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Chow Groups, Intersection Products, and Grothendieck-Riemann-Roch — Examples

1 · Prerequisites

2 · Summary

The companion page to Chow Groups, Intersection Products, and Grothendieck-Riemann-Roch records one counterexample and one worked computation. The counterexample refutes the naive scheme-theoretic preimage recipe on Chow groups: for the blowup of the projective plane at a rational point, the rationally equivalent classes of two distinct rational points have preimages in different degrees, and applying proper pushforward gives the contradiction $[q] = 0$ in $A_{0}$ ; the discussion limits the refutation to the uncorrected recipe and points to flat pullback and refined Gysin as the well-defined replacements. The example computes the Chow ring of projective space as $Z [h] / (h^{n + 1})$ with $h = c_{1} (O (1))$ , identifies the powers of $h$ with linear subspaces, and obtains the Bezout degree formula for products of hypersurface classes, with an explicit disclaimer that the identification with the scheme-theoretic intersection cycle and its local multiplicities is the classical proper-intersection theorem and is not claimed here.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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5 · Examples, counterexamples and false statements

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passOpen item page →

Scheme-theoretic preimages do not define a pullback on Chow groups

Statement refuted

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. Statement refuted. For every morphism $f : Y \to X$ of smooth projective $k$ -varieties, the scheme-theoretic preimage recipe — send an integral closed subscheme $V \subseteq X$ to the cycle $[f^{- 1} (V)]$ of its scheme-theoretic preimage (Cycles of coherent sheaves and of closed subschemes, with flat pullback) and extend the assignment linearly to all cycles — descends to a well-defined homomorphism of abelian groups $A_{*} (X) \to A_{*} (Y)$ ; that is, preimage cycles of rationally equivalent cycles are rationally equivalent.

Facts & Assumptions

Given: the Axiom of Choice; a field $k$ ; the projective plane $X = P_{k}^{2}$ with a rational point $p$ , a second rational point $q \neq p$ , and the blowup $π : Y = {Bl}_{p} X \to X$ with exceptional curve $E = π^{- 1} (p)$ .

[F1]

The blowup is proper and birational; projectivity in this example is verified by the incidence model in step 1.1, rather than inferred from local H-projectivity. Its source is smooth by [F2]; the exceptional divisor is $π^{- 1} (p)$ (Blowup of a scheme along an ideal sheaf, Exceptional subscheme of a blowup, Blowups of finite type ideals are locally H-projective, and proper, Blowing up a nonzero ideal on an integral scheme is birational).

[F2]

Because $p$ is a rational point of the regular surface $X$ , the blowup $Y$ is a smooth projective surface and $E ≅ P_{k}^{1}$ ; $π$ restricts to an isomorphism over $X ∖ {p}$ (Point blowups of regular surfaces stay regular, with rational exceptional fibre over the residue field, The blowup is an isomorphism off the center).

[F3]

In $A_{0} (P_{k}^{2}) ≅ Z$ the class of a closed point $x$ is $[κ (x) : k] [P^{0}]$ ; in particular all $k$ -rational points have the same class, and the degree homomorphism is injective (Chow groups of projective space).

[F4]

The scheme-theoretic preimage of an integral subscheme is its cycle under the fundamental-cycle convention; proper pushforward of cycles is defined by the norm-degree formula and descends to rational equivalence (Cycles of coherent sheaves and of closed subschemes, with flat pullback, Proper pushforward of cycles and the norm formula).

[F5]

For a flat morphism of fixed pure relative dimension the preimage recipe is the flat pullback and is well defined on Chow groups; in general the correction is given by the refined Gysin construction (Flat pullback of cycles and of rational equivalence, Refined Gysin pullback for regular embeddings).

Counterexample

1.1F1F2given

The geometric set-up. Choose homogeneous coordinates $[x : y : z]$ with $p = [0 : 0 : 1]$ . The incidence subscheme $H = {x v = y u} \subset P_{k}^{2} \times P_{k}^{1}$ , with coordinates $[u : v]$ on the second factor, is the blowup: on $z \neq 0$ , its $u \neq 0$ and $v \neq 0$ charts are $Spec k [x, v / u]$ with $y = x (v / u)$ and $Spec k [y, u / v]$ with $x = y (u / v)$ , the two Rees charts for $(x, y)$ ; away from $p$ the incidence projection is an isomorphism. These identifications glue to $H ≅ Y$ . The Segre embedding The Segre-Veronese map is a closed embedding therefore embeds $Y$ as a closed subscheme of projective space; the incidence embedding also proves that $π$ is projective. By [F2] the blowup $π : Y \to X$ at the rational point $p$ is a birational morphism of smooth projective surfaces, its exceptional curve $E = π^{- 1} (p)$ is isomorphic to $P_{k}^{1}$ , and $π$ restricts to an isomorphism $π^{- 1} (X ∖ {p}) \to X ∖ {p}$ ; in particular for the second rational point $q \neq p$ the scheme-theoretic preimage $π^{- 1} (q) = {q^{'}}$ is a single reduced point.

