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Closed subschemes of projective space and saturated ideals

Statement

Assume the Axiom of Choice as inherited from the affine quotient and gluing suppliers (The Axiom of Choice). Let $A$ be a commutative ring, $n \geq 0$ , $B = A [x_{0}, \dots, x_{n}]$ graded by total degree, and $b = (x_{0}, \dots, x_{n}) = B_{+}$ . For a homogeneous ideal $I \subseteq B$ let $I^{s a t} = ⋃_{r \geq 0} (I : b^{r})$ be its saturation (Saturation detected on projective charts).

Then:

Every homogeneous ideal $I \subseteq B$ determines a closed subscheme $V_{+} (I) ↪ P_{A}^{n},$ whose intersection with the chart $D_{+} (x_{i})$ is $Spec (B_{(x_{i})} / I_{(x_{i})})$ , where $I_{(x_{i})} = (I [x_{i}^{- 1}])_{0}$ ; equivalently $V_{+} (I) = Proj (B / I)$ under the canonical closed immersion $Proj (B / I) \to Proj B = P_{A}^{n}$ .
For homogeneous ideals $I, J \subseteq B$ one has $V_{+} (I) = V_{+} (J)$ as closed subschemes of $P_{A}^{n}$ if and only if $I^{s a t} = J^{s a t}$ .
Every closed subscheme $Z ↪ P_{A}^{n}$ is of the form $V_{+} (I)$ for a unique $b$ -saturated homogeneous ideal $I = I^{s a t}$ ; it is recovered from the chart ideals of $Z$ by the saturation criterion.

Facts & Assumptions

Given: A commutative ring $A$ , an integer $n \geq 0$ , the graded polynomial ring $B = A [x_{0}, \dots, x_{n}]$ , the irrelevant ideal $b = (x_{0}, \dots, x_{n})$ , and homogeneous ideals $I, J \subseteq B$ .

[A1]

The Axiom of Choice states that every family of nonempty sets has a choice function. (The Axiom of Choice)

[F1]

$X = Proj B = P_{A}^{n}$ is covered by the affine charts $D_{+} (x_{i}) = Spec B_{(x_{i})}$ , and $D_{+} (x_{i}) \cap D_{+} (x_{j}) = D_{+} (x_{i} x_{j})$ with coordinate ring $B_{(x_{i} x_{j})}$ , so the overlap of the $i$ -th and $j$ -th charts is the distinguished open of $x_{j} / x_{i}$ in the $i$ -th chart. (Projective space is Proj of a polynomial ring, Saturation detected on projective charts)

[F2]

A homogeneous ideal $I$ has chart ideals $I_{(x_{i})} = (I [x_{i}^{- 1}])_{0}$ , and these localisations are compatible: $(I_{(x_{i})})_{x_{j} / x_{i}} = (I [x_{i}^{- 1}, x_{j}^{- 1}])_{0} = (I_{(x_{j})})_{x_{i} / x_{j}}$ . (Associated sheaf of a graded module on Proj)

[F3]

For a homomorphism $B \to B / I$ of graded rings the induced map $Proj (B / I) \to Proj B$ has on the chart $D_{+} (x_{i})$ the ring map $B_{(x_{i})} \to (B / I)_{(x_{i})} = B_{(x_{i})} / I_{(x_{i})}$ , the chart ideals satisfying $((B / I) [x_{i}^{- 1}])_{0} = B_{(x_{i})} / I_{(x_{i})}$ . (Saturation detected on projective charts)

[F4]

Let $i : Z \to Y$ be a closed immersion. For every affine open $U = Spec R \subseteq Y$ there is a unique ideal $K \subseteq R$ with $i^{- 1} (U) ≅ Spec (R / K)$ ; conversely each quotient $R \to R / K$ induces a closed immersion. Every base change of a closed immersion is a closed immersion, and ideals glue: a family of closed subschemes $Z_{i} ↪ U_{i}$ of an affine cover whose restrictions to the overlaps $U_{i} \cap U_{j}$ agree glues to a closed subscheme $Z ↪ Y$ with $Z \cap U_{i} = Z_{i}$ . (Closed immersions are affine quotients and survive base change, Gluing affine schemes along compatible open isomorphisms)

