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Article items · 17 — Affine étale vanishing and the actual localized evaluation
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 17–20.
1 · Prerequisites
- The Axiom of Choice
- The source qcqs cdh site
- Slice localization and the actual root-of-unity coefficient map
- An independent full equivalence into the truncated cdh localization
- The full top-weight map with its exact factorization
- Cancellation at the required dimension-bounded evaluation objects
- Affine Artin bound and its consequence after localization
- Derived global sections on a scheme site
- Roots of unity and étale Tate twists
- Motivic complexes and cohomology
- Affine schemes and their coordinate rings
- Finite-type field schemes lie in the qcqs source domain
- Canonical truncation of a complex
- Canonical truncation is a complex and has the claimed cohomology
- Elden Elmanto, Zero cycles on singular varieties, Oberwolfach Report 36/2025 · Theorem 3(1) and footnote 1, printed p. 1994; Theorem 4 and its proof, printed p. 1995 (PDF pages 46-47)
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Section 2.1, p. 11; section 3.3, pp. 14-15, Theorem 3.5(3); Theorems 4.10(4), p. 21, and 4.24(3), p. 28; Theorem 4.33(1) and its proof, p. 31
- J. S. Milne, Lectures on Etale Cohomology, version 2.21, 22 March 2013 · Section 15, Theorem 15.1 and preceding definition, p. 105; Schemes with nilpotents, p. 22
- Motivic and filtered K theory with support
- Perfect K theory localization and open excision
- Perfect complexes on schemes and their support
- Perfect residue classes at smooth closed points
- Dévissage of K zero with regular ambient support
- Nonconnective K theory of perfect complexes
- Stable spectra and exact triangles
- Schemes
- Schemes and morphisms over a base
- Morphisms of schemes
- Affine open subschemes
- Open immersions of schemes
- Quasi-compact and quasi-separated schemes
- Closed points of an affine scheme
- The residue field at a point of an affine scheme
- Closed immersions of schemes
- A sheaf on a topological space
- Restricting fibre products to open subschemes
- Smooth morphism of schemes
- Smoothness over a field by geometric regularity
- Locally finite presentation morphisms
- The smooth locus is open
- Every point of a Zariski-open set has a distinguished-open neighbourhood inside it
- A principal localization identifies its spectrum with a distinguished open
- Every affine scheme is quasi-compact
- Subalgebra generated by a subset, algebras of finite type, and module-finite algebras
- Locally finite type and finite type morphisms
- Every algebra of finite type over a Noetherian ring is a Noetherian ring
- Locally Noetherian and Noetherian schemes
- The spectrum of a Noetherian ring is a Noetherian topological space
- Chain dimension and the empty-space convention
- A polynomial ring in n variables over a field has dimension n
- Passing to a quotient does not increase Krull dimension
- A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point
- Regular points of locally Noetherian schemes
- Rank-nullity: dim_F V=nullityT+rankT
- Two finite-dimensional vector spaces over F are linearly isomorphic if and only if they have the same dimension
- Grothendieck universe closure convention
- Separable algebraic elements and separable extensions
- Coherent module sheaves
- Finite type and finitely presented module sheaves
- Kerodon: adjoints and cofinality · Corollary 7.2.3.7(1), tag 02P3
- Kerodon: restriction of limit cones · Corollary 7.2.2.2, tag 02XQ, absolute case with U the functor to a point
- Kerodon: pointwise limits in diagram infinity-categories · Proposition 7.1.8.3(1)-(2), tag 02X9, applied by passage to opposite infinity-categories
- Kerodon: right adjoints preserve limits · Corollary 7.1.4.28, tag 02KE, assertion for the right adjoint G
- Kerodon: universal property of a coherent limit · Definition 7.1.1.1, tag 02VZ, limit clause
- The actual exterior-dual class in affine perfect K zero
- A split trivial line annihilates the exterior and top Chow classes
- Affine bundles and finite projective modules
- Krull dimension of a nonzero ring
- The height of a prime ideal
- Smooth morphisms via local standard smooth presentations
- Projective modules and the lifting property
- Fields of characteristic zero, finite fields, and algebraically closed fields are perfect
- A perfect affine module sheaf admits a finite projective resolution
- Finite-variable polynomial algebras over fields are Noetherian by finite generators
- Every quotient and every localisation of a Noetherian ring is Noetherian
- Finitely generated modules over a left Noetherian ring are Noetherian
- Projective dimension at most n iff the nth syzygy is projective
- The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
- Quotient module M/N with scalar multiplication on additive cosets
