The Stacks project

31.31 Closed subschemes of relative proj

Some auxiliary lemmas about closed subschemes of relative proj.

Lemma 31.31.1. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme. Denote $\mathcal{I} \subset \mathcal{A}$ the kernel of the canonical map

\[ \mathcal{A} \longrightarrow \bigoplus \nolimits _{d \geq 0} p_*\left((i_*\mathcal{O}_ Z)(d)\right). \]

If $p$ is quasi-compact, then there is an isomorphism $Z = \underline{\text{Proj}}_ S(\mathcal{A}/\mathcal{I})$.

Proof. The morphism $p$ is separated by Constructions, Lemma 27.16.9. As $p$ is quasi-compact, $p_*$ transforms quasi-coherent modules into quasi-coherent modules, see Schemes, Lemma 26.24.1. Hence $\mathcal{I}$ is a quasi-coherent $\mathcal{O}_ S$-module. In particular, $\mathcal{B} = \mathcal{A}/\mathcal{I}$ is a quasi-coherent graded $\mathcal{O}_ S$-algebra. The functoriality morphism $Z' = \underline{\text{Proj}}_ S(\mathcal{B}) \to \underline{\text{Proj}}_ S(\mathcal{A})$ is everywhere defined and a closed immersion, see Constructions, Lemma 27.18.3. Hence it suffices to prove $Z = Z'$ as closed subschemes of $X$.

Having said this, the question is local on the base and we may assume that $S = \mathop{\mathrm{Spec}}(R)$ and that $X = \text{Proj}(A)$ for some graded $R$-algebra $A$. Assume $\mathcal{I} = \widetilde{I}$ for $I \subset A$ a graded ideal. By Constructions, Lemma 27.8.9 there exist $f_0, \ldots , f_ n \in A_{+}$ such that $A_{+} \subset \sqrt{(f_0, \ldots , f_ n)}$ in other words $X = \bigcup D_{+}(f_ i)$. Therefore, it suffices to check that $Z \cap D_{+}(f_ i) = Z' \cap D_{+}(f_ i)$ for each $i$. By renumbering we may assume $i = 0$. Say $Z \cap D_{+}(f_0)$, resp. $Z' \cap D_{+}(f_0)$ is cut out by the ideal $J$, resp. $J'$ of $A_{(f_0)}$.

The inclusion $J' \subset J$. Let $d$ be the least common multiple of $\deg (f_0), \ldots , \deg (f_ n)$. Note that each of the twists $\mathcal{O}_ X(nd)$ is invertible, trivialized by $f_ i^{nd/\deg (f_ i)}$ over $D_{+}(f_ i)$, and that for any quasi-coherent module $\mathcal{F}$ on $X$ the multiplication maps $\mathcal{O}_ X(nd) \otimes _{\mathcal{O}_ X} \mathcal{F}(m) \to \mathcal{F}(nd + m)$ are isomorphisms, see Constructions, Lemma 27.10.2. Observe that $J'$ is the ideal generated by the elements $g/f_0^ e$ where $g \in I$ is homogeneous of degree $e\deg (f_0)$ (see proof of Constructions, Lemma 27.11.3). Of course, by replacing $g$ by $f_0^ lg$ for suitable $l$ we may always assume that $d | e$. Then, since $g$ vanishes as a section of $\mathcal{O}_ X(e\deg (f_0))$ restricted to $Z$ we see that $g/f_0^ d$ is an element of $J$. Thus $J' \subset J$.

Conversely, suppose that $g/f_0^ e \in J$. Again we may assume $d | e$. Pick $i \in \{ 1, \ldots , n\} $. Then $Z \cap D_{+}(f_ i)$ is cut out by some ideal $J_ i \subset A_{(f_ i)}$. Moreover,

\[ J \cdot A_{(f_0f_ i)} = J_ i \cdot A_{(f_0f_ i)}. \]

The right hand side is the localization of $J_ i$ with respect to $f_0^{\deg (f_ i)}/f_ i^{\deg (f_0)}$. It follows that

\[ f_0^{e_ i}g/f_ i^{(e_ i + e)\deg (f_0)/\deg (f_ i)} \in J_ i \]

for some $e_ i \gg 0$ sufficiently divisible. This proves that $f_0^{\max (e_ i)}g$ is an element of $I$, because its restriction to each affine open $D_{+}(f_ i)$ vanishes on the closed subscheme $Z \cap D_{+}(f_ i)$. Hence $g/f_0^ e \in J'$ and we conclude $J \subset J'$ as desired. $\square$