[F5]

(Saturation criterion.) For a homogeneous $h \in B$ of degree $d$ one has $h \in I^{s a t}$ if and only if $h / x_{i}^{d} \in I_{(x_{i})}$ for every $i$ ; consequently $I$ and $I^{s a t}$ have the same chart ideals, and two saturated homogeneous ideals with equal chart ideals are equal. (Saturation detected on projective charts)

[F6]

The extension of an ideal along a localisation is saturated for the localised elements: if $J = (I_{(x_{j})}) B_{(x_{i} x_{j})}$ and $u = x_{i} / x_{j}$ , then $u^{m} y \in J$ for some $m \geq 0$ implies $y \in J$ . [algebra]

Proof

technique · direct: glue the chartwise quotients of a homogeneous ideal, compare two ideals through their chart ideals and the saturation criterion, and recover any closed subscheme from its chart ideals

1.1F2F3F4

Chartwise closed subschemes of an ideal. Let $I \subseteq B$ be homogeneous. For every $i$ , [F3] identifies the $i$ -th chart of $Proj (B / I)$ with the closed subscheme $Spec (B_{(x_{i})} / I_{(x_{i})}) ↪ Spec B_{(x_{i})} = D_{+} (x_{i})$ , a closed immersion by [F4].

1.2F1F4algebra

Recovering a closed subscheme from its chart ideals. Let $Z ↪ P_{A}^{n}$ be a closed subscheme. By [F1] the affine charts $D_{+} (x_{i})$ cover $P_{A}^{n}$ , and by [F4] applied to them, $Z \cap D_{+} (x_{i}) = Spec (B_{(x_{i})} / K_{i})$ for a unique ideal $K_{i} \subseteq B_{(x_{i})}$ . On the overlap $D_{+} (x_{i} x_{j})$ the two descriptions agree, so $(K_{i})_{x_{j} / x_{i}} = K_{i j} = (K_{j})_{x_{i} / x_{j}}$ inside $B_{(x_{i} x_{j})}$ . Define the homogeneous ideal degreewise by $I = ⨁_{d \geq 0} I_{d}, I_{d} = {h \in B_{d} : h / x_{i}^{d} \in K_{i} for all i} .$ Each $I_{d}$ is an additive subgroup of $B_{d}$ , and if $h \in I_{d}$ and $g \in B_{e}$ are homogeneous, then $(g h) / x_{i}^{d + e} = (g / x_{i}^{e}) (h / x_{i}^{d}) \in K_{i}$ for every $i$ ; hence $g I_{d} \subseteq I_{d + e}$ , and distributivity extends this to arbitrary elements of $B$ . Thus $I$ is a homogeneous ideal.

2.1F1F2F3F4step 1.1

Compatibility on overlaps. For $i \neq j$ the restrictions of the two chartwise subschemes of step 1.1 to the overlaps $D_{+} (x_{i} x_{j})$ of [F1] are cut out by the ideals $I_{(x_{i} x_{j})}$ computed from either side: localising $B_{(x_{i})} / I_{(x_{i})}$ at $x_{j} / x_{i}$ gives $B_{(x_{i} x_{j})} / I_{(x_{i} x_{j})}$ by [F2], and symmetrically from $j$ ; hence the restrictions agree. Therefore by the gluing clause of [F4] the chartwise subschemes of step 1.1 glue to a closed subscheme $V_{+} (I) ↪ P_{A}^{n}$ whose intersection with $D_{+} (x_{i})$ is $Spec (B_{(x_{i})} / I_{(x_{i})})$ , and which is $Proj (B / I)$ under the canonical map of [F3]. This proves (1).

