$. Then $\mathcal{A}$ satisfies the conditions of (8). $\square$

Lemma 36.43.2. Let $S$ be a scheme which has an ample invertible sheaf. Let $\text{P}_S$ be the full subcategory of the category of schemes over $S$ satisfying the equivalent conditions of Lemma 36.43.1.

  1. if $S' \to S$ is a morphism of schemes and $S'$ has an ample invertible sheaf, then base change determines a functor $\text{P}_S \to \text{P}_{S'}$,
  2. if $X \in \text{P}_S$ and $Y \in \text{P}_X$, then $Y \in \text{P}_S$,
  3. the category $\text{P}_S$ is closed under fibre products,
  4. the category $\text{P}_S$ is closed under finite disjoint unions,
  5. if $X \to S$ is finite, then $X$ is in $\text{P}_S$,
  6. add more here.

Proof. Part (1) follows from Morphisms, Lemma 28.41.9.

Part (2) follows from the fifth characterization of Lemma 36.43.1 and the fact that compositions of proper morphisms are proper (Morphisms, Lemma 28.39.4).

If $X \to S$ and $Y \to S$ are projective, then $X \times_S Y \to Y$ is projective by Morphisms, Lemma 28.41.9. Hence (3) follows from (2).

If $X = Y \amalg Z$ is a disjoint union of schemes and $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module such that $\mathcal{L}|_Y$ and $\mathcal{L}|_Z$ are ample, then $\mathcal{L}$ is ample (details omitted). Thus part (4) follows from the fifth characterization of Lemma 36.43.1.

Part (5) follows from Morphisms, Lemma 28.42.16. $\square$

Here is a slightly different type of result.

Lemma 36.43.3. Let $f : X \to Y$ be a proper morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $y \in Y$ be a point such that $\mathcal{L}_y$ is ample on $X_y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.

Proof. We may assume $Y$ is affine. Then we find a directed set $I$ and an inverse system of morphisms $X_i \to Y_i$ of schemes with $Y_i$ of finite type over $\mathbf{Z}$, with affine transition morphisms $X_i \to X_{i'}$ and $Y_i \to Y_{i'}$, with $X_i \to Y_i$ proper, such that $X \to Y = \mathop{\mathrm{lim}}\nolimits (X_i \to Y_i)$. See Limits, Lemma 31.13.3. After shrinking $I$ we can assume we have a compatible system of invertible $\mathcal{O}_{X_i}$-modules $\mathcal{L}_i$ pulling back to $\mathcal{L}$, see Limits, Lemma 31.10.3. Let $y_i \in Y_i$ be the image of $y$. Then $\kappa(y) = \mathop{\mathrm{colim}}\nolimits \kappa(y_i)$. Hence for some $i$ we have $\mathcal{L}_{i, y_i}$ is ample on $X_{i, y_i}$ by Limits, Lemma 31.4.15. By Cohomology of Schemes, Lemma 29.21.4 we find an open neigbourhood $V_i \subset Y_i$ of $y_i$ such that $\mathcal{L}_i$ restricted to $f_i^{-1}(V_i)$ is ample relative to $V_i$. Letting $V \subset Y$ be the inverse image of $V_i$ finishes the proof (hints: use Morphisms, Lemma 28.35.9 and the fact that $X \to Y \times_{Y_i} X_i$ is affine and the fact that the pullback of an ample invertible sheaf by an affine morphism is ample by Morphisms, Lemma 28.35.7). $\square$

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 11941–12126 (see updates for more information).

