Conventions for the Chow ring and Grothendieck-Riemann-Roch

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. (i) Rational equivalence is the version of Rational equivalence and the Chow group of cycles (Stacks 42.19.1, tag 02RW; Fulton 1.3), with order functions on possibly singular integral closed subschemes supplied by The order function of a one-dimensional Noetherian local domain; the equivalent ${\mathbb{P}}^1$-parametrized definition (Fulton 1.6; Stacks 43.8-43.9) is not used but is consistent with it. (ii) $A^{\ast }(X)$ always denotes the codimension-graded intersection ring of a smooth equidimensional $k$-scheme (The intersection product and Chow ring of a smooth scheme); on general schemes $A_{\ast }(X)$ and operational Chern cap maps are used; ring-valued Chern classes, character and Todd classes are obtained through the operational identification on smooth schemes. (iii) The intersection product is the diagonal Gysin construction (Stacks 42.62, tag 0FC0); the moving-lemma/Serre Tor-formula construction of Stacks Chapter 43 is an independent route to the same product for nonsingular projective varieties over an algebraically closed field and is not needed here. (iv) In $K$-theory, $K_0$ and $K^0$ are identified on regular quasi-projective finite-type schemes (Grothendieck groups of coherent sheaves and of vector bundles on a scheme), and pushforward is defined via higher direct images with the AC/DC inheritance declared in Pushforward of coherent sheaves in algebraic K-theory.

Grothendieck-Riemann-Roch for projective morphisms is stated for an algebraically closed base field $k$, nonsingular irreducible quasi-projective $k$-schemes $X,Y$, and a projective (hence proper) morphism $f$; the field hypothesis agrees with the Borel-Serre proof scope. Vakil Classes 14/16/17/19 are partial comparisons with omitted details; the complete proof used here is the current local supplier chain, with the Borel-Serre introduction and Sections 7-16 as the independently retrieved comparison. The design-named Fulton source entry was dropped by the owner with confidence certain after five item-level alternatives per page were recorded; Fulton was not retrieved or read and is retained only as a bibliographical comparison, never as a proof premise. No claim is made here for singular targets, for proper non-projective morphisms, for pairs over a non-algebraically-closed field, or in the $K$-theory of perfect complexes; the singular and bivariant versions (Fulton, Intersection Theory, Chapters 18 and 17) are outside this pair.

The Axiom of Choice (and, where the coherence of higher direct images is invoked, Dependent Choice) is inherited from the published coherent-cohomology suppliers throughout; it is used to form the resolutions and direct images in Pushforward of coherent sheaves in algebraic K-theory and is recorded in Grothendieck groups of coherent sheaves and of vector bundles on a scheme. The order function and the finiteness of the lengths it computes also use the Axiom of Choice, as recorded in The order function of a one-dimensional Noetherian local domain, so the whole Chow-theoretic chain of this page carries the assumption explicitly. No choice-free claim is made stronger here; the definitional part of the Chow groups uses no choice beyond the free abelian group on a set, as recorded in Algebraic cycles and the cycle group of a scheme of finite type over a field.