Example 31.31.2. Let $A$ be a graded ring. Let $X = \text{Proj}(A)$ and $S = \mathop{\mathrm{Spec}}(A_0)$. Given a graded ideal $I \subset A$ we obtain a closed subscheme $V_+(I) = \text{Proj}(A/I) \to X$ by Constructions, Lemma 27.11.3. Translating the result of Lemma 31.31.1 we see that if $X$ is quasi-compact, then any closed subscheme $Z$ is of the form $V_+(I(Z))$ where the graded ideal $I(Z) \subset A$ is given by the rule

\[ I(Z) = \mathop{\mathrm{Ker}}(A \longrightarrow \bigoplus \nolimits _{n \geq 0} \Gamma (Z, \mathcal{O}_ Z(n))) \]

Then we can ask the following two natural questions:

  1. Which ideals $I$ are of the form $I(Z)$?

  2. Can we describe the operation $I \mapsto I(V_+(I))$?

We will answer this when $A$ is Noetherian.

First, assume that $A$ is generated by $A_1$ over $A_0$. In this case, for any ideal $I \subset A$ the kernel of the map $A/I \to \bigoplus \Gamma (\text{Proj}(A/I), \mathcal{O})$ is the set of torsion elements of $A/I$, see Cohomology of Schemes, Proposition 30.14.4. Hence we conclude that

\[ I(V_+(I)) = \{ x \in A \mid A_ n x \subset I\text{ for some }n \geq 0\} \]

The ideal on the right is sometimes called the saturation of $I$. This answers (2) and the answer to (1) is that an ideal is of the form $I(Z)$ if and only if it is saturated, i.e., equal to its own saturation.

If $A$ is a general Noetherian graded ring, then we use Cohomology of Schemes, Proposition 30.15.3. Thus we see that for $d$ equal to the lcm of the degrees of generators of $A$ over $A_0$ we get

\[ I(V_+(I)) = \{ x \in A \mid (Ax)_{nd} \subset I\text{ for all }n \gg 0\} \]

This can be different from the saturation of $I$ if $d \not= 1$. For example, suppose that $A = \mathbf{Q}[x, y]$ with $\deg (x) = 2$ and $\deg (y) = 3$. Then $d = 6$. Let $I = (y^2)$. Then we see $y \in I(V_+(I))$ because for any homogeneous $f \in A$ such that $6 | \deg (fy)$ we have $y | f$, hence $fy \in I$. It follows that $I(V_+(I)) = (y)$ but $x^ n y \not\in I$ for all $n$ hence $I(V_+(I))$ is not equal to the saturation.

Lemma 31.31.3. Let $R$ be a UFD. Let $Z \subset \mathbf{P}^ n_ R$ be a closed subscheme which has no embedded points such that every irreducible component of $Z$ has codimension $1$ in $\mathbf{P}^ n_ R$. Then the ideal $I(Z) \subset R[T_0, \ldots , T_ n]$ corresponding to $Z$ is principal.

Proof. Observe that the local rings of $X = \mathbf{P}^ n_ R$ are UFDs because $X$ is covered by affine pieces isomorphic to $\mathbf{A}^ n_ R$ and $R[x_1, \ldots , x_ n]$ is a UFD (Algebra, Lemma 10.120.10). Thus $Z$ is an effective Cartier divisor by Lemma 31.15.9. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaf of ideals corresponding to $Z$. Choose an isomorphism $\mathcal{O}(m) \to \mathcal{I}$ for some $m \in \mathbf{Z}$, see Lemma 31.28.5. Then the composition

\[ \mathcal{O}_ X(m) \to \mathcal{I} \to \mathcal{O}_ X \]

is nonzero. We conclude that $m \leq 0$ and that the corresponding section of $\mathcal{O}_ X(m)^{\otimes -1} = \mathcal{O}_ X(-m)$ is given by some $F \in R[T_0, \ldots , T_ n]$ of degree $-m$, see Cohomology of Schemes, Lemma 30.8.1. Thus on the $i$th standard open $U_ i = D_+(T_ i)$ the closed subscheme $Z \cap U_ i$ is cut out by the ideal

\[ (F(T_0/T_ i, \ldots , T_ n/T_ i)) \subset R[T_0/T_ i, \ldots , T_ n/T_ i] \]

Thus the homogeneous elements of the graded ideal $I(Z) = \mathop{\mathrm{Ker}}(R[T_0, \ldots , T_ n] \to \bigoplus \Gamma (\mathcal{O}_ Z(m)))$ is the set of homogeneous polynomials $G$ such that

\[ G(T_0/T_ i, \ldots , T_ n/T_ i) \in (F(T_0/T_ i, \ldots , T_ n/T_ i)) \]

for $i = 0, \ldots , n$. Clearing denominators, we see there exist $e_ i \geq 0$ such that

\[ T_ i^{e_ i}G \in (F) \]

for $i = 0, \ldots , n$. As $R$ is a UFD, so is $R[T_0, \ldots , T_ n]$. Then $F | T_0^{e_0}G$ and $F | T_1^{e_1}G$ implies $F | G$ as $T_0^{e_0}$ and $T_1^{e_1}$ have no factor in common. Thus $I(Z) = (F)$. $\square$

In case the closed subscheme is locally cut out by finitely many equations we can define it by a finite type ideal sheaf of $\mathcal{A}$.