- Generated submodule, cyclic and finitely generated modules, module basis and free module
- Modules over a ring form an abelian category
- Noetherian modules: every submodule is finitely generated
- A coherent closed-point skyscraper
- M. Pavaman Murthy, Zero cycles and projective modules, ICTP SMR.761/7 preliminary lecture-note scan · Introduction, printed pp. 1–2 / PDF pp. 3–4; Definition 3.5, Remark 3.6(i), Theorem 3.7 and its short proof, printed p. 17 / PDF p. 19
- Charles Weibel, The K-book, author-hosted Chapter II PDF (cached version identified by SHA256) · Resolution Theorem 7.6 and its proof, chapter/PDF pp. 65–66; Definition 7.7, pp. 66–67; Corollary 7.7.2 and the preceding truncated-cone proof, p. 67
- The ordinary Chow Chern map on isomorphism classes
- Cylinder and zero section of a smooth affine surface
- Chern data survive addition of trivial bundles
- Surface uniqueness from the cylinder and cancellation
- Surface existence from rank-three existence on the cylinder
- Local application interface for a critical-rank cancellation theorem
- Suslin cancellation over affine algebras
- Mohan Kumar–Murthy rank-three Chow classification
- Isomorphism classes of vector bundles
- Article scheme conventions
- Chern classes of a vector bundle on a smooth scheme
- Effective cartier divisor
- Invertible sheaf of cartier divisor
- A field has only the zero ideal and itself, hence is Noetherian
- Dimension and purity of an affine polynomial cylinder
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · Printed p. 2, Section 1, contextual paragraph immediately after Theorem 1, displayed rank-three Chern map; printed p. 4, opening preliminaries for the finite-type convention.
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds (v1) · Printed p. 2, Section 1, opening paragraph before Theorem 1.
- Images of support groups in global groups
- The origin of the affine line has zero perfect K class
- The actual smooth open pair and its choice scope
- Excision and the exact fixed-support cycle fiber
- Normalization of an affine integral curve
- Top cycles supported on an integral curve
- Divisibility of the Picard group of an affine curve
- Cycle complexes with fixed support
- Bloch cycle complexes and higher Chow groups
- Relative dimension of a smooth morphism at a point
- An algebraically closed field: every nonconstant polynomial has a root in the field
- External EM low-weight maps with fixed positive normalization
- Affine perfect K zero and bundle K zero
- Motivic filtration and image groups
- Stage projections, kernels and image groups
- Complex shifts and spectral homotopy indices
- Field-valued points and local-ring points
- The spectrum of a field is a one-point affine scheme
- The Grothendieck ring of finite-dimensional vector spaces
- Quasi-compact and quasi-separated morphisms
- Noetherian topological spaces via ACC on opens or DCC on closed subsets
- Noetherian open subsets are quasi-compact
- Polynomial rings are flat and smooth
- Generic rank and point-supported classes on a curve
- Field
- Integral schemes
- Composition series and length of a module
- Finite type algebras over fields have bounded dimension
- Closed immersions are affine quotients and survive base change
- Nonconnective K theory in degree zero
- Cofiber and triangle presentations of K zero
- Restriction of a sheaf to an open subspace
- External actual EM graded algebra, Picard class and complex PBF
- External symmetric P¹ model, orientation criterion and fundamental stabilization
- Classical and derived smooth Zariski sites over a classical base
- A smooth point has the exact finite-type excision pair
- The actual smooth restriction arrow gives the point-supported cycle fiber
- Perfect-field cancellation removes exactly d coefficient twists
- External classical coefficient, derived Yoneda and effective-localization interface
- Gysin purity for a smooth closed pair
- Support fibers with explicit cohomological indexing
- Perfect fields: every irreducible polynomial is separable
- Separated S-scheme
- Spectral sequence of a tower
- Actual cofiber triangles and source grading of the motivic tower
- Cohomological indexing of the actual graded groups
- The product on graded motivic groups is the source cup product
- External actual graded algebra and multiplicative tower spectral sequence
- Derived exact couple
- A map of exact couples induces a map of spectral sequences
- Multiplicative filtrations and graded products
- Homotopy exactness of a spectral fiber sequence
- Alice Petronella Hedenlund, Multiplicative Tate Spectral Sequences (PhD thesis, 2020), Part II · Section II.2.1 and Proposition II.2.1 (PDF pp.218–219); Definition II.2.15 (p.230); Theorem II.3.2 (pp.234–235); Definition II.3.4 and Theorem II.3.5 with proof (pp.236–239). PDF pages are one-based.