    \section{Projective schemes}
    \label{section-projective}
    
    \noindent
    This section is the analogue of Section \ref{section-quasi-projective}
    for projective morphisms.
    
    \begin{lemma}
    \label{lemma-projective}
    Let $S$ be a scheme which has an ample invertible sheaf.
    Let $f : X \to S$ be a morphism of schemes. The following are
    equivalent
    \begin{enumerate}
    \item $X \to S$ is projective,
    \item $X \to S$ is H-projective,
    \item $X \to S$ is quasi-projective and proper,
    \item $X \to S$ is H-quasi-projective and proper,
    \item $X \to S$ is proper and $X$ has an ample invertible sheaf,
    \item $X \to S$ is proper and there exists an $f$-ample invertible sheaf,
    \item $X \to S$ is proper and there exists an $f$-very ample invertible sheaf,
    \item there is a quasi-coherent graded $\mathcal{O}_S$-algebra $\mathcal{A}$
    generated by $\mathcal{A}_1$ over $\mathcal{A}_0$ with $\mathcal{A}_1$ a
    finite type $\mathcal{O}_S$-module such that
    $X = \underline{\text{Proj}}_S(\mathcal{A})$. 
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Observe first that in each case the morphism $f$ is proper, see
    Morphisms, Lemmas \ref{morphisms-lemma-H-projective} and
    \ref{morphisms-lemma-locally-projective-proper}.
    Hence it suffices to prove the equivalence of the notions in
    case $f$ is a proper morphism. We will use this without further
    mention in the following.
    
    \medskip\noindent
    The equivalences (1) $\Leftrightarrow$ (3) and
    (2) $\Leftrightarrow$ (4) are
    Morphisms, Lemma \ref{morphisms-lemma-projective-is-quasi-projective-proper}.
    
    \medskip\noindent
    The implication (2) $\Rightarrow$ (1) is
    Morphisms, Lemma \ref{morphisms-lemma-H-projective}.
    
    \medskip\noindent
    The implications (1) $\Rightarrow$ (2) and (3) $\Rightarrow$ (4) are
    Morphisms, Lemma
    \ref{morphisms-lemma-projective-over-quasi-projective-is-H-projective}.
    
    \medskip\noindent
    The implication (1) $\Rightarrow$ (7) is immediate from
    Morphisms, Definitions \ref{morphisms-definition-projective} and
    \ref{morphisms-definition-very-ample}.
    
    \medskip\noindent
    The conditions (3) and (6) are equivalent by
    Morphisms, Definition \ref{morphisms-definition-quasi-projective}.
    
    \medskip\noindent
    Thus (1) -- (4), (6) are equivalent and imply (7). By
    Lemma \ref{lemma-quasi-projective}
    conditions (3), (5), and (7) are equivalent.
    Thus we see that (1) -- (7) are equivalent.
    
    \medskip\noindent
    By Divisors, Lemma \ref{divisors-lemma-relative-proj-projective}
    we see that (8) implies (1). Conversely, if (2) holds, then
    we can choose a closed immersion
    $$
    i :
    X
    \longrightarrow
    \mathbf{P}^n_S = \underline{\text{Proj}}_S(\mathcal{O}_S[T_0, \ldots, T_n]).
    $$
    See Constructions, Lemma \ref{constructions-lemma-projective-space-bundle}
    for the equality. By
    Divisors, Lemma \ref{divisors-lemma-closed-subscheme-proj}
    we see that $X$ is the relative Proj of a quasi-coherent graded quotient
    algebra $\mathcal{A}$ of $\mathcal{O}_S[T_0, \ldots, T_n]$.
    Then $\mathcal{A}$ satisfies the conditions of (8).
    \end{proof}
    
    \begin{lemma}
    \label{lemma-category-projective}
    Let $S$ be a scheme which has an ample invertible sheaf.
    Let $\text{P}_S$ be the full subcategory of the
    category of schemes over $S$ satisfying the equivalent
    conditions of Lemma \ref{lemma-projective}.
    \begin{enumerate}
    \item if $S' \to S$ is a morphism of schemes and $S'$ has
    an ample invertible sheaf, then base change determines
    a functor $\text{P}_S \to \text{P}_{S'}$,
    \item if $X \in \text{P}_S$ and $Y \in \text{P}_X$, then $Y \in \text{P}_S$,
    \item the category $\text{P}_S$ is closed under fibre products,
    \item the category $\text{P}_S$ is closed under
    finite disjoint unions,
    \item if $X \to S$ is finite, then $X$ is in $\text{P}_S$,
    \item add more here.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Part (1) follows from Morphisms, Lemma
    \ref{morphisms-lemma-base-change-projective}.
    
