Alphabeta MathPersonal research
Article items · 15 — Stable triviality fixes the K-zero class and determinant
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 14–16.
1 · Prerequisites
- Stable triviality and stable bundle equivalence
- Isomorphism classes of vector bundles
- Exterior powers and determinant of an algebraic bundle
- The Axiom of Choice
- Affine schemes and their coordinate rings
- Locally Noetherian and Noetherian schemes
- Quasi-compact and quasi-separated schemes
- Every affine scheme is quasi-compact
- Affine schemes are quasi-separated
- Exact sequences of sheaves
- A sheaf on a topological space
- Grothendieck universe closure convention
- Chern classes of a vector bundle on a smooth scheme
- Rational equivalence and the Chow group of cycles
- Projective bundle in the quotient convention
- Symmetric algebra of a quasi-coherent module
- Relative Proj of a graded quasi-coherent algebra
- Intersection with an invertible sheaf and the first Chern class
- The projective bundle formula for Chow groups
- The intersection product and Chow ring of a smooth scheme
- Smooth morphism of schemes
- Locally finite type and finite type morphisms
- Finite-type schemes over a field are qcqs
- Article scheme conventions
- Motivic complexes and cohomology
- A chosen root trivializes the actual Tate coefficient on the slice
- External canonical top-weight motivic-to-syntomic comparison
- External affine étale vanishing at invertible primes
- Derived finite coefficients and the Bockstein sequence
- Primewise detection of torsion
- Roots of unity and étale Tate twists
- Derived global sections on a scheme site
- The characteristic of a ring: the least n ≥ 1 with n · 1_R = 0 when one exists, and 0 otherwise
- A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension
- tⁿ-1 is separable over K exactly when the characteristic does not divide n, and then a splitting field carries n distinct n-th roots of unity
- Cohomology factors through the derived category
- Restriction of the base field for weight one
- Elmanto–Morrow weight-one comparison input
- Low-degree consequences of a weight-one comparison
- Nisnevich covers and distinguished squares
- A field's prime subfield is isomorphic to ℚ in characteristic zero and to Fₚ in characteristic p
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Introduction prime-field convention; Lemma 5.1 statement; Definition 5.2 and map (5.1); Corollary 7.11 statement and short proof
- Finite-dimensional families of generated bundle functionals
- Finite global generation on an affine scheme
- Kernels of surjections of vector bundles
- Normal affine threefolds have controlled nonsmooth strata
- Coordinate multiples of dual generators span all first jets
- The scheme of nontransverse zero jets has small dimension
- One nonempty parameter open gives smooth-locus avoidance and transversality
- The selected zero fibers are finite etale schemes
- Weil divisor normal noetherian scheme
- Coherent module sheaves
- Noetherian commutative rings and modules
- Every algebra of finite type over a Noetherian ring is a Noetherian ring
- Coherent sheaves on a locally Noetherian scheme
- The internal Hom sheaf of two module sheaves
- Kernel sheaves are objectwise, while cokernels and images are sheafified
- Restriction of a sheaf to an open subspace
- Finite morphisms of schemes
- Étale morphism of schemes
- The Stacks Project, method of first-order bad incidence · Situation33.47.2 and Lemma33.47.3, complete proof
- Motivic filtration and image groups
- Nonconnective K theory of perfect complexes
- Smoothness over a field by geometric regularity
- A field has only the zero ideal and itself, hence is Noetherian
- Quasi-compact and quasi-separated morphisms
- The spectrum of a Noetherian ring is a Noetherian topological space
- Noetherian open subsets are quasi-compact
- Stable spectra and exact triangles
- Decreasing spectral filtrations and associated graded
- Support fibers commute with graded triangles
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Sections 2.1–2.2, pp.11–12; Section 3.1, (3.2), pp.12–13; Theorem 3.5(1),(8) and Remark 3.6, pp.14–16; Constructions 4.34 (4.8)–(4.9), 4.36 and Remark 4.37, pp.31–32; Lemma 5.1 and Definition 5.2, pp.33–34; Construction 5.5, p.34; Lemma 5.10, Construction 5.11 and Lemma 5.12, p.36; Theorem 6.1(2), Remark 6.3, Corollary 6.4 with proof and footnote 12, pp.44–45.
