## 32.6 Limits and morphisms of finite presentation

The following is a generalization of Algebra, Lemma 10.127.3.

Proposition 32.6.1. 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

$\mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{lim}}\nolimits _ i T_ i, X) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, X)$
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

$\mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{lim}}\nolimits _ i T_ i, X) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, X)$

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 $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$. According to Algebra, Lemma 10.127.3 we have to show that

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ S(V)}(\mathcal{O}_ X(U), A) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ S(V)}(\mathcal{O}_ X(U), A_ i)$

Consider the schemes $T_ i = \mathop{\mathrm{Spec}}(A_ i)$. 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 $T := \mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{lim}}\nolimits _ i T_ i$. The formula above becomes in terms of morphism sets of schemes

$\mathop{\mathrm{Mor}}\nolimits _ V(\mathop{\mathrm{lim}}\nolimits _ i T_ i, U) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _ V(T_ i, U).$

We first observe that $\mathop{\mathrm{Mor}}\nolimits _ V(T_ i, U) = \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, U)$ and $\mathop{\mathrm{Mor}}\nolimits _ V(T, U) = \mathop{\mathrm{Mor}}\nolimits _ S(T, U)$. Hence we have to show that

$\mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{lim}}\nolimits _ i T_ i, U) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, U)$

and we are given that

$\mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{lim}}\nolimits _ i T_ i, X) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, X).$

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 32.4.8 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 $T = \mathop{\mathrm{lim}}\nolimits _ i T_ i$. Denote $f_ i : T \to T_ i$ the projection morphisms. We have to show:

1. 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. 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 26.21.2. 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 32.4.8 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 10.127.3 and the fact that the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is of finite presentation (see Morphisms, Lemma 29.21.2).

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 32.4.11 and 32.4.13 (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 $W_ j = \mathop{\mathrm{lim}}\nolimits _ i W_{j, i}$. We apply Algebra, Lemma 10.127.3 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 29.21.2). 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 $T_{i'} \to X$. $\square$

Remark 32.6.2. Let $S$ be a scheme. Let us say that a functor $F : (\mathit{Sch}/S)^{opp} \to \textit{Sets}$ is limit preserving if for every directed inverse system $\{ T_ i\} _{i \in I}$ of affine schemes with limit $T$ we have $F(T) = \mathop{\mathrm{colim}}\nolimits _ i F(T_ i)$. Let $X$ be a scheme over $S$, and let $h_ X : (\mathit{Sch}/S)^{opp} \to \textit{Sets}$ be its functor of points, see Schemes, Section 26.15. In this terminology Proposition 32.6.1 says that a scheme $X$ is locally of finite presentation over $S$ if and only if $h_ X$ is limit preserving.

Lemma 32.6.3. Let $f : X \to S$ be a morphism of schemes. If for every directed limit $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ of affine schemes over $S$ the map

$\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, X) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _ S(T, X)$

is surjective, then $f$ is locally of finite presentation. In other words, in Proposition 32.6.1 parts (2) and (3) it suffices to check surjectivity of the map.

Proof. The proof is exactly the same as the proof of the implication “(2) implies (1)” in Proposition 32.6.1. 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 $A = \mathop{\mathrm{colim}}\nolimits _ i A_ i$. According to Algebra, Lemma 10.127.3 it suffices to show that

$\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ S(V)}(\mathcal{O}_ X(U), A_ i) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ S(V)}(\mathcal{O}_ X(U), A)$

is surjective. Consider the schemes $T_ i = \mathop{\mathrm{Spec}}(A_ i)$. 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 $T := \mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{lim}}\nolimits _ i T_ i$. The formula above becomes in terms of morphism sets of schemes

$\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _ V(T_ i, U) \to \mathop{\mathrm{Mor}}\nolimits _ V(\mathop{\mathrm{lim}}\nolimits _ i T_ i, U)$

We first observe that $\mathop{\mathrm{Mor}}\nolimits _ V(T_ i, U) = \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, U)$ and $\mathop{\mathrm{Mor}}\nolimits _ V(T, U) = \mathop{\mathrm{Mor}}\nolimits _ S(T, U)$. Hence we have to show that

$\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, U) \to \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{lim}}\nolimits _ i T_ i, U)$

is surjective and we are given that

$\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, X) \to \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{lim}}\nolimits _ i T_ i, X)$

is surjective. 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 32.4.8 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. $\square$

The following is an example application of Proposition 32.6.1.

Lemma 32.6.4. Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Assume $Y$ is locally of finite presentation over $S$. Let $x \in X$ be a closed point such that $U = X \setminus \{ x\} \to X$ is quasi-compact. With $V = \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \setminus \{ x\}$ there is a bijection

$\left\{ \begin{matrix} \text{morphisms }X \to Y\text{ over }S \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} (a, b)\text{ where } a : U \to Y\text{ and }b : \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to Y \\ \text{ are morphisms over }S \text{ which agree over }V \end{matrix} \right\}$

Proof. Let $W \subset X$ be an open neighbourhood of $x$. By glueing of schemes, see Schemes, Section 26.14 the result holds if we consider pairs of morphisms $a : U \to Y$ and $c : W \to Y$ which agree over $U \cap W$. We have $\mathcal{O}_{X, x} = \mathop{\mathrm{colim}}\nolimits \mathcal{O}_ W(W)$ where $W$ runs over the affine open neighbourhoods of $x$ in $X$. Hence $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) = \mathop{\mathrm{lim}}\nolimits W$ where $W$ runs over the affine open neighbourhoods of $s$. Thus by Proposition 32.6.1 any morphism $b : \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to Y$ over $S$ comes from a morphism $c : W \to Y$ for some $W$ as above (and $c$ is unique up to further shrinking $W$). For every affine open $x \in W$ we see that $U \cap W$ is quasi-compact as $U \to X$ is quasi-compact. Hence $V = \mathop{\mathrm{lim}}\nolimits W \cap U = \mathop{\mathrm{lim}}\nolimits W \setminus \{ x\}$ is a limit of quasi-compact and quasi-separated schemes (see Lemma 32.2.2). Thus if $a$ and $b$ agree over $V$, then after shrinking $W$ we see that $a$ and $c$ agree over $U \cap W$ (by the same proposition). The lemma follows. $\square$

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 toolbar).