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Powers and sections of a rational surface exceptional ideal

Statement

Assume AC and DC. Let $A$ be a rational permitted normal local surface domain and $X \to Spec A$ a normal projective modification. A coherent globally generated sheaf $F$ on $X$ has $H^{1} (X, F) = 0$ . If the scheme-theoretic closed fibre is Cartier with ideal $I = m O_{X}$ , then $H^{0} (X, I^{n}) = m^{n}$ and $H^{1} (X, I^{n}) = 0$ for all $n \geq 0$ .

Facts & Assumptions

Given: A rational permitted normal local surface domain $A$ , a normal projective modification $X \to Spec A$ , a coherent globally generated sheaf $F$ on $X$ , and the ideal $I = m O_{X}$ when the closed fibre is Cartier.

[F1]

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $F$ all of whose members are nonempty, there exists a function $g$ with domain $F$ satisfying $g (S) \in S$ for all $S \in F$ . (The Axiom of Choice)

[F2]

def-dependent-choice. Let $X$ be a set and let $R \subseteq X \times X$ be a binary relation on $X$ . Call $R$ entire on $X$ when $for every x \in X there is y \in X with x R y .$ The Axiom of Dependent Choice, written $D C$ , is the following statement. (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain)

[F3]

def-globally-generated-sheaf. Let $X$ be a scheme and let $F$ be an $O_{X}$ -module, for instance a quasi-coherent sheaf (def-quasi-coherent-module-scheme). (Global generation by the evaluation map)

[F4]

def-rational-normal-surface-singularity-and-bounded-modification-h1. Assume AC and DC. A normal two-dimensional Noetherian local domain $A$ essentially of finite type over a field or complete equicharacteristic local base defines a rational singularity if $H^{1} (Y, O_{Y}) = 0$ for every normal integral proper modification $Y \to Spec A$ . Bounded modification H1 means these modules have uniformly bounded $A$ -length. (Rational normal surface singularities and bounded modification cohomology)

[F5]

lem-local-normal-surface-modification-dimension-and-projective-cohomology. Assume AC and DC. Let $(A, m)$ be a normal Noetherian local domain of dimension two and $f : X \to Spec A$ an integral modification. Then $X$ has dimension two, all closed points have local dimension two, $f$ is an isomorphism off the closed point, $f_{*} O_{X} = O_{Spec A}$ , and its special fibre has dimension at most one. (Dimension and cohomology of local normal surface modifications)

[F6]

thm-long-exact-sequence-sheaf-cohomology. Assume the Axiom of Choice. Let $0 \to F^{'} \to F \to F^{''} \to 0$ be a short exact sequence of abelian sheaves on a topological space $X$ , and let $H^{q} (X, -)$ be sheaf cohomology computed from the supplied functorial injective resolution datum $I$ on $A b (X)$ (def-sheaf-cohomology-derived-global-sections). (Long exact sequence of sheaf cohomology)

Proof

1.1F3F4F5F6

A globally generated coherent sheaf on the quasi-compact $X$ admits a finite global generating family, so there is a surjection $O_{X}^{r} \to F$ ; the kernel is coherent and has vanishing $H^{2}$ by the two-affine cover of the projective modification, while rationality gives $H^{1} (X, O_{X}) = 0$ , so the long exact sequence gives $H^{1} (X, F) = 0$ .

2.1F4F6step 1.1

Every power $I^{n}$ is globally generated by the finite products of generators of $m$ , hence has vanishing $H^{1}$ ; and $H^{0} (X, I) = m$ , because $H^{0} (X, O_{X}) = A$ and restriction to the nonempty closed fibre has kernel exactly $m$ since units stay units there.

3.1F6step 2.1

Let $x_{1}, \dots, x_{r}$ generate $I$ and let $K$ be the kernel of $O_{X}^{r} \to I$ ; locally one generator is a unit in a frame of $I$ , so the Koszul relations $x_{i} e_{j} - x_{j} e_{i}$ generate $K \otimes I$ , and these are global images of $⋀^{2} O_{X}^{r}$ . Hence $K \otimes I$ is globally generated, and tensoring with the globally generated $I^{n - 2}$ makes $K \otimes I^{n - 1}$ globally generated for $n \geq 2$ , with vanishing $H^{1}$ .

4.1F1F2F4step 3.1∎

The sequence $0 \to K \otimes I^{n - 1} \to (I^{n - 1})^{r} \to I^{n} \to 0$ is then surjective on $H^{0}$ , and induction gives $H^{0} (X, I^{n}) = \sum x_{i} m^{n - 1} = m^{n}$ for every $n \geq 1$ , with $H^{1} (X, I^{n}) = 0$ ; the case $n = 0$ is $H^{0} (O_{X}) = A$ and rationality. The Axiom of Choice and the Axiom of Dependent Choice are inherited from the cited suppliers.

Remarks

Global generation of the powers of the exceptional ideal comes from the finite generating family of the maximal ideal of A.
The Koszul relations are used only to keep the kernels globally generated so that the vanishing theorem applies inductively.

Depends on

Used by

Normality and fibre cohomology of a rational surface point blowup Lemma

Dependency tree · two levels

38 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, Resolution of Surfaces, Sections 54.8–54.9: complete source arguments with local prerequisite replacements (standard reference, not scraped)