2.2step 1.2F6algebra

The recovered ideal has the prescribed charts. Let $I$ be as in step 1.2. Every element of $(I [x_{i}^{- 1}])_{0}$ has the form $h / x_{i}^{d}$ with $h \in I_{d}$ , hence lies in $K_{i}$ , giving $(I [x_{i}^{- 1}])_{0} \subseteq K_{i}$ . Conversely let $b / x_{i}^{d} \in K_{i}$ with $b \in B_{d}$ . On the overlap with chart $j$ , the same element is $(x_{i} / x_{j})^{- d} (b / x_{j}^{d})$ in $B_{(x_{i} x_{j})}$ , so the equality of localised ideals in step 1.2 shows that $b / x_{j}^{d}$ belongs to $(K_{j})_{x_{i} / x_{j}}$ . By the localisation criterion [F6], there is an exponent $N_{j} \geq 0$ with $(x_{i} / x_{j})^{N_{j}} (b / x_{j}^{d}) = x_{i}^{N_{j}} b / x_{j}^{d + N_{j}} \in K_{j}$ . Since there are only $n + 1$ charts, choose $N \geq N_{j}$ for every $j$ . Then $h = x_{i}^{N} b \in B_{d + N}$ satisfies $h / x_{j}^{d + N} \in K_{j}$ for every $j$ ; in the $i$ -th chart the same follows from $b / x_{i}^{d} \in K_{i}$ . Hence $h \in I_{d + N}$ , and $b / x_{i}^{d} = h / x_{i}^{d + N} \in (I [x_{i}^{- 1}])_{0}$ . Thus $(I [x_{i}^{- 1}])_{0} = K_{i}$ for every $i$ .

3.1F4F5step 2.1

Equality of closed subschemes forces equal saturations. Suppose $V_{+} (I) = V_{+} (J)$ . On the chart $D_{+} (x_{i})$ the two closed subschemes of the affine scheme $Spec B_{(x_{i})}$ coincide, so their ideals coincide: $I_{(x_{i})} = J_{(x_{i})}$ for every $i$ , by the uniqueness of the quotient ideal in [F4]. Then, for homogeneous $h$ of degree $d$ , the criterion [F5] gives $h \in I^{s a t} ⟺ h / x_{i}^{d} \in I_{(x_{i})} \forall i ⟺ h / x_{i}^{d} \in J_{(x_{i})} \forall i ⟺ h \in J^{s a t}$ , so $I^{s a t} = J^{s a t}$ .

4.1F5step 2.1step 3.1

Equal saturations give equal subschemes. Conversely, if $I^{s a t} = J^{s a t}$ then the chart ideals agree, $I_{(x_{i})} = (I^{s a t})_{(x_{i})} = (J^{s a t})_{(x_{i})} = J_{(x_{i})}$ , by the last clause of [F5]; hence the chartwise descriptions of steps 1.1 and 2.1 coincide, and $V_{+} (I) = V_{+} (J)$ . Together with step 3.1 this proves (2).

4.2F4F5step 1.2step 2.2step 3.1cases: saturated and unsaturated ideals

Uniqueness. The ideal $I$ of steps 1.2 and 2.2 is saturated: for homogeneous $h \in B_{d}$ , [F5] and step 2.2 give $h \in I^{s a t} ⟺ h / x_{i}^{d} \in I_{(x_{i})} = K_{i}$ for every $i$ , which is exactly the defining condition $h \in I_{d}$ . Its chart ideals are the $K_{i}$ by step 2.2, so $V_{+} (I) = Z$ by the uniqueness of the affine quotient ideals and gluing [F4]. If $I^{'}$ is another saturated homogeneous ideal with $V_{+} (I^{'}) = Z = V_{+} (I)$ , then $I^{' s a t} = I^{s a t}$ by step 3.1, that is $I^{'} = I$ by saturation. This proves (3).

5.1

Conclusion. Step 2.1 gives statement (1), steps 3.1 and 4.1 give the saturation criterion (2), and steps 1.2, 2.2 and 4.2 show that every closed subscheme arises from a unique saturated homogeneous ideal. The Axiom of Choice [A1] is inherited through the affine quotient lemma and the gluing theorem [F4]; no further choice is made. [A1, F4, step 1.2, step 2.1, step 2.2, step 3.1, step 4.1, step 4.2] \qed

Depends on

Used by

A projective hypersurface as a homogeneous quotient Example
The conic map from O(2) Example
A projective morphism has a relative Proj presentation Lemma
Hypersurface cohomology sequence Lemma
Segre embedding and its line bundle Theorem

Dependency tree · two levels

45 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Constructions of Schemes, Sections 27.8-27.21 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Sections 4.5, 7.4, 9.3, 10.6, 17.4, 17.6, 18.2 (standard reference, not scraped)