The Stacks Project


Tag 01ZC

Chapter 31: Limits of Schemes > Section 31.6: Limits and morphisms of finite presentation

Proposition 31.6.1. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

  1. The morphism $f$ is locally of finite presentation.
  2. For any directed set $I$, and any inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$ with each $T_i$ affine, we have $$ \mathop{\rm Mor}\nolimits_S(\mathop{\rm lim}\nolimits_i T_i, X) = \mathop{\rm colim}\nolimits_i \mathop{\rm Mor}\nolimits_S(T_i, X) $$
  3. For any directed set $I$, and any inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$ with each $f_{ii'}$ affine and every $T_i$ quasi-compact and quasi-separated as a scheme, we have $$ \mathop{\rm Mor}\nolimits_S(\mathop{\rm lim}\nolimits_i T_i, X) = \mathop{\rm colim}\nolimits_i \mathop{\rm Mor}\nolimits_S(T_i, X) $$

Proof. It is clear that (3) implies (2).

Let us prove that (2) implies (1). Assume (2). Choose any affine opens $U \subset X$ and $V \subset S$ such that $f(U) \subset V$. We have to show that $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation. Let $(A_i, \varphi_{ii'})$ be a directed system of $\mathcal{O}_S(V)$-algebras. Set $A = \mathop{\rm colim}\nolimits_i A_i$. According to Algebra, Lemma 10.126.3 we have to show that $$ \mathop{\rm Hom}\nolimits_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A) = \mathop{\rm colim}\nolimits_i \mathop{\rm Hom}\nolimits_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A_i) $$ Consider the schemes $T_i = \mathop{\rm Spec}(A_i)$. They form an inverse system of $V$-schemes over $I$ with transition morphisms $f_{ii'} : T_i \to T_{i'}$ induced by the $\mathcal{O}_S(V)$-algebra maps $\varphi_{i'i}$. Set $T := \mathop{\rm Spec}(A) = \mathop{\rm lim}\nolimits_i T_i$. The formula above becomes in terms of morphism sets of schemes $$ \mathop{\rm Mor}\nolimits_V(\mathop{\rm lim}\nolimits_i T_i, U) = \mathop{\rm colim}\nolimits_i \mathop{\rm Mor}\nolimits_V(T_i, U). $$ We first observe that $\mathop{\rm Mor}\nolimits_V(T_i, U) = \mathop{\rm Mor}\nolimits_S(T_i, U)$ and $\mathop{\rm Mor}\nolimits_V(T, U) = \mathop{\rm Mor}\nolimits_S(T, U)$. Hence we have to show that $$ \mathop{\rm Mor}\nolimits_S(\mathop{\rm lim}\nolimits_i T_i, U) = \mathop{\rm colim}\nolimits_i \mathop{\rm Mor}\nolimits_S(T_i, U) $$ and we are given that $$ \mathop{\rm Mor}\nolimits_S(\mathop{\rm lim}\nolimits_i T_i, X) = \mathop{\rm colim}\nolimits_i \mathop{\rm Mor}\nolimits_S(T_i, X). $$ Hence it suffices to prove that given a morphism $g_i : T_i \to X$ over $S$ such that the composition $T \to T_i \to X$ ends up in $U$ there exists some $i' \geq i$ such that the composition $g_{i'} : T_{i'} \to T_i \to X$ ends up in $U$. Denote $Z_{i'} = g_{i'}^{-1}(X \setminus U)$. Assume each $Z_{i'}$ is nonempty to get a contradiction. By Lemma 31.4.8 there exists a point $t$ of $T$ which is mapped into $Z_{i'}$ for all $i' \geq i$. Such a point is not mapped into $U$. A contradiction.

Finally, let us prove that (1) implies (3). Assume (1). Let an inverse directed system $(T_i, f_{ii'})$ of $S$-schemes be given. Assume the morphisms $f_{ii'}$ are affine and each $T_i$ is quasi-compact and quasi-separated as a scheme. Let $T = \mathop{\rm lim}\nolimits_i T_i$. Denote $f_i : T \to T_i$ the projection morphisms. We have to show:

  1. (a)    Given morphisms $g_i, g'_i : T_i \to X$ over $S$ such that $g_i \circ f_i = g'_i \circ f_i$, then there exists an $i' \geq i$ such that $g_i \circ f_{i'i} = g'_i \circ f_{i'i}$.
  2. (b)    Given any morphism $g : T \to X$ over $S$ there exists an $i \in I$ and a morphism $g_i : T_i \to X$ such that $g = f_i \circ g_i$.