    \medskip\noindent
    Part (2) follows from the fifth characterization of
    Lemma \ref{lemma-projective} and the fact that compositions
    of proper morphisms are proper
    (Morphisms, Lemma \ref{morphisms-lemma-composition-proper}).
    
    \medskip\noindent
    If $X \to S$ and $Y \to S$ are projective, then
    $X \times_S Y \to Y$ is projective by
    Morphisms, Lemma \ref{morphisms-lemma-base-change-projective}.
    Hence (3) follows from (2).
    
    \medskip\noindent
    If $X = Y \amalg Z$ is a disjoint union of schemes
    and $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module
    such that $\mathcal{L}|_Y$ and $\mathcal{L}|_Z$ are ample, then
    $\mathcal{L}$ is ample (details omitted). Thus
    part (4) follows from the fifth characterization of
    Lemma \ref{lemma-projective}.
    
    \medskip\noindent
    Part (5) follows from
    Morphisms, Lemma \ref{morphisms-lemma-finite-projective}.
    \end{proof}
    
    \noindent
    Here is a slightly different type of result.
    
    \begin{lemma}
    \label{lemma-ample-in-neighbourhood}
    \begin{reference}
    \cite[IV Corollary 9.6.4]{EGA}
    \end{reference}
    Let $f : X \to Y$ be a proper morphism of schemes.
    Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
    Let $y \in Y$ be a point such that $\mathcal{L}_y$ is ample
    on $X_y$. Then there is an open neighbourhood $V \subset Y$
    of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.
    \end{lemma}
    
    \begin{proof}
    We may assume $Y$ is affine. Then we find a directed set $I$
    and an inverse system of morphisms $X_i \to Y_i$ of schemes
    with $Y_i$ of finite type over $\mathbf{Z}$, with affine
    transition morphisms $X_i \to X_{i'}$ and $Y_i \to Y_{i'}$,
    with $X_i \to Y_i$ proper, such that $X \to Y = \lim (X_i \to Y_i)$.
    See Limits, Lemma
    \ref{limits-lemma-proper-limit-of-proper-finite-presentation-noetherian}.
    After shrinking $I$ we can assume we have a compatible system of
    invertible $\mathcal{O}_{X_i}$-modules $\mathcal{L}_i$
    pulling back to $\mathcal{L}$, see
    Limits, Lemma \ref{limits-lemma-descend-invertible-modules}.
    Let $y_i \in Y_i$ be the image of $y$.
    Then $\kappa(y) = \colim \kappa(y_i)$.
    Hence for some $i$ we have $\mathcal{L}_{i, y_i}$
    is ample on $X_{i, y_i}$ by
    Limits, Lemma \ref{limits-lemma-limit-ample}.
    By Cohomology of Schemes, Lemma \ref{coherent-lemma-ample-in-neighbourhood}
    we find an open neigbourhood
    $V_i \subset Y_i$ of $y_i$ such that
    $\mathcal{L}_i$ restricted to $f_i^{-1}(V_i)$
    is ample relative to $V_i$.
    Letting $V \subset Y$ be the inverse image of
    $V_i$ finishes the proof (hints: use
    Morphisms, Lemma \ref{morphisms-lemma-ample-base-change} and
    the fact that $X \to Y \times_{Y_i} X_i$ is affine
    and the fact that the pullback of an
    ample invertible sheaf by an affine morphism is ample by
    Morphisms, Lemma \ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}).
    \end{proof}

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