- Elmanto–Morrow filtration input
- Elmanto–Morrow motivic complex input
- Spectral sequence of a tower
- Multiplicative filtrations and graded products
- Valuative dimension and degreewise bounded towers
- External stable enhancement and smash product of spectra
- Affine perfect K zero and bundle K zero
- Affine bundles and finite projective modules
- The Grothendieck ring of finite-dimensional vector spaces
- Affine schemes are contravariantly equivalent to commutative rings
- A polynomial ring in n variables over a field has dimension n
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Version 1 (15 September 2023): §2.2 pp.11–12; Definition4.9 and Theorem4.10(1)–(2) pp.21–22; Definition4.23 pp.27–28 and Theorem4.24 p.28; Theorem4.12 pp.22–23 and Theorem4.27 p.29; §3.1 p.13.
- Daniel Dugger, Multiplicative structures on homotopy spectral sequences I · v1, Section 6: Theorem 6.1 p.13; Theorem 6.2, homotopy/local realizability definitions and Proposition 6.4 p.14
- Alice Petronella Hedenlund, Multiplicative Tate Spectral Sequences (PhD thesis, 2020), Part II: Multiplicative spectral sequences via décalage · Part II: Definition II.1.1 (PDF p.204), Section II.1.3 (PDF p.207), Theorem II.1.21 (PDF pp.214–215), Section II.2.1 including Proposition II.2.1 (PDF pp.218–219), Definition II.2.15 (PDF p.230), Proposition II.2.19 (PDF p.231), Theorem II.2.22 (PDF pp.233–234), Theorem II.3.2 (PDF pp.234–235), Definition II.3.4 and Theorem II.3.5 with proof (PDF pp.236–239); PDF page numbers are one-based
- Benjamin Antieau, Spectral sequences, décalage, and the Beilinson t-structure · v1 (14 November 2024), Example 2.13(a) p.6; Definition 3.25 p.10; Construction 4.5 and Remark 4.7 p.14; Definitions 4.8/4.11, Construction 4.10 and Proposition 4.12 pp.15–16; Theorem 4.13, Corollary 4.14 and Remark 4.15 p.16
- Schemes and morphisms over a base
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §2.2 pp.11–12; Theorem4.12(1) p.22 and Theorem4.27(1) p.29; Theorem1.1(1) p.3
- Bloch cycle complexes and higher Chow groups
- Field
- Derived category of an abelian category
- Abelian groups form an abelian category
- Cochain complex in an abelian category
- Finite-type schemes over fields are qcqs
- Motivic cohomology of equicharacteristic schemes · Theorem 1.1(9), PDF page 4 of 74; §3.1, equation (3.1) and slice-filtration discussion, PDF pages 12–13 of 74
- Motivic and filtered K theory with support
- Actual raw simplicial cycles satisfy descent without separatedness
- A coherent spectral realization of derived abelian complexes
- Perfect fields: every irreducible polynomial is separable
- Open immersions of schemes
- Affine open subschemes
- Separated morphism of schemes
- Affine schemes and affine morphisms are separated
- Closed immersions are affine quotients and survive base change
- Localization of a category at a class of morphisms
- From simplicial sequential spectra to the fixed coherent realization
- The prime subfield as the intersection of all subfields
- Schemes
- Open immersions are etale
- Open restriction of admissible cycles
- Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, v1 · Definition 3.1, p.13; §3.3 (3.5), p.14; §3.1 (3.1)–(3.2), pp.12–13; Theorem 3.5(1),(2),(8), Remark 3.6, pp.14–16; Construction 4.36, p.32; Theorem 6.1(2), Remark 6.3, Corollary 6.4 and footnote 12, pp.44–45.
- Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, served v1 HTML · §1.6; (1.9) and Corollary 1.11; §3.1 and §3.2, mapping-spectrum construction, (3.7), and the graded-right-adjoint identity immediately following it; (4.7), Remark 4.12, Definition 4.15 and Remark 4.21; proof of Theorem 6.3, three construction paragraphs immediately preceding (6.4), and the qualification following (6.4); Definition 7.1, Lemma 7.3(1), Theorem 7.5, Remark 7.7, Definition 7.8 and Remark 7.9.
- Marc Levine, The homotopy coniveau tower, arXiv:math/0510334v1 · §§1.1–1.4, pp.7–8 (sequential simplicial spectra and pointwise versus Nisnevich localizations); §1.6 and §§2.1–2.2, pp.9–12 (total spectra and support construction); Theorem 4.1.1 with its construction, pp.20–22; Corollary 4.1.5, p.23; Theorem 5.3.1(3) and its proof, pp.28–29; Corollary 5.3.2 and its proof, pp.29–30; §6.2, pp.31–32; Theorem 6.4.1, pp.35–36; Theorem 6.4.2 and proof, pp.36–37; §6.5 and the first sentence of the proof of Theorem 6.5.1, pp.37–38; Definition 8.1.1, pp.42–43, the suspension formula at the end of §8.1, §8.2 and Remark 8.2.1(2), pp.43–44; Example 8.2.2(3), pp.45–46; Remark 8.3.1, p.46; Theorem 9.0.3 and Remark 9.0.4, p.47; §§10.4–10.5, pp.59–61.
- Jacob Lurie, Higher Topos Theory, homotopy categories and equivalences · Proposition 1.2.3.9 and its proof, printed pp.32–33; §1.2.4, opening definition of equivalence, printed p.33 (PDF indices 49–50).
- Kerodon, Natural Isomorphisms · Subsection 4.4.4, Definition 4.4.4.1 and Theorem 4.4.4.4, with the evaluation-functor paragraph immediately preceding the theorem.
- External actual motivic stagewise boundedness
- General equicharacteristic scope and the standing filtration restriction
- Every algebra of finite type over a Noetherian ring is finitely presented
- Locally finite presentation morphisms
- The vanishing sets define the Zariski topology on the prime spectrum
- The prime spectrum and vanishing sets
- Principal distinguished subsets of the prime spectrum
- A principal localization identifies its spectrum with a distinguished open
- Open and closed immersions are separated
- Separated morphisms compose
- The smooth locus of a morphism
- The smooth locus is open
- Relative Jacobian criterion with its presentation hypothesis
- Smooth morphisms via local standard smooth presentations
- A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective
- Regular equals smooth over a perfect field
- Relative dimension of a smooth morphism at a point
- Differentials of a smooth morphism
- Restricting fibre products to open subschemes
- Integral schemes
- External finite-type dimension, rational-point and etale-fiber inputs
- External Swan finite controlling-family Bertini theorem
- An algebraically closed field: every nonconstant polynomial has a root in the field
- Subalgebra generated by a subset, algebras of finite type, and module-finite algebras
- Locally free sheaves of finite rank
- Affine quasi-coherent sheaves are modules
- Over a Noetherian ring a module is Noetherian exactly when it is finitely generated, exactly when it is finitely presented
- Finite type and finitely presented module sheaves
- Internal Hom from a finitely presented sheaf is quasi-coherent
- Cotangent space at a rational point
- Fibre of a module sheaf at a point
- The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums
- The underlying space of an affine spectrum
- A finite spanning list contains a basis, without choice
- A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit
- Base change of objects, morphisms and properties
- Affine fibre products are spectra of tensor products
- A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point
- Krull dimension of a nonzero ring
- The Stacks Project, geometric integrality after algebraic closure · Lemma 10.49.3, equivalence (1) with (3); Definition 10.49.1 in Section 10.49 (tag 05DW).
