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Dualizing traces compose and become isomorphisms on rational modifications

Statement

Assume AC and DC. Let $R$ be regular local of dimension two, $A$ finite normal local over $R$ , and let $g : X^{'} \to X$ be a morphism of projective normal modifications over $A$ . Their regular-base dualizing complexes are independent of the chosen projective embeddings up to the unique isomorphism preserving their duality pairings. There is a canonical trace $R g_{*} D_{X^{'}} \to D_{X}$ , and traces compose. If every closed local ring of $X$ is rational, this trace is an isomorphism, giving $g_{*} ω_{X^{'}} = ω_{X}$ , $R^{q} g_{*} ω_{X^{'}} = 0$ for $q > 0$ , and an adjoint evaluation $g^{*} ω_{X} \to ω_{X^{'}}$ . These evaluations compose along rational modifications.

Facts & Assumptions

Given: A regular Noetherian local ring $R$ of dimension two, a finite normal local $R$ -domain $A$ in the permitted class, projective normal modifications $X$ and $X^{'}$ over $A$ with a morphism $g ⁣ : X^{'} \to X$ over $A$ , and the regular-base dualizing complexes $D_{X} = ω_{X} [2]$ , $D_{X^{'}} = ω_{X^{'}} [2]$ .

[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]

lem-projective-regular-local-base-coherent-duality-by-embedding. Assume AC and DC. Let $R$ be regular Noetherian local of dimension $d$ , let $X$ be projective over $R$ , and fix $i : X ↪ P = P_{R}^{N}$ . Put $D_{X} = i^{!} (O_{P} (- N - 1) [N + d])$ . Then $D_{X}$ is a dualizing complex on $X$ , with coherent biduality. (Projective coherent duality over a regular local base)

[F4]

lem-normal-projective-surface-dualizing-module-over-regular-local-base. Assume AC and DC. Let $R$ be a regular Noetherian local ring of dimension two, let $A$ be a finite normal local $R$ -domain of dimension two, with $R ↪ A$ local, and let $X$ be a normal integral scheme of dimension two projective over $R$ , with a proper birational map $f : X \to Spec A$ . Put $ω_{A} = {Hom}_{R} (A, R)$ . (Dualizing modules and trace pairing for normal projective surface modifications)

[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]

lem-surface-modification-isomorphism-in-codimension-one. Assume AC. Let $f : X \to S$ be a modification of integral Noetherian schemes and let $S$ be normal of dimension two. Then $f$ is an isomorphism over an open subset containing every point of codimension at most one in $S$ . The complement is a finite set of closed points. If every fibre is zero-dimensional, $f$ is an isomorphism. (A normal-surface modification is an isomorphism in codimension one)

[F7]

lem-surface-flat-base-change-coherent-cohomology-by-cech. Assume AC and DC. For a quasi-compact separated scheme $X$ over a ring $A$ , a quasi-coherent sheaf $F$ and a flat $A$ -algebra $C$ , the canonical maps $H^{q} (X, F) \otimes_{A} C \to H^{q} (X_{C}, F_{C})$ are isomorphisms for every $q$ . No flatness of $F$ over $A$ is required. (Flat base change for quasi-coherent surface cohomology by Čech)

[F8]

thm-proper-pushforward-coherent. Assume the Axiom of Choice and the Axiom of Dependent Choice, inherited from the affine localization theorem, the Čech comparison and the dévissage lemma cited below (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). (Coherent higher direct images under proper morphisms)

[F9]

thm-serre-vanishing. Assume the Axiom of Choice as inherited from the cited suppliers (The Axiom of Choice). Let $A$ be a Noetherian commutative ring with $1$ , let $X$ be a scheme projective over $A$ in the finite-dimensional H-projective convention (def-projective-morphism-pre-proj): the structure morphism $X \to Spec A$ factors as a closed immersion (Serre vanishing for coherent sheaves and ample twists)

Proof

1.1F3F4given

Evaluation duality on a projective model gives, for every bounded coherent $K$ on $X$ and every $j$ , an isomorphism ${Hom}_{X} (K, D_{X} [j]) ≅ {Hom}_{R} (R Γ (X, K), R [2 + j])$ natural in $K$ ; since the right side is determined by the pair $(R Γ (X, \cdot), R)$ alone, the dualizing objects from any two projective embeddings of this fixed $X$ represent the same duality functor on $D_{C o h}^{b} (X)$ and Yoneda's lemma gives a unique identification preserving the pairings.