- Benjamin Antieau, Spectral sequences, décalage, and the Beilinson t-structure, arXiv:2411.09115v1 · Definition 3.25 and Remark 3.26, printed pp.10–11; Construction 4.5, p.14; Construction 4.10, Definition 4.11 and Proposition 4.12, pp.15–16; Theorem 4.13, Corollary 4.14 and Remark 4.15, p.16.
- Motivic weight zero on all qcqs schemes
- Etale morphisms are universally open and quasi-finite at every point
- Proper morphisms are closed
- A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps
- For a quotient map q : X → Y, a map out of Y is continuous iff its composite with q is; a continuous map on X constant on the fibres of q factors uniquely through q; and a composite of quotient maps is a quotient map
- Fibre product of schemes
- Existence of all scheme fibre products
- Base change of immersions
- Decreasing spectral filtrations and associated graded
- External stable enhancement and smash product of spectra
- Constant integer coefficients and exact derived evaluation
- Scheme sites and sheafification
- Nisnevich covers and distinguished squares
- A finite spanning list contains a basis, without choice
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- A coherent spectral realization of derived abelian complexes
- Pointed simplicial constructions and sequential model structures
- Weak homotopy equivalence
- The Stacks Project, Lemma 7.47.11 (00Z9) · Lemma 7.47.11 (tag 00Z9), statement and displayed proof, accessed 12 November 2026.
- The Stacks Project, Remark 26.21.18 (0816) · Remark 26.21.18(3) and its derivation, with the affine-intersection criterion in Lemma 26.21.6 of Section 26.21; accessed 12 November 2026.
- Motivic Chern classes
- Integral closure in an extension ring and integrally closed domains
- embedding dimension and regular local ring
- Affine schemes are quasi-separated
- The stalk of the affine structure sheaf at a prime is the localization at that prime
- Every actual motivic stage on the empty scheme is zero
- Valuative dimension and degreewise bounded towers
- The local valuative convention matches the source convention
- General equicharacteristic scope and the standing filtration restriction
- The precise source stage bound and its homotopy interpretation
- Explicit eventual-stage thresholds in every fixed homotopy degree
- Degreewise boundedness and finite collections of eventual vanishings
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · arXiv v1 (15 September 2023; internal date 18 September 2023), Introduction footnote 1 p. 2; Section 2.2 pp. 11–12; Theorem 4.12(1) p. 22 and boundedness paragraph p. 23; Theorem 4.27(1) and boundedness paragraph p. 29
- Quasi-projective morphisms before Proj
- Generic points of irreducible closed subsets
- Classical supported K zero and the current Euler map agree
- Bruno Kahn and Marc Levine, Motives of Azumaya algebras, JIMJ 9 (2010), 481–599, published final PDF · Notation p.487; §§1.1–1.2 pp.487–490; Proposition 2.2.2 p.496; Lemmas 2.3.1–2.3.2 pp.499–500; Definition 3.1.5 p.503.
- Kahn–Levine, published construction of the cycle map and its K/G comparison · Definitions 3.1.9–3.1.15 pp.505–507; Theorem 3.2.4 and its complete proof pp.508–510; Lemma 6.2.1 p.535; Lemmas 6.4.1–6.4.3 pp.539–540.
- Kahn–Levine, positive cycle coefficient and face convention · Section 5.9, printed pp.531–532, before Lemma 5.9.1; for orientation context Theorem 6.5.5 and its proof pp.542–543.
- Marc Levine, The homotopy coniveau tower, arXiv math/0510334v1 · Section8.3 pp.46–47, equation(8.3.1); Theorem9.0.3 and Remark9.0.4 p.47; model conventions pp.44–45
- Flag towers with line-bundle quotients
- Detection on a simultaneous flag tower
- Motivic projective bundle formula
- Locally free sheaves of finite rank
- Pullback of a module along a morphism of ringed spaces
- Rank-two Chern map and its fixed-invariant fibers
- Unique lifting from vanishing groups yields the Chern classification
- Asok–Fasel Chern-compatible comparison and lifting input
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · Theorem 1 and preceding notation, printed p. 2 (PDF page 2); lift-fiber discussion, pp. 23–24; Proposition 6.2, p. 24; Propositions 6.3 and 6.5, p. 25; Theorem 6.6 and its complete proof, p. 26.