- The actual big cdh slice over a field
- Restriction to the slice commutes with big-site localization
- Coherent derived étale sections for the constant and Tate coefficients
- Canonical t structure on a derived category
- Truncation functors, t-cohomology, amplitude and bounded t-structures
- Canonical truncation of a complex
- Canonical truncation is a complex and has the claimed cohomology
- External input on derived enhancements
- Dual conventions for the exterior Euler class
- Euler class of a regular bundle functional
- Localisation of a module at a multiplicative subset
- Localisation commutes with quotient modules and arbitrary direct sums
- Universal property of the tensor product for balanced maps into abelian groups
- Universal property of localisation for modules
- Sections of the associated sheaf on basic opens
- Sections and restrictions on distinguished opens of an affine scheme
- Exterior Algebra Of A Finite Free Module
- Exterior Algebra Basis Monomials
- Over a commutative ring, M⊗_RN is an R-module with r(m⊗ n)=(rm)⊗ n=m⊗(rn)
- External K-flat sheaf resolution and pullback input
- Tensor product of sheaves of modules
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk
- Sheafification is left adjoint to the inclusion of sheaves into presheaves
- The full top-weight map with its exact factorization
- An independent full equivalence into the truncated cdh localization
- Slice localization and the actual root-of-unity coefficient map
- Étale derived cdh locality on the actual qcqs site
- Exactness of ordinary derived cdh sheafification and evaluation
- Canonical truncations fit a distinguished triangle
- External perfect-field Tate cancellation
- External classical coefficient, derived Yoneda and effective-localization interface
- Purity twists and cohomological shifts with explicit premises
- Finite field extensions and etaleness
- Affine morphisms are separated
- Open excision for a descent theory
- Standard smooth presentations and locally standard smooth maps
- Fibres of standard smooth algebras are regular of relative dimension
- Fields are regular local rings
- Existence and basic properties of irreducible components
- The spectrum of a quotient is a closed subspace
- The stalk of the affine structure sheaf at a prime is the localization at that prime
- Rₚ/pRₚcongFrac(R/p) is the residue field at p
- Frac(D) is a field and d↦ d/1 embeds the integral domain D
- A ring map induces a contraction map on prime spectra
- A field finitely generated as a k-algebra is a finite extension of k
- Closed immersions into affine schemes are quotient spectra
- The degree [K:F]=dim_F K of a finite field extension
- Every algebraic extension of a perfect field is separable
- Separable algebraic elements and separable extensions
- Finite-type field schemes lie in the qcqs source domain
- The Stacks Project, Algebraic schemes · Lemma 33.20.3(4), (6), and (10); statement and displayed proof paragraphs (4), (6), and (10).
- Jacob Lurie, Higher Algebra (September 18, 2017) · Definition 7.1.1.2, p. 1199; Proposition 7.1.1.4, p. 1200; Remark 7.1.1.16, p. 1206; Remark 7.1.2.9, p. 1211; Theorem 7.1.2.13 and its proof, pp. 1212–1213 (printed pages).
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes, arXiv:2309.08463v1 · Theorem 1.1(1), p. 3; Section 2.1, p. 11; Definition 4.9 and Theorem 4.10(1)–(2), pp. 21–22; Definition 4.23 and Theorem 4.24(1)–(2), pp. 27–28 (printed pages).