First let us prove the uniqueness part (a). Let $g_i, g'_i : T_i \to X$ be morphisms such that $g_i \circ f_i = g'_i \circ f_i$. For any $i' \geq i$ we set $g_{i'} = g_i \circ f_{i'i}$ and $g'_{i'} = g'_i \circ f_{i'i}$. We also set $g = g_i \circ f_i = g'_i \circ f_i$. Consider the morphism $(g_i, g'_i) : T_i \to X \times_S X$. Set $$ W = \bigcup\nolimits_{U \subset X\text{ affine open}, V \subset S\text{ affine open}, f(U) \subset V} U \times_V U. $$ This is an open in $X \times_S X$, with the property that the morphism $\Delta_{X/S}$ factors through a closed immersion into $W$, see the proof of Schemes, Lemma 25.21.2. Note that the composition $(g_i, g'_i) \circ f_i : T \to X \times_S X$ is a morphism into $W$ because it factors through the diagonal by assumption. Set $Z_{i'} = (g_{i'}, g'_{i'})^{-1}(X \times_S X \setminus W)$. If each $Z_{i'}$ is nonempty, then by Lemma 31.4.8 there exists a point $t \in T$ which maps to $Z_{i'}$ for all $i' \geq i$. This is a contradiction with the fact that $T$ maps into $W$. Hence we may increase $i$ and assume that $(g_i, g'_i) : T_i \to X \times_S X$ is a morphism into $W$. By construction of $W$, and since $T_i$ is quasi-compact we can find a finite affine open covering $T_i = T_{1, i} \cup \ldots \cup T_{n, i}$ such that $(g_i, g'_i)|_{T_{j, i}}$ is a morphism into $U \times_V U$ for some pair $(U, V)$ as in the definition of $W$ above. Since it suffices to prove that $g_{i'}$ and $g'_{i'}$ agree on each of the $f_{i'i}^{-1}(T_{j, i})$ this reduces us to the affine case. The affine case follows from Algebra, Lemma 10.126.3 and the fact that the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation (see Morphisms, Lemma 28.20.2).

Finally, we prove the existence part (b). Let $g : T \to X$ be a morphism of schemes over $S$. We can find a finite affine open covering $T = W_1 \cup \ldots \cup W_n$ such that for each $j \in \{1, \ldots, n\}$ there exist affine opens $U_j \subset X$ and $V_j \subset S$ with $f(U_j) \subset V_j$ and $g(W_j) \subset U_j$. By Lemmas 31.4.11 and 31.4.13 (after possibly shrinking $I$) we may assume that there exist affine open coverings $T_i = W_{1, i} \cup \ldots \cup W_{n, i}$ compatible with transition maps such that $W_j = \mathop{\rm lim}\nolimits_i W_{j, i}$. We apply Algebra, Lemma 10.126.3 to the rings corresponding to the affine schemes $U_j$, $V_j$, $W_{j, i}$ and $W_j$ using that $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_j)$ is of finite presentation (see Morphisms, Lemma 28.20.2). Thus we can find for each $j$ an index $i_j \in I$ and a morphism $g_{j, i_j} : W_{j, i_j} \to X$ such that $g_{j, i_j} \circ f_i|_{W_j} : W_j \to W_{j, i} \to X$ equals $g|_{W_j}$. By part (a) proved above, using the quasi-compactness of $W_{j_1, i} \cap W_{j_2, i}$ which follows as $T_i$ is quasi-separated, we can find an index $i' \in I$ larger than all $i_j$ such that $$ g_{j_1, i_{j_1}} \circ f_{i'i_{j_1}}|_{W_{j_1, i'} \cap W_{j_2, i'}} = g_{j_2, i_{j_2}} \circ f_{i'i_{j_2}}|_{W_{j_1, i'} \cap W_{j_2, i'}} $$ for all $j_1, j_2 \in \{1, \ldots, n\}$. Hence the morphisms $g_{j, i_j} \circ f_{i'i_j}|_{W_{j, i'}}$ glue to given the desired morphism $T_{i'} \to X$. $\square$

    The code snippet corresponding to this tag is a part of the file limits.tex and is located in lines 1223–1248 (see updates for more information).

