29.21 Morphisms of finite presentation
Recall that a ring map $R \to A$ is of finite presentation if $A$ is isomorphic to $R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ as an $R$-algebra for some $n, m$ and some polynomials $f_ j$, see Algebra, Definition 10.6.1.
Definition 29.21.1. Let $f : X \to S$ be a morphism of schemes.
We say that $f$ is of finite presentation at $x \in X$ if there exists an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset X$ of $x$ and affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is of finite presentation.
We say that $f$ is locally of finite presentation if it is of finite presentation at every point of $X$.
We say that $f$ is of finite presentation if it is locally of finite presentation, quasi-compact and quasi-separated.
Note that a morphism of finite presentation is not just a quasi-compact morphism which is locally of finite presentation. Later we will characterize morphisms which are locally of finite presentation as those morphisms such that
\[ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Mor}}\nolimits _ S(T_ i, X) = \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{lim}}\nolimits T_ i, X) \]
for any directed system of affine schemes $T_ i$ over $S$. See Limits, Proposition 32.6.1. In Limits, Section 32.10 we show that, if $S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$ is a limit of affine schemes, any scheme $X$ of finite presentation over $S$ descends to a scheme $X_ i$ over $S_ i$ for some $i$.
Lemma 29.21.2. Let $f : X \to S$ be a morphism of schemes. The following are equivalent
The morphism $f$ is locally of finite presentation.
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 of finite presentation.
There exist 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 locally of finite presentation.
There exist 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 the ring map $\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_ i)$ is of finite presentation, for all $j\in J, i\in I_ j$.
Moreover, if $f$ is locally of finite presentation then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_ U : U \to V$ is locally of finite presentation.
Proof.
This follows from Lemma 29.14.4 if we show that the property “$R \to A$ is of finite presentation” is local. We check conditions (a), (b) and (c) of Definition 29.14.1. By Algebra, Lemma 10.14.2 being of finite presentation is stable under base change and hence we conclude (a) holds. By Algebra, Lemma 10.6.2 being of finite presentation is stable under composition and trivially for any ring $R$ the ring map $R \to R_ f$ is of finite presentation. We conclude (b) holds. Finally, property (c) is true according to Algebra, Lemma 10.23.3.
$\square$
Lemma 29.21.3. The composition of two morphisms which are locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation.
Proof.
In the proof of Lemma 29.21.2 we saw that being of finite presentation is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 29.14.5 combined with the fact that being of finite presentation is a property of ring maps that is stable under composition, see Algebra, Lemma 10.6.2. By the above and the fact that compositions of quasi-compact, quasi-separated morphisms are quasi-compact and quasi-separated, see Schemes, Lemmas 26.19.4 and 26.21.12 we see that the composition of morphisms of finite presentation is of finite presentation.
$\square$
Lemma 29.21.4. The base change of a morphism which is locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation.
Proof.
In the proof of Lemma 29.21.2 we saw that being of finite presentation is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 29.14.5 combined with the fact that being of finite presentation is a property of ring maps that is stable under base change, see Algebra, Lemma 10.14.2. By the above and the fact that a base change of a quasi-compact, quasi-separated morphism is quasi-compact and quasi-separated, see Schemes, Lemmas 26.19.3 and 26.21.12 we see that the base change of a morphism of finite presentation is a morphism of finite presentation.
$\square$
Lemma 29.21.5. Any open immersion is locally of finite presentation.
Proof.
This is true because an open immersion is a local isomorphism.
$\square$
Lemma 29.21.6. Any open immersion is of finite presentation if and only if it is quasi-compact.
Proof.
We have seen (Lemma 29.21.5) that an open immersion is locally of finite presentation. We have seen (Schemes, Lemma 26.23.8) that an immersion is separated and hence quasi-separated. From this and Definition 29.21.1 the lemma follows.
$\square$
slogan
Lemma 29.21.7. A closed immersion $i : Z \to X$ is of finite presentation if and only if the associated quasi-coherent sheaf of ideals $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to i_*\mathcal{O}_ Z)$ is of finite type (as an $\mathcal{O}_ X$-module).
Proof.
On any affine open $\mathop{\mathrm{Spec}}(R) \subset X$ we have $i^{-1}(\mathop{\mathrm{Spec}}(R)) = \mathop{\mathrm{Spec}}(R/I)$ and $\mathcal{I} = \widetilde{I}$. Moreover, $\mathcal{I}$ is of finite type if and only if $I$ is a finite $R$-module for every such affine open (see Properties, Lemma 28.16.1). And $R/I$ is of finite presentation over $R$ if and only if $I$ is a finite $R$-module. Hence we win.
$\square$
Lemma 29.21.8. A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type.
Proof.
Omitted.
$\square$
slogan
Lemma 29.21.9. Let $f : X \to S$ be a morphism.
If $S$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation.
If $S$ is locally Noetherian and $f$ of finite type then $f$ is of finite presentation.
Proof.
The first statement follows from the fact that a ring of finite type over a Noetherian ring is of finite presentation, see Algebra, Lemma 10.31.4. Suppose that $f$ is of finite type and $S$ is locally Noetherian. Then $f$ is quasi-compact and locally of finite presentation by (1). Hence it suffices to prove that $f$ is quasi-separated. This follows from Lemma 29.15.7 (and Lemma 29.21.8).
$\square$
Lemma 29.21.10. Let $S$ be a scheme which is quasi-compact and quasi-separated. If $X$ is of finite presentation over $S$, then $X$ is quasi-compact and quasi-separated.
Proof.
Omitted.
$\square$
Lemma 29.21.11. Let $f : X \to Y$ be a morphism of schemes over $S$.
If $X$ is locally of finite presentation over $S$ and $Y$ is locally of finite type over $S$, then $f$ is locally of finite presentation.
If $X$ is of finite presentation over $S$ and $Y$ is quasi-separated and locally of finite type over $S$, then $f$ is of finite presentation.
Proof.
Proof of (1). Via Lemma 29.21.2 this translates into the following algebra fact: Given ring maps $A \to B \to C$ such that $A \to C$ is of finite presentation and $A \to B$ is of finite type, then $B \to C$ is of finite presentation. See Algebra, Lemma 10.6.2.
Part (2) follows from (1) and Schemes, Lemmas 26.21.13 and 26.21.14.
$\square$
Lemma 29.21.12. Let $f : X \to Y$ be a morphism of schemes with diagonal $\Delta : X \to X \times _ Y X$. If $f$ is locally of finite type then $\Delta $ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta $ is of finite presentation.
Proof.
Note that $\Delta $ is a morphism of schemes over $X$ (via the second projection $X \times _ Y X \to X$). Assume $f$ is locally of finite type. Note that $X$ is of finite presentation over $X$ and $X \times _ Y X$ is locally of finite type over $X$ (by Lemma 29.15.4). Thus the first statement holds by Lemma 29.21.11. The second statement follows from the first, the definitions, and the fact that a diagonal morphism is a monomorphism, hence separated (Schemes, Lemma 26.23.3).
$\square$
Comments (0)