- Quotient and line projectivizations agree after dualizing
- Natural unital bigraded motivic ring with negative zero extension
- The actual EM first Chern class is additive and natural
- The exact projective bundle map in every integer bidegree
- Bigraded cohomology and the projective bundle contract
- Morphisms of schemes
- Category, object, morphism, domain, codomain, identity, composition, and hom-collection
- Pullback of a module along a morphism of ringed spaces
- Affine bundle classes are actual projective-module classes
- Spectral K-zero equality gives an actual projective stabilization
- Common projective complements and cancellation by free modules
- Commutative ring
- The Stacks Project, Noetherian schemes · Lemma 28.5.2, Tag 01OW, implication (1) => (2)
- The actual smooth open pair and its Nisnevich square
- A coherent comparison of the particular open restriction
- Cycle complexes with fixed support
- Support fibers with explicit cohomological indexing
- Affine bundle classes are actual projective-module classes
- Spectral K-zero equality gives an actual projective stabilization
- Common projective complements and cancellation by free modules
- Projective modules and the lifting property
- Sections, restrictions, and global sections of a presheaf
- The Stacks Project, Lemma 28.5.2 (Tag 01OW) · Lemma 28.5.2, implication (1) to (2), and its proof paragraph
- Degree zero and negative homology of the cycle complex
- Cohomology of equivalent derived complexes
- Cycle-complex degree translation
- Filtered equivalences preserve image filtrations
- Extending a nonnegative filtration to integer indices
- Rank-two Chern map and its fixed-invariant fibers
- Group and abelian group
- Monoid homomorphism and group homomorphism
- The kernel and image of a group homomorphism
- Aravind Asok and Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds · Theorem 1 and introduction, pp. 2–3; §2 preliminaries and twists, pp. 4–5; Theorem 3.3, p. 10; Definition 4.1, p. 11; Notation 4.12 and Proposition 4.16, p. 17; Propositions 5.1 and 5.4 and Corollary 5.3, pp. 20–21; Theorem 6.1, Proposition 6.2 and its preceding lift-fiber discussion, pp. 22–24; Propositions 6.3 and 6.5, p. 25; complete proof of Theorem 6.6, p. 26.
- Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, served v1 HTML · §3.2.1 (3.29) and its following smooth-adjunction paragraph; §3.3 (3.36)–(3.43), especially Remark 3.41; Definition 4.2, Remark 4.3 and Remark 4.21.
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
LemmaDependency level 14
Stable triviality fixes the K-zero class and determinant
LemmaDependency level 14
The ordinary Chow Chern map on isomorphism classes
LemmaDependency level 14
Characteristic-zero coefficient vanishing and torsion freeness
PropositionDependency level 14
Motivic weight one
PropositionDependency level 14
Bertini functional with finite smooth zero locus
TheoremDependency level 14
External actual augmented EM and smooth slice tower comparison
TheoremDependency level 14
External actual graded algebra and multiplicative tower spectral sequence
TheoremDependency level 14
External actual motivic stagewise boundedness
TheoremDependency level 14
Elmanto–Morrow smooth comparison input
DefinitionDependency level 15
Images of support groups in global groups
LemmaDependency level 15
Open-natural smooth cycle layers by separated-open descent
LemmaDependency level 15
The precise source stage bound and its homotopy interpretation
LemmaDependency level 15
The actual smooth open pair and its choice scope
LemmaDependency level 15
Augmented tower comparisons preserve supported graded projections
LemmaDependency level 15
Coherent dual extension supplies global Swan controlling sections
LemmaDependency level 15
The root map through derived sections, truncation and localization
LemmaDependency level 15
The actual exterior-dual class in affine perfect K zero
LemmaDependency level 15
Cancellation at the required dimension-bounded evaluation objects
LemmaDependency level 15
Perfect-field cancellation removes exactly d coefficient twists
LemmaDependency level 15
A smooth point has the exact finite-type excision pair
LemmaDependency level 15
Actual cofiber triangles and source grading of the motivic tower
LemmaDependency level 15
Projective bundle closure and naturality of the actual decomposition
LemmaDependency level 15
Local application interface for a critical-rank cancellation theorem
LemmaDependency level 15
Excision and the exact fixed-support cycle fiber
LemmaDependency level 15
Local application interface for a critical-rank cancellation theorem
PropositionDependency level 15
Smooth motivic comparison
TheoremDependency level 15
Asok–Fasel Chern-compatible comparison and lifting input
LemmaDependency level 16