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Article items · 02 — Maps to finite disjoint unions of the base
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 0–1.
1 · Prerequisites
- Schemes and morphisms over a base
- Morphisms of schemes
- A Noetherian space is a finite union of irreducible closed subsets
- The Axiom of Choice
- Derived category of an abelian category
- Annihilators, torsion elements and the torsion subset of a module
- Divisible modules over an integral domain
- Exact sequences and short exact sequences of modules
- The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums
- Locally free sheaves of finite rank
- Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated 14 April 2025 (third version) · Section 2.1.1, p. 14; Lemma 3.1.8(i), p. 22; Definition 4.4.1 and Lemma 4.4.3(i), complete lemma proof, pp. 30–31
- Bruno Kahn and Marc Levine, Motives of Azumaya algebras, JIMJ 9 (2010), 481–599, published final PDF · Notation p.487; Proposition 2.2.2 p.496 and compatible tower diagram p.498; 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
- Winfried Bruns and Udo Vetter, Determinantal Rings, LNM1327, Springer1988 · Section2.A, Theorem(2.1), printed p10/PDF18, and proof; Lemma(2.3), printed p11/PDF19.
- The Stacks Project, standard smooth presentations and openness · Algebra Definition10.137.5 and Lemma10.137.6, complete proof; Morphisms Lemmas29.35.10–29.35.11 at https://stacks.math.columbia.edu/tag/01V4.
- Richard G. Swan, A cancellation theorem for projective modules in the metastable range, Invent. Math.27 (1974),23–43 · Printed p23 variety convention; Theorems1.3–1.4 p26, complete proof pp26–28; Corollary1.6 p28. GDZ scans29–34. DOI10.1007/BF01389963; EuDML https://eudml.org/doc/142306.
- Algebraic K-theory of stable ∞-categories via binary complexes · Section 2.2, paragraph immediately preceding Lemma 2.12
- A universal characterization of higher algebraic K-theory · Definition 9.6 and Remark 9.33
- The Stacks Project, More on Algebra, Section 15.129: Splitting off a free module · Theorem 15.129.5 (tag 0GVC), statement and displayed proof
- Mazza–Voevodsky–Weibel, Lecture notes on motivic cohomology: derived coefficient evaluation · Theorem13.1 and Lemma13.4 with full printed proof pp101–102/PDF112–113; Exercise13.5 p102; Proposition13.10 and Example13.11 p103/PDF114; Definition14.6/equation14.6.1 pp115–116; Propositions14.8/14.9, Theorem14.11 and Remark14.12 pp116–117; Proposition14.16 p118/PDF129.
- MVW, Tate coefficients and negative-weight convention · Definition3.1 p23/PDF34; Example8.10 p60/PDF71; Definition14.2 p113/PDF124 and Lemma14.4 p114/PDF125; Sm/k convention p5/PDF16.
- Vladimir Voevodsky, Cancellation Theorem, Documenta Mathematica Extra Volume Suslin(2010),671–685 · Corollary4.10 and complete printed proof, p684/PDF14; adjacent Corollary4.9 and its proof; introduction p671/PDF1.
- Manoj Kumar Keshari, Cancellation problem for projective modules over affine algebras · p. 1, Introduction: definition of cancellative and Theorem 1.1(i); p. 16, reference [20]
- Amalendu Krishna, Murthy’s conjecture on 0-cycles, arXiv1511.04221v3,18March2019 · Section1.9 and opening Section2.1, printed/PDF p6; Cartier curves and quotient equation(2.1),p7; cycle map equation(2.2) and K-theory convention,p8. Intro p1 height-d regular maximal ideals and p2 §1.3 cycle-map/image statement.
- The Stacks Project, Proposition 39.9.11: Abelian varieties · Proposition 39.9.11, parts (3)–(4), and opening proof of part (4)
- The Stacks Project, Section 59.69: Picard groups of curves · Opening paragraphs, before Lemma 59.69.1
- Artin affine étale vanishing, as explicitly stated and applied in Elmanto’s published Zero cycles report · Theorem4 proof, printed1995/PDF47, and affine application printed1996/PDF48; cited primary source SGA4 tome3, Exposé XIV Corollaire3.2
- J.S. Milne, Lectures on Étale Cohomology, version2.21 (22 March2013) · Theorem15.1, p.105; nilpotent invariance of the étale category, p.22
- Manoj Kumar Keshari, Cancellation problem for projective modules over affine algebras · p. 1, Introduction: definition of cancellative and Theorem 1.1(i); p. 16, reference [20]
- The Stacks Project, finite-type dimensions and image closures · Lemmas33.20.3(2),(4),(6),(10), 33.20.4(2) and33.20.5, full indicated proofs; also Lemma33.20.2 for zero-dimensional context.
- The Stacks Project, closed points on finite-type field schemes · Lemma33.14.1, full statement/proof.
- The Stacks Project, etale finite-type algebras over fields · Definition10.143.1; Lemma10.143.3(7), proof paragraph(7); Lemma10.143.4, full statement/proof. Stable individual tags00U2 and00U3.
- The Stacks Project, normal components and Serre normality criterion · Lemma28.7.5 (0357), complete statement/proof; also Section28.12, Lemma28.12.5, https://stacks.math.columbia.edu/tag/033P
- The Stacks Project, regular and smooth loci over perfect fields · Lemma33.25.8 and its complete proof; Lemmas33.25.3–33.25.4 for smoothness/regularity context.
- Affine bundles and finite projective modules
- Exact hypercomplete arc descent for finite constructible coefficients
- Stable triviality and stable bundle equivalence
- Exact cdh factorization inputs for the source étale map
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 0
Maps to finite disjoint unions of the base
LemmaDependency level 0
Derived finite coefficients and the Bockstein sequence
LemmaDependency level 0
Quotient by a divisible subgroup
LemmaDependency level 0
Detecting injectivity by rationalization
LemmaDependency level 0
Kernels of surjections of vector bundles
TheoremDependency level 0
External nilpotence of oriented projective-relation Chern coefficients
TheoremDependency level 0
External supported slice projection with positive generic coefficient
TheoremDependency level 0
External maximal-minor height bound for the component-scope example
TheoremDependency level 0
External relative Jacobian criterion and openness of smooth maps
TheoremDependency level 0
External Swan finite controlling-family Bertini theorem
TheoremDependency level 0
Nonconnective K theory in degree zero
TheoremDependency level 0
Serre free-summand theorem for modules
TheoremDependency level 0
External classical coefficient, derived Yoneda and effective-localization interface
TheoremDependency level 0
External perfect-field Tate cancellation
TheoremDependency level 0
Suslin cancellation over affine algebras
TheoremDependency level 0
External Levine–Weibel quotient cycle map with exact generators
TheoremDependency level 0
Divisibility of rational points of an abelian variety
TheoremDependency level 0
Degree-zero Picard group as points of an abelian variety
TheoremDependency level 0
External affine étale vanishing at invertible primes
TheoremDependency level 0
Suslin cancellation over affine algebras
TheoremDependency level 0
External finite-type dimension, rational-point and etale-fiber inputs
TheoremDependency level 0
External normality and perfect-field smooth-locus inputs
DefinitionDependency level 1
Unimodular elements and trivial summands
LemmaDependency level 1
Local ranks in the affine bundle correspondence
LemmaDependency level 1
Projectivity and rank of a free-summand complement
LemmaDependency level 1
From constructible-category hyperdescent to the actual cohomology complex
LemmaDependency level 1
Stable triviality as the kernel of a right-invertible matrix
LemmaDependency level 1