Lemma 31.31.4. Let $S$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme. If $p$ is quasi-compact and $i$ of finite presentation, then there exists a $d > 0$ and a quasi-coherent finite type $\mathcal{O}_ S$-submodule $\mathcal{F} \subset \mathcal{A}_ d$ such that $Z = \underline{\text{Proj}}_ S(\mathcal{A}/\mathcal{F}\mathcal{A})$.

Proof. By Lemma 31.31.1 we know there exists a quasi-coherent graded sheaf of ideals $\mathcal{I} \subset \mathcal{A}$ such that $Z = \underline{\text{Proj}}(\mathcal{A}/\mathcal{I})$. Since $S$ is quasi-compact we can choose a finite affine open covering $S = U_1 \cup \ldots \cup U_ n$. Say $U_ i = \mathop{\mathrm{Spec}}(R_ i)$. Let $\mathcal{A}|_{U_ i}$ correspond to the graded $R_ i$-algebra $A_ i$ and $\mathcal{I}|_{U_ i}$ to the graded ideal $I_ i \subset A_ i$. Note that $p^{-1}(U_ i) = \text{Proj}(A_ i)$ as schemes over $R_ i$. Since $p$ is quasi-compact we can choose finitely many homogeneous elements $f_{i, j} \in A_{i, +}$ such that $p^{-1}(U_ i) = D_{+}(f_{i, j})$. The condition on $Z \to X$ means that the ideal sheaf of $Z$ in $\mathcal{O}_ X$ is of finite type, see Morphisms, Lemma 29.21.7. Hence we can find finitely many homogeneous elements $h_{i, j, k} \in I_ i \cap A_{i, +}$ such that the ideal of $Z \cap D_{+}(f_{i, j})$ is generated by the elements $h_{i, j, k}/f_{i, j}^{e_{i, j, k}}$. Choose $d > 0$ to be a common multiple of all the integers $\deg (f_{i, j})$ and $\deg (h_{i, j, k})$. By Properties, Lemma 28.22.3 there exists a finite type quasi-coherent $\mathcal{F} \subset \mathcal{I}_ d$ such that all the local sections

\[ h_{i, j, k}f_{i, j}^{(d - \deg (h_{i, j, k}))/\deg (f_{i, j})} \]

are sections of $\mathcal{F}$. By construction $\mathcal{F}$ is a solution. $\square$

The following version of Lemma 31.31.4 will be used in the proof of Lemma 31.34.2.

Lemma 31.31.5. Let $S$ be a quasi-compact and quasi-separated scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme. Let $U \subset X$ be an open. Assume that

  1. $p$ is quasi-compact,

  2. $i$ of finite presentation,

  3. $U \cap p(i(Z)) = \emptyset $,

  4. $U$ is quasi-compact,

  5. $\mathcal{A}_ n$ is a finite type $\mathcal{O}_ S$-module for all $n$.

Then there exists a $d > 0$ and a quasi-coherent finite type $\mathcal{O}_ S$-submodule $\mathcal{F} \subset \mathcal{A}_ d$ with (a) $Z = \underline{\text{Proj}}_ S(\mathcal{A}/\mathcal{F}\mathcal{A})$ and (b) the support of $\mathcal{A}_ d/\mathcal{F}$ is disjoint from $U$.

Proof. Let $\mathcal{I} \subset \mathcal{A}$ be the sheaf of quasi-coherent graded ideals constructed in Lemma 31.31.1. Let $U_ i$, $R_ i$, $A_ i$, $I_ i$, $f_{i, j}$, $h_{i, j, k}$, and $d$ be as constructed in the proof of Lemma 31.31.4. Since $U \cap p(i(Z)) = \emptyset $ we see that $\mathcal{I}_ d|_ U = \mathcal{A}_ d|_ U$ (by our construction of $\mathcal{I}$ as a kernel). Since $U$ is quasi-compact we can choose a finite affine open covering $U = W_1 \cup \ldots \cup W_ m$. Since $\mathcal{A}_ d$ is of finite type we can find finitely many sections $g_{t, s} \in \mathcal{A}_ d(W_ t)$ which generate $\mathcal{A}_ d|_{W_ t} = \mathcal{I}_ d|_{W_ t}$ as an $\mathcal{O}_{W_ t}$-module. To finish the proof, note that by Properties, Lemma 28.22.3 there exists a finite type $\mathcal{F} \subset \mathcal{I}_ d$ such that all the local sections

\[ h_{i, j, k}f_{i, j}^{(d - \deg (h_{i, j, k}))/\deg (f_{i, j})} \quad \text{and}\quad g_{t, s} \]

are sections of $\mathcal{F}$. By construction $\mathcal{F}$ is a solution. $\square$