2.1F3step 1.1

Taking $K = R g_{*} D_{X^{'}}$ , the same evaluation on $X$ gives ${Hom}_{X} (R g_{*} D_{X^{'}}, D_{X}) ≅ {Hom}_{R} (R Γ (X^{'}, D_{X^{'}}), R [2])$ , while evaluation on $X^{'}$ with $K = D_{X^{'}}$ identifies the right side with ${Hom}_{X^{'}} (D_{X^{'}}, D_{X^{'}})$ ; the image of the identity under this composite is by definition the canonical trace $tr ⁣ : R g_{*} D_{X^{'}} \to D_{X}$ .

3.1F8step 2.1

For a composable second morphism $h ⁣ : X^{''} \to X^{'}$ the identities of $D_{X^{''}}$ , $D_{X^{'}}$ and $D_{X}$ correspond under the same evaluations to ${tr}_{g h}$ , ${tr}_{h}$ and ${tr}_{g}$ , and $R (g h)_{*} = R g_{*} R h_{*}$ identifies the two routes; both composites therefore represent the same element of ${Hom}_{R} (R Γ (X^{''}, D_{X^{''}}), R [2])$ , so traces compose.

4.1F5F6step 2.1step 3.1

Assume now every closed local ring of $X$ is rational. At a point of codimension at most one, $g$ is an isomorphism, so $(R^{q} g_{*} O_{X^{'}})_{x} = 0$ for $q > 0$ ; at a closed point $x$ , the localized modification of the rational local ring $O_{X, x}$ has vanishing $H^{1}$ , while projective normal modifications of a two-dimensional base have no cohomology above degree one, so again $R^{q} g_{*} O_{X^{'}}$ vanishes at $x$ for $q > 0$ ; hence $R g_{*} O_{X^{'}} = O_{X}$ because $g$ is birational and both surfaces are normal.

5.1F3F4F7F8F9step 4.1

For a locally free $F = O_{X} (m)$ the pullback/direct-image adjunction and $R g_{*} O_{X^{'}} = O_{X}$ compute $R Γ (X^{'}, g^{*} F) = R Γ (X, F)$ , so the trace induces isomorphisms ${Hom}_{X} (F, R g_{*} D_{X^{'}} [j]) \to {Hom}_{X} (F, D_{X} [j])$ for all $j$ , by the same evaluation identities; applying the long exact sequence of the cone to negative twists and using Serre vanishing to detect a nonzero highest cohomology sheaf forces that cone to be zero.

6.1F3F4step 5.1

Thus the trace is an isomorphism $R g_{*} ω_{X^{'}} [2] ≅ ω_{X} [2]$ : extracting cohomology sheaves gives $g_{*} ω_{X^{'}} = ω_{X}$ and $R^{q} g_{*} ω_{X^{'}} = 0$ for $q > 0$ , and dualizing the inverse trace through the two evaluation pairings produces the adjoint evaluation $g^{*} ω_{X} \to ω_{X^{'}}$ .

7.1F1F2step 3.1step 6.1∎

When $g$ and $h$ are both morphisms between surfaces with rational closed local rings, the evaluations of step 6.1 compose because the traces of step 3.1 do, giving $(g h)^{*} ω_{X} \to ω_{X^{''}}$ as the composite $(h^{*}) (g^{*})$ ; the Axiom of Choice and the Axiom of Dependent Choice are inherited from the cited duality and modification suppliers.

Remarks

All identifications are fixed by the regular base $R$ and the evaluation pairings, not merely by abstract quasi-isomorphism classes.
The rationality hypothesis enters only through the vanishing of $R^{1} g_{*} O_{X^{'}}$ at closed points.

Depends on

Used by

Dependency tree · two levels

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Sources

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