Alphabeta MathPersonal research
Article items · 05 — External BEM big-site finite Beilinson–Lichtenbaum comparison
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 6–9.
1 · Prerequisites
- Scheme sites and sheafification
- Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes · §2.2, Definition2.5, abstract blowup square(2.6), Remark2.7 and Proposition2.8; Corollary7.20 and its full proof, with Theorem7.16 and Example5.45(1)
- Nisnevich covers and distinguished squares
- Bloch cycle complexes and higher Chow groups
- Face boundaries preserve closed support
- Open restriction of admissible cycles
- The kernel of restriction is the supported subcomplex
- Restriction modulo its kernel
- Kernel models the homotopy fiber when the cokernel is acyclic
- Higher-cycle localization including nonseparated schemes
- Valuative dimension and degreewise bounded towers
- External valuative dimension comparison on locally Noetherian schemes
- The source qcqs cdh site
- Exact valuation inputs for the topology comparison
- Article scheme conventions
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Introduction, pp. 2–3, especially Theorem 1.1 and its opening statement; Section 4 for the construction
- The degree-zero homology quotient
- Negative homology of the cycle complex
- Reindexing a comparison with higher Chow groups
- Classical Chow relations and Bloch boundaries
- Cohomology descends to the derived category
- Functorial finite coefficients for derived complexes
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem 1.1(6); Corollary 7.11 and its proof; comparison map (5.1)
- The dimension formula for affine domains
- The Axiom of Choice
- Roots of unity and étale Tate twists
- Étale morphism of schemes
- From constructible-category hyperdescent to the actual cohomology complex
- Exact hypercomplete arc descent for finite constructible coefficients
- Prime quotients of noetherian rings
- Finite type algebras over fields have bounded dimension
- Valuative dimension can be computed on an affine cover
- Noetherian domains have equal Krull and valuative dimensions
- Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture notes on motivic cohomology · Lecture1 convention, printed p.5; Lecture12 pp.91,96–98: Nisnevich lifting property, Definition12.23, Examples12.24–12.25, Lemma12.26, Proposition12.27 and Proposition12.28.
- Seminormality and seminormalization
- Differential forms in positive characteristic, II: cdh-descent via functorial Riemann–Zariski spaces · Published 2018 version, Algebra & Number Theory 12(3), 649–692; Theorem 1.4 and comparisons (2)–(3), pp. 651–652; Section 2A, Definition 2.1 and Remark 2.2(1), pp. 653–654; Remark 2.7 and Proposition 2.8 statement, pp. 654–655
- Applying nonconnective K-theory to the enhanced interface
- Essential smallness of perfect and supported categories
- Coherent derived tensor and pullback on perfect complexes
- Support ideals, mixed products and pullback
- External nonconnective lax symmetric monoidal K-theory
- Perfect complexes on schemes and their support
- Stable spectra and exact triangles
- One coherent K-flat family and its supplied perfect fibers
- External prime-field cdh comparison and cdh dimension bound
- Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Theorem 4.10(2)–(4), printed pp. 20–21, and proof of part (4)
- Completely decomposed proper maps are abstract-blowup covers
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
TheoremDependency level 6
External BEM big-site finite Beilinson–Lichtenbaum comparison
DefinitionDependency level 7
Completely decomposed proper covers and the cdh topology
DefinitionDependency level 7
Milnor K symbols and the Nisnevich sheaf
DefinitionDependency level 7
Cycle complexes with fixed support
LemmaDependency level 7
The local valuative convention matches the source convention
LemmaDependency level 7
The actual big cdh slice over a field
LemmaDependency level 7
Every qcqs cdh generator is an arc covering family
LemmaDependency level 7
The actual smooth open pair and its Nisnevich square
LemmaDependency level 7
Elmanto–Morrow motivic complex input
LemmaDependency level 7
Degree zero and negative homology of the cycle complex
DefinitionDependency level 8
Motivic complexes and cohomology
LemmaDependency level 8
Elmanto–Morrow weight-one comparison input
LemmaDependency level 8
Bottom supported Bloch groups on a codimension two curve
LemmaDependency level 8
A chosen root trivializes the actual Tate coefficient on the slice
LemmaDependency level 8
Restriction to the slice commutes with big-site localization
LemmaDependency level 8
Pullback and composition of the cdh generating families
LemmaDependency level 8
Finite subfamilies of Nisnevich covers on the finite-type site
LemmaDependency level 8
Restriction of hyperdescent and arbitrary abstract blowup squares
TheoremDependency level 8
Finite valuative dimension for noetherian and finite type schemes
TheoremDependency level 8
External MVW proper-cover identification and two-stage refinement
TheoremDependency level 8
Huber–Kelly representable cdh sheafification input
DefinitionDependency level 9
Nonconnective K theory of perfect complexes
LemmaDependency level 9
Prime-field evaluation and compatibility with the standing theory
LemmaDependency level 9
Restriction of the base field for weight one
LemmaDependency level 9
Elmanto–Morrow rational comparison cofiber
LemmaDependency level 9
A finite-family basis for the generated cdh topology
LemmaDependency level 9
Constant cdh sections on the two article sites