TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Grothendieck-Riemann-Roch for projective morphisms

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. Let $k$ be an algebraically closed field and let $f : X \to Y$ be a projective morphism of nonsingular irreducible quasi-projective $k$ -schemes of finite type (Projective morphisms before Proj, Smooth morphism of schemes, Proper morphisms); projectivity is exactly the hypothesis that $f$ factors as a closed immersion $i : X ↪ P_{k}^{n} \times_{k} Y$ followed by the projection $p : P_{k}^{n} \times_{k} Y \to Y$ for some $n \geq 0$ (A projective morphism has a relative Proj presentation). Then for every $α \in K_{0} (X)$ (Grothendieck groups of coherent sheaves and of vector bundles on a scheme, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes): $ch (f_{!} (α)) \cdot td (T_{Y}) = f_{*} (ch (α) \cdot td (T_{X})) in A^{*} (Y)_{Q},$ where $ch, td$ are the Chern character and Todd class of The Chern character and the Todd class, $f_{!}$ is the $K$ -theory pushforward (Pushforward of coherent sheaves in algebraic K-theory) and $f_{*}$ is the proper Chow pushforward (Proper pushforward of cycles and the norm formula). Equivalently $τ_{Y} \circ f_{!} = f_{*} \circ τ_{X}$ for $τ_{S} (α) = ch (α) td (T_{S})$ . For $Y = Spec k$ this is Hirzebruch-Riemann-Roch $χ (X, F) = \deg (ch (F) td (T_{X}))$ (Euler characteristic of a coherent sheaf).

Facts & Assumptions

Given: the Axiom of Choice; an algebraically closed field $k$ ; a projective morphism $f : X \to Y$ of nonsingular irreducible quasi-projective finite type $k$ -schemes; a factorization $f = p \circ i$ with $i : X ↪ P_{k}^{n} \times_{k} Y$ a closed immersion and $p$ the projection.

[L1]

Riemann-Roch holds for the closed immersion $i$ : $ch (i_{*} E) = i_{*} (ch (E) td (N)^{- 1})$ , equivalently $τ \circ i_{*} = i_{*} \circ τ$ (Riemann-Roch for regular embeddings).

[L2]

Riemann-Roch holds for the projection $p$ : $τ_{Y} \circ p_{!} = p_{*} \circ τ_{P^{n} \times_{k} Y}$ (Riemann-Roch for projective-space projections).

[L3]

$K$ -theory pushforward is functorial under composition, $(p \circ i)_{!} = p_{!} \circ i_{!}$ , and Chow proper pushforward is functorial, $(p \circ i)_{*} = p_{*} i_{*}$ ; $K_{0}$ on the smooth quasi-projective source is generated by vector bundles (Pushforward of coherent sheaves in algebraic K-theory, Proper pushforward of cycles and the norm formula, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes).

[L4]

The projection formula and naturality of the Chern character and Todd class for flat morphisms identify the two composites; the target of $τ$ is the codimension-graded Chow group of a smooth scheme (Naturality of the Chow ring and the projection formula, Additivity and multiplicativity of the Chern character and Todd class, Projection formula for higher direct images and K-theory pushforward).

Proof

technique · direct; factor the projective morphism into a closed immersion followed by a projective-space projection and compose the two Riemann-Roch squares

1.1L3givenalgebra

The factorization. By the relative Proj presentation of a projective morphism, $f$ factors as $f = p \circ i$ with $i : X ↪ P_{k}^{n} \times_{k} Y$ a closed immersion of nonsingular varieties and $p$ the projection; both factors are morphisms of nonsingular irreducible quasi-projective $k$ -schemes, and $p$ is the projection to the second factor.

2.1L1L2L3step 1.1algebra

Composition of the two squares. Consider the two commutative squares: $τ \circ i_{*} = i_{*} \circ τ$ on $X \to P^{n} \times_{k} Y$ by [L1], and $τ \circ p_{!} = p_{*} \circ τ$ on $P^{n} \times_{k} Y \to Y$ by [L2]. Composing them gives $τ_{Y} \circ p_{!} \circ i_{!} = p_{*} \circ i_{*} \circ τ_{X}$ , and by the functoriality of both pushforwards, $p_{!} \circ i_{!} = f_{!}$ and $p_{*} i_{*} = f_{*}$ , this is exactly $τ_{Y} \circ f_{!} = f_{*} \circ τ_{X}$ , the displayed formula.

3.1L3L4step 2.1givenalgebra∎

Coherent classes and Hirzebruch-Riemann-Roch. Since $X$ is smooth and quasi-projective, $K_{0} (X) = K^{0} (X)$ and the identity proved for vector bundles extends to all coherent classes by additivity of $ch$ , of $f_{!}$ and of $f_{*}$ along finite resolutions [L3]. For $Y = Spec k$ , the Chow pushforward is the degree and the $K$ -theory pushforward is the Euler characteristic, so the formula becomes Hirzebruch-Riemann-Roch $χ (X, F) = \deg (ch (F) td (T_{X}))$ ; the transport of $ch$ and $td$ to coherent classes uses the naturality and multiplicativity of [L4].

Depends on

Used by

Conventions for the Chow ring and Grothendieck-Riemann-Roch Remark

Dependency tree · two levels

99 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

Borel and Serre, Le theoreme de Riemann-Roch (1958), §7, Lemma 15(a), together with §9 and §15 (standard reference, not scraped)
Ravi Vakil, Math 245 Topics in Algebraic Geometry, Introduction to Intersection Theory, Class 19 (standard reference, not scraped)