    \begin{proposition}
    \label{proposition-characterize-locally-finite-presentation}
    \begin{reference}
    \cite[IV, Proposition 8.14.2]{EGA}
    \end{reference}
    Let $f : X \to S$ be a morphism of schemes.
    The following are equivalent:
    \begin{enumerate}
    \item The morphism $f$ is locally of finite presentation.
    \item For any directed set $I$, and any
    inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$
    with each $T_i$ affine, we have
    $$
    \Mor_S(\lim_i T_i, X) =
    \colim_i \Mor_S(T_i, X)
    $$
    \item For any directed set $I$, and any
    inverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$
    with each $f_{ii'}$ affine and every $T_i$ quasi-compact and
    quasi-separated as a scheme, we have
    $$
    \Mor_S(\lim_i T_i, X) =
    \colim_i \Mor_S(T_i, X)
    $$
    \end{enumerate}
    \end{proposition}
    
    \begin{proof}
    It is clear that (3) implies (2).
    
    \medskip\noindent
    Let us prove that (2) implies (1). Assume (2).
    Choose any affine opens $U \subset X$ and $V \subset S$ such that
    $f(U) \subset V$. We have to show that
    $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation.
    Let $(A_i, \varphi_{ii'})$ be a directed system of
    $\mathcal{O}_S(V)$-algebras. Set $A = \colim_i A_i$.
    According to
    Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation}
    we have to show that
    $$
    \Hom_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A) =
    \colim_i \Hom_{\mathcal{O}_S(V)}(\mathcal{O}_X(U), A_i)
    $$
    Consider the schemes $T_i = \Spec(A_i)$. They
    form an inverse system of $V$-schemes over $I$
    with transition morphisms $f_{ii'} : T_i \to T_{i'}$
    induced by the $\mathcal{O}_S(V)$-algebra maps $\varphi_{i'i}$.
    Set $T := \Spec(A) = \lim_i T_i$.
    The formula above becomes in terms of morphism sets of schemes
    $$
    \Mor_V(\lim_i T_i, U) =
    \colim_i \Mor_V(T_i, U).
    $$
    We first observe that
    $\Mor_V(T_i, U) = \Mor_S(T_i, U)$
    and
    $\Mor_V(T, U) = \Mor_S(T, U)$.
    Hence we have to show that
    $$
    \Mor_S(\lim_i T_i, U) =
    \colim_i \Mor_S(T_i, U)
    $$
    and we are given that
    $$
    \Mor_S(\lim_i T_i, X) =
    \colim_i \Mor_S(T_i, X).
    $$
    Hence it suffices to prove that given a morphism $g_i : T_i \to X$ over $S$
    such that the composition $T \to T_i \to X$ ends up in $U$ there exists some
    $i' \geq i$ such that the composition $g_{i'} : T_{i'} \to T_i \to X$ ends up
    in $U$. Denote $Z_{i'} = g_{i'}^{-1}(X \setminus U)$.
    Assume each $Z_{i'}$ is nonempty
    to get a contradiction. By Lemma \ref{lemma-limit-closed-nonempty}
    there exists a point $t$ of $T$ which is mapped into $Z_{i'}$ for all
    $i' \geq i$. Such a point is not mapped into $U$. A contradiction.
    
    \medskip\noindent
    Finally, let us prove that (1) implies (3). Assume (1). Let an inverse directed
    system $(T_i, f_{ii'})$ of $S$-schemes be given. Assume the morphisms $f_{ii'}$
    are affine and each $T_i$ is quasi-compact and quasi-separated as a scheme. Let
    $T = \lim_i T_i$. Denote $f_i : T \to T_i$ the projection morphisms.
    We have to show:
    \begin{enumerate}
    \item[(a)] Given morphisms $g_i, g'_i : T_i \to X$ over $S$ such that
    $g_i \circ f_i = g'_i \circ f_i$, then there exists an $i' \geq i$
    such that $g_i \circ f_{i'i} = g'_i \circ f_{i'i}$.
    \item[(b)] Given any morphism
    $g : T \to X$ over $S$ there exists an $i \in I$ and a morphism
    $g_i : T_i \to X$ such that $g = f_i \circ g_i$.
    \end{enumerate}
    