2.1F3step 1.1

The rationally equivalent cycles. In $A_{0} (X) ≅ Z$ the classes of $k$ -rational points are all equal because the degree homomorphism sends each to $[κ (x) : k] = 1$ and is injective: $[p] = [q]$ in $A_{0} (X)$ . The preimage cycles, however, lie in different Chow degrees: $π^{- 1} (p) = E$ is a curve, so $[E] \in Z_{1} (Y)$ , while $π^{- 1} (q) = {q^{'}}$ is a point, so $[q^{'}] \in Z_{0} (Y)$ .

3.1F3F4F5step 2.1∎

The contradiction. If the preimage recipe descended to a homomorphism $φ : A_{*} (X) \to A_{*} (Y)$ , then $[p] = [q]$ would force $φ [p] = φ [q]$ , that is $[E] = [q^{'}]$ in $A_{*} (Y)$ . Apply the proper pushforward $π_{*}$ , which is well defined on rational equivalence by [F4]. Since $π (E) = {p}$ has dimension $0 < 1 = \dim E$ , the norm-degree formula gives $π_{*} [E] = 0$ ; and since $π$ is an isomorphism over $q$ , $π_{*} [q^{'}] = [q]$ . Together with $[E] = [q^{'}]$ this gives $[q] = 0$ in $A_{0} (X)$ , contradicting that $[q]$ has degree $1$ by [F3]. By contrast the flat case of [F5] is well defined on Chow groups, so the failure is exactly the dimension jump of the non-flat morphism. Hence the preimage cycles of the rationally equivalent cycles $[p]$ and $[q]$ are not rationally equivalent, and the scheme-theoretic preimage recipe is not well defined on Chow groups.

Discussion. The obstruction is the jump of fibre dimension at $p$ ; for flat morphisms of fixed pure relative dimension the recipe is the flat pullback of Flat pullback of cycles and of rational equivalence and is well defined, while in general one needs the expected-dimension correction provided by the refined Gysin construction (Refined Gysin pullback for regular embeddings, and the Gysin construction for complete-intersection morphisms beyond the scope of this page). The counterexample refutes only the preimage recipe: it makes no claim about whether some corrected operation can define a pullback for the morphism above.

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The Chow ring of projective space and Bezout degrees

Example

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. Let $k$ be a field and $n \geq 0$ . In the Chow ring $A^{*} (P_{k}^{n})$ (The intersection product and Chow ring of a smooth scheme, Chow groups of projective space) put $h : = c_{1} (O_{P^{n}} (1)) \in A^{1} (P^{n})$ (Intersection with an invertible sheaf and the first Chern class, Twisting sheaf on Proj). Then:

$A^{d} (P_{k}^{n}) = Z \cdot h^{d}$ for $0 \leq d \leq n$ and $A^{d} (P_{k}^{n}) = 0$ for $d > n$ ; under the identification $A^{d} (X) = A_{n - d} (X)$ of the intersection product with the cycle groups, the class $h^{d}$ corresponds to the class $[Λ^{n - d}]$ of a linear subspace of codimension $d$ : equivalently $h^{d} \cap [P^{n}] = [Λ^{n - d}]$ . In particular $A^{*} (P_{k}^{n}) ≅ Z [h] / (h^{n + 1}),$ the isomorphism sending $h$ to $c_{1} (O (1))$ .
(Degrees of products) The degree isomorphism $\deg : A^{n} (P_{k}^{n}) \to Z$ of Chow groups of projective space satisfies $\deg (h^{n}) = 1$ . If $H_{1}, \dots, H_{n} \subseteq P^{n}$ are reduced hypersurfaces of degrees $d_{1}, \dots, d_{n}$ (degree projective hypersurface) with $H_{i} = V (f_{i})$ for nonconstant square-free forms $f_{i}$ of degree $d_{i}$ , then $[H_{i}] = d_{i} h$ in $A^{1} (P^{n})$ and $[H_{1}] \cdot [H_{2}] \dots [H_{n}] = (d_{1} d_{2} \dots d_{n}) h^{n}, \deg ([H_{1}] \dots [H_{n}]) = d_{1} d_{2} \dots d_{n} .$

Discussion. This is the Chow-ring form of Bezout's theorem: the degree of the product of the hypersurface classes is the Bezout number. The identification of this product class with the cycle of the scheme-theoretic intersection of the $H_{i}$ , with its local intersection multiplicities, is the classical proper-intersection theorem and is not claimed here; for plane curves ( $n = 2$ , curves without common components) the corresponding local-multiplicity statement is developed on the plane-curves page.

Verification

Given: the Axiom of Choice; a field $k$ ; $n \geq 0$ ; the projective space $P_{k}^{n}$ with its ample generator $O (1)$ and $h = c_{1} (O (1))$ ; reduced hypersurfaces $H_{1}, \dots, H_{n}$ of degrees $d_{1}, \dots, d_{n}$ .

[L1] The projective bundle formula gives, for a rank- $(n + 1)$ bundle $E$ on a scheme, the isomorphism $⨁_{i = 0}^{n} A_{d + i} (X) \to A_{d + n} (P (E))$ via $ξ$ -caps (The projective bundle formula for Chow groups); the projective space $P_{k}^{n}$ is the projectivization of the free rank- $(n + 1)$ bundle on $Spec k$ (Twisting sheaf on Proj, Invertible twists for degree-one generated rings).

[L2] The Chow ring structure and the identification $A^{d} = A_{n - d}$ are as in The intersection product and Chow ring of a smooth scheme; the cycle groups of projective space are $Z$ in each dimension with $\deg (h^{n}) = 1$ (Chow groups of projective space).

[L3] The cap action of $c_{1} (O (1))$ is cutting with a hyperplane $H$ : for an integral $V \subseteq̸ H$ one has $c_{1} (O (1)) \cap [V] = [V \cap H]$ (Intersection with an invertible sheaf and the first Chern class).

[L4] A reduced hypersurface $H = V (f)$ of degree $d$ has $[H] = d h$ : $c_{1} (O (1))$ is additive and normalized so that the divisor of a degree- $d$ form is $d$ times a hyperplane class (degree projective hypersurface, Chern classes of a vector bundle on a smooth scheme, Additivity, naturality and the splitting principle for Chern classes).

1.1L1L2givenalgebra

The ring. Apply [L1] to the free rank- $(n + 1)$ bundle on $Spec k$ , whose projectivization is $P_{k}^{n}$ : the formula gives $A_{d + n} (P^{n}) = ⨁_{i = 0}^{n} ξ^{i} \cap π^{*} A_{d + i} (k)$ with $ξ = c_{1} (O (1))$ , so $A^{d} (P^{n}) = Z \cdot h^{d}$ for $0 \leq d \leq n$ and $A^{d} = 0$ for $d > n$ ; the relation $h^{n + 1} = 0$ and the absence of other relations give $A^{*} (P_{k}^{n}) ≅ Z [h] / (h^{n + 1})$ with $h \mapsto c_{1} (O (1))$ .

1.2L2L3givenalgebra

Identification with linear subspaces. By induction on $d$ : for a linear subspace $Λ^{n - d + 1}$ and a general hyperplane $H$ of complementary position, $H \supseteq̸ Λ$ and $H \cap Λ = Λ^{n - d}$ is a linear subspace, so [L3] gives $c_{1} (O (1)) \cap [Λ^{n - d + 1}] = [Λ^{n - d}]$ ; starting from $h^{0} \cap [P^{n}] = [P^{n}]$ this shows $h^{d} \cap [P^{n}] = [Λ^{n - d}]$ under the identification $A^{d} = A_{n - d}$ . In particular $\deg (h^{n}) = \deg [Λ^{0}] = 1$ by [L2].

2.1L2L4step 1.2algebra∎

Degrees of products. By [L4] each reduced hypersurface of degree $d_{i}$ has class $[H_{i}] = d_{i} h$ ; multiplicativity of the Chow ring product gives $[H_{1}] \dots [H_{n}] = (d_{1} \dots d_{n}) h^{n}$ , and the degree homomorphism of [L2] sends $h^{n}$ to $1$ , so $\deg ([H_{1}] \dots [H_{n}]) = d_{1} d_{2} \dots d_{n}$ . This is the Bezout number in the Chow ring; the identification with the cycle of the scheme-theoretic intersection with local multiplicities is deliberately not asserted here.