Lemma 31.31.6. Let $X$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ X$-module. There is a bijection

\[ \left\{ \begin{matrix} \text{sections }\sigma \text{ of the } \\ \text{morphism } \mathbf{P}(\mathcal{E}) \to X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{surjections }\mathcal{E} \to \mathcal{L}\text{ where} \\ \mathcal{L}\text{ is an invertible }\mathcal{O}_ X\text{-module} \end{matrix} \right\} \]

In this case $\sigma $ is a closed immersion and there is a canonical isomorphism

\[ \mathop{\mathrm{Ker}}(\mathcal{E} \to \mathcal{L}) \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes -1} \longrightarrow \mathcal{C}_{\sigma (X)/\mathbf{P}(\mathcal{E})} \]

Both the bijection and isomorphism are compatible with base change.

Proof. Recall that $\pi : \mathbf{P}(\mathcal{E}) \to X$ is the relative proj of the symmetric algebra on $\mathcal{E}$, see Constructions, Definition 27.21.1. Hence the descriptions of sections $\sigma $ follows immediately from the description of the functor of points of $\mathbf{P}(\mathcal{E})$ in Constructions, Lemma 27.16.11. Since $\pi $ is separated, any section is a closed immersion (Constructions, Lemma 27.16.9 and Schemes, Lemma 26.21.11). Let $U \subset X$ be an affine open and $k \in \mathcal{E}(U)$ and $s \in \mathcal{E}(U)$ be local sections such that $k$ maps to zero in $\mathcal{L}$ and $s$ maps to a generator $\overline{s}$ of $\mathcal{L}$. Then $f = k/s$ is a section of $\mathcal{O}_{\mathbf{P}(\mathcal{E})}$ defined in an open neighbourhood $D_+(s)$ of $s(U)$ in $\pi ^{-1}(U)$. Moreover, since $k$ maps to zero in $\mathcal{L}$ we see that $f$ is a section of the ideal sheaf of $s(U)$ in $\pi ^{-1}(U)$. Thus we can take the image $\overline{f}$ of $f$ in $\mathcal{C}_{\sigma (X)/\mathbf{P}(\mathcal{E})}(U)$. We claim (1) that the image $\overline{f}$ depends only on the sections $k$ and $\overline{s}$ and not on the choice of $s$ and (2) that we get an isomorphism over $U$ in this manner (see below). However, once (1) and (2) are established, we see that the construction is compatible with base change by $U' \to U$ where $U'$ is affine, which proves that these local maps glue and are compatible with arbitrary base change.

To prove (1) and (2) we make explicit what is going on. Namely, say $U = \mathop{\mathrm{Spec}}(A)$ and say $\mathcal{E} \to \mathcal{L}$ corresponds to the map of $A$-modules $M \to N$. Then $k \in K = \mathop{\mathrm{Ker}}(M \to N)$ and $s \in M$ maps to a generator $\overline{s}$ of $N$. Hence $M = K \oplus A s$. Thus

\[ \text{Sym}(M) = \text{Sym}(K)[s] \]

Consider the identification $\text{Sym}(K) \to \text{Sym}(M)_{(s)}$ via the rule $g \mapsto g/s^ n$ for $g \in \text{Sym}^ n(K)$. This gives an isomorphism $D_+(s) = \mathop{\mathrm{Spec}}(\text{Sym}(K))$ such that $\sigma $ corresponds to the ring map $\text{Sym}(K) \to A$ mapping $K$ to zero. Via this isomorphism we see that the quasi-coherent module corresponding to $K$ is identified with $\mathcal{C}_{\sigma (U)/D_+(s)}$ proving (2). Finally, suppose that $s' = k' + s$ for some $k' \in K$. Then

\[ k/s' = (k/s) (s/s') = (k/s) (s'/s)^{-1} = (k/s) (1 + k'/s)^{-1} \]

in an open neighbourhood of $\sigma (U)$ in $D_+(s)$. Thus we see that $s'/s$ restricts to $1$ on $\sigma (U)$ and we see that $k/s'$ maps to the same element of the conormal sheaf as does $k/s$ thereby proving (1). $\square$

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