    \noindent
    First let us prove the uniqueness part (a). Let $g_i, g'_i : T_i \to X$ be
    morphisms such that $g_i \circ f_i = g'_i \circ f_i$. For any $i' \geq i$
    we set $g_{i'} = g_i \circ f_{i'i}$ and $g'_{i'} = g'_i \circ f_{i'i}$.
    We also set $g = g_i \circ f_i = g'_i \circ f_i$.
    Consider the morphism
    $(g_i, g'_i) : T_i \to X \times_S X$. Set
    $$
    W =
    \bigcup\nolimits_{U \subset X\text{ affine open},
    V \subset S\text{ affine open}, f(U) \subset V}
    U \times_V U.
    $$
    This is an open in $X \times_S X$, with the property that the morphism
    $\Delta_{X/S}$ factors through a closed immersion into $W$, see the proof
    of Schemes, Lemma \ref{schemes-lemma-diagonal-immersion}.
    Note that the composition
    $(g_i, g'_i) \circ f_i : T \to X \times_S X$ is a morphism into $W$
    because it factors through the diagonal by assumption.
    Set $Z_{i'} = (g_{i'}, g'_{i'})^{-1}(X \times_S X \setminus W)$.
    If each $Z_{i'}$ is nonempty, then by Lemma \ref{lemma-limit-closed-nonempty}
    there exists a point $t \in T$ which maps to $Z_{i'}$ for all
    $i' \geq i$. This is a contradiction with the fact that $T$ maps into $W$.
    Hence we may increase $i$ and assume that $(g_i, g'_i) : T_i \to X \times_S X$
    is a morphism into $W$. By construction of $W$, and since $T_i$ is
    quasi-compact we can find a finite affine open covering
    $T_i = T_{1, i} \cup \ldots \cup T_{n, i}$ such that
    $(g_i, g'_i)|_{T_{j, i}}$ is a morphism into $U \times_V U$ for
    some pair $(U, V)$ as in the definition of $W$ above.
    Since it suffices to prove that $g_{i'}$ and $g'_{i'}$ agree
    on each of the $f_{i'i}^{-1}(T_{j, i})$ this reduces us to the affine case.
    The affine case follows from
    Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation}
    and the fact that the ring map
    $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite presentation
    (see Morphisms,
    Lemma \ref{morphisms-lemma-locally-finite-presentation-characterize}).
    
    \medskip\noindent
    Finally, we prove the existence part (b).
    Let $g : T \to X$ be a morphism of schemes over $S$.
    We can find a finite affine open covering
    $T = W_1 \cup \ldots \cup W_n$ such that for
    each $j \in \{1, \ldots, n\}$ there exist affine opens
    $U_j \subset X$ and $V_j \subset S$ with $f(U_j) \subset V_j$
    and $g(W_j) \subset U_j$. By Lemmas \ref{lemma-descend-opens}
    and \ref{lemma-limit-affine}
    (after possibly shrinking $I$) we may assume that
    there exist affine open coverings $T_i = W_{1, i} \cup \ldots \cup W_{n, i}$
    compatible with transition maps such that $W_j = \lim_i W_{j, i}$.
    We apply Algebra, Lemma \ref{algebra-lemma-characterize-finite-presentation}
    to the rings corresponding to the affine schemes $U_j$, $V_j$, $W_{j, i}$ and
    $W_j$ using that $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_j)$ is of finite
    presentation (see Morphisms,
    Lemma \ref{morphisms-lemma-locally-finite-presentation-characterize}).
    Thus we can find for each $j$ an index $i_j \in I$ and a morphism
    $g_{j, i_j} : W_{j, i_j} \to X$ such that
    $g_{j, i_j} \circ f_i|_{W_j} : W_j \to W_{j, i} \to X$
    equals $g|_{W_j}$. By part (a) proved above, using the quasi-compactness of
    $W_{j_1, i} \cap W_{j_2, i}$ which follows as $T_i$ is quasi-separated,
    we can find an index $i' \in I$ larger than all $i_j$ such that
    $$
    g_{j_1, i_{j_1}} \circ f_{i'i_{j_1}}|_{W_{j_1, i'} \cap W_{j_2, i'}} =
    g_{j_2, i_{j_2}} \circ f_{i'i_{j_2}}|_{W_{j_1, i'} \cap W_{j_2, i'}}
    $$
    for all $j_1, j_2 \in \{1, \ldots, n\}$. Hence the morphisms
    $g_{j, i_j} \circ f_{i'i_j}|_{W_{j, i'}}$ glue to given the
    desired morphism $T_{i'} \to X$.
    \end{proof}

    References

    [EGA, IV, Proposition 8.14.2]

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 01ZC

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?