- Smooth affine components and gluing a trivial line
- Determinants detect vanishing of the rank-one exterior-dual class
- The actual first Chern map is the Picard isomorphism on an affine curve
- The zero-dimensional critical-rank boundary
- Murthy top Chow cycle comparison with its actual exterior-dual map
- Unimodular elements and trivial summands
- An affine scheme of finite type over a field has a finitely generated coordinate ring
- regular noetherian ring
- Scheme-theoretic fibre
- Integral affine schemes
- Rational equivalence and the Chow group of cycles
- Group and abelian group
- Monoid homomorphism and group homomorphism
- Divisible modules over an integral domain
- Bottom supported Bloch groups on a codimension two curve
- Projective bundle in the quotient convention
- Classical-base comparison of the native smooth site and bundle constructions
- Compactness of the smooth Zariski stack coefficients
- External actual motivic stagewise boundedness
- Degreewise eventually zero spectral towers have zero limit
- External normality and perfect-field smooth-locus inputs
- Irreducible components as schemes
- Weil divisor normal noetherian scheme
- A clopen decomposition of the spectrum comes from a nontrivial idempotent
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk
- External Levine–Weibel quotient cycle map with exact generators
- External determinant and pointed one-type inputs
- Motivic weight one
- Units split the K one determinant
- Units and Picard groups in Nisnevich degree zero and one
- Morphisms to an affine scheme and global sections
- Two-affine projective line and its twists
- The Picard group of the projective line
- Point-supported cycles give all-weight chain purity and low-weight zero
- The degree [K:F]=dim_F K of a finite field extension
- The EM weight-one Picard normalization is positive identity
- Pointed spectrum lifts through a discrete target are unique
- Jacob Lurie, Higher Topos Theory: discrete spaces · Section 5.5.6 opening and Definition 5.5.6.1, pp.492–493; Remark 5.5.6.4, p.494; Proposition 5.5.6.18, p.497 (PDF indices 509–511,514).
- An actual non-A¹ P¹-spectrum for the supplied EM complexes
- Actual EM orientation and projective bundle maps on that model
- External non-A¹ projective-relation filtered and Whitney theorem
- Chern coefficients of the projective relation
- Relative projective space from standard charts
- Relative Jacobian criterion with its presentation hypothesis
- Smoothness survives base change and composition
- Annala–Iwasa, Motivic spectra and universality of K-theory (author PDF dated April 14, 2025) · Definition 4.4.1 and Lemma 4.4.3, printed pp. 30–31; sections 2.1.1 and 2.1.5, p. 14
- External non-A¹ projective-relation filtered and Whitney theorem
- Annala-Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · Definition 4.4.1 and Lemma 4.4.3(ii), including the full proof of Lemma 4.4.3, printed pp. 30–31; geometric conventions §§2.1.1–2.1.5, p. 14; linear orientation §§3.1.3–3.1.5, pp. 21–22; bundle PBF Lemma 3.2.3, p. 23.
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
PropositionDependency level 17
Affine étale vanishing and the actual localized evaluation
PropositionDependency level 17
Excision and K theory purity at a smooth closed point
TheoremDependency level 17
Murthy splitting criterion for the exterior-dual obstruction
TheoremDependency level 17
Classical smooth critical-rank classifications
DefinitionDependency level 18
Smooth-point K theory class map
LemmaDependency level 18
The divisor presentation and actual Picard surjection
LemmaDependency level 18
Weight zero is the locally constant rank function
LemmaDependency level 18
The actual supported Euler class has the generic length coordinate
LemmaDependency level 18
An actual non-A¹ P¹-spectrum for the supplied EM complexes
LemmaDependency level 18
The identified support fiber is the classical relative Hom before Gysin
LemmaDependency level 18
Identifying all spectral-sequence pages with the source multiplicative pages
LemmaDependency level 18
The actual first EM stage is the rank-zero K space on all qcqs schemes
Open problemDependency level 18
Rank-two uniqueness on singular threefolds
PropositionDependency level 18
Eventual homotopy vanishing of the motivic tower
TheoremDependency level 18
External supported slice projection with positive generic coefficient
TheoremDependency level 18
Motivic splitting principle
TheoremDependency level 18
Smooth threefold rank-two classification
TheoremDependency level 18
Smooth critical-rank splitting criterion
LemmaDependency level 19
Divisibility survives the quotient and the support image
LemmaDependency level 19
Positive generic cycles determine the curve-supported homomorphism
LemmaDependency level 19
Actual EM orientation and projective bundle maps on that model
LemmaDependency level 19
Stage bounds imply completeness and finite multiplicative convergence
LemmaDependency level 19
Normal dimension strata and the actual residue class model
LemmaDependency level 19
The globally normalized first-layer Picard map is determinant
LemmaDependency level 19
Motivic purity and lower-weight vanishing at a smooth closed point
LemmaDependency level 20
Actual first-Chern and determinant maps of the first EM layer
LemmaDependency level 20
The source Chern classes are the supplied projective-relation classes
LemmaDependency level 20