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Naturality of the Chow ring and the projection formula

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. For every morphism $f : X \to Y$ of smooth equidimensional finite type schemes over $k$ , $f^{*} : A^{*} (Y) \to A^{*} (X)$ is the graph Gysin pullback, preserves codimension and the unit, is a ring homomorphism, and obeys $(g f)^{*} = f^{*} g^{*}$ . If $f$ is flat it is precisely the previously defined flat pullback. For proper $f$ , proper pushforward shifts codimension by $\dim Y - \dim X$ , obeys $(g f)_{*} = g_{*} f_{*}$ for proper composites, and $f_{*} (f^{*} α \cdot β) = α \cdot f_{*} β$ . Flat/proper composition laws are asserted under their respective hypotheses, not by identifying the two kinds of operation.

Facts & Assumptions

Given: the Axiom of Choice; smooth equidimensional finite type $k$ -schemes $X, Y$ and a morphism $f : X \to Y$ ; for the last assertions a composable morphism $g : Y \to Z$ .

[L1]

The Chow ring structure: $A^{p} (X) = A_{\dim X - p} (X)$ is a commutative graded ring with product $Δ_{X}^{!} (α \times β)$ , unit $[X]$ , graph pullback $f^{*} = Γ_{f}^{!} {pr}_{Y}^{*}$ , and for proper $f$ the projection formula $f_{*} (f^{*} α β) = α f_{*} β$ (The intersection product and Chow ring of a smooth scheme).

[L2]

Graph Gysin pullback is a regular-section Gysin, hence commutes with flat pullback and composes; the graph is a regular immersion and a section of the smooth projection to the source. When $f$ is flat, the regular-immersion-followed-by-smooth-projection identity applied to $Γ_{f} : X ↪ X \times_{k} Y$ and ${pr}_{Y}$ gives $Γ_{f}^{!} {pr}_{Y}^{*} = f^{*}$ for the flat pullback (Refined Gysin operations commute and compose).

[L3]

Proper pushforward of cycles is functorial under composition and shifts degrees by the dimension difference; flat pullback is functorial and preserves codimension (Proper pushforward of cycles and the norm formula, Flat pullback of cycles and of rational equivalence).

Proof

technique · direct; translate each statement into the operational description of the pullback and apply the composition and commutation theorems for refined Gysin

1.1L1L2givenalgebra

Pullback is a unital ring homomorphism. By [L1] the graph pullback $f^{*} = Γ_{f}^{!} {pr}_{Y}^{*}$ is a composite of refined Gysin operations, which preserve codimension by [L2]; it is unital because $Γ_{f}^{!} {pr}_{Y}^{*} [Y] = [Γ_{f}] = [X]$ up to the canonical identification of the graph with $X$ , and the smooth-section identity gives $Γ_{f}^{!} {pr}_{X}^{*} = 1$ . To see that $f^{*}$ is multiplicative, write $α = c [Y]$ and $β = d [Y]$ in the operational description of [L1]: restriction of operational classes is a ring map by [L2], so $f^{*} (α β) = f^{*} (c d [Y]) = c d [X] = f^{*} α f^{*} β$ . Composition $(g f)^{*} = f^{*} g^{*}$ follows from the composition theorem for refined Gysin applied to the composable graphs, with the projection identity of [L2].

2.1L2L3step 1.1algebra

Agreement with flat pullback. If $f$ is flat, apply the regular-immersion/smooth-projection identity of [L2] to $Γ_{f} : X ↪ X \times_{k} Y$ followed by ${pr}_{Y} : X \times_{k} Y \to Y$ . Its composite is $f$ , flat of relative dimension $\dim X - \dim Y$ , so the identity gives $Γ_{f}^{!} {pr}_{Y}^{*} = f^{*}$ for the flat pullback; hence $f^{*}$ agrees with the flat pullback of [L3] on classes of the stated dimensions.

3.1L1L3step 1.1algebra∎

Proper pushforward. If $f$ is proper, [L3] gives functoriality $(g f)_{*} = g_{*} f_{*}$ and the codimension shift $\dim Y - \dim X$ on nonzero cycle classes, since a proper pushforward of a $p$ -dimensional class is supported on images of dimension at most $p$ and the norm-degree formula is multiplicative in towers. The projection formula is the corresponding statement of [L1], with the identification $A^{p} = A_{\dim - p}$ : writing $α = c [Y]$ in operational form, $f_{*} (c β) = c f_{*} β$ is the proper axiom of bivariant classes, which is exactly $f_{*} (f^{*} α β) = α f_{*} β$ .

Naturality of the Chow ring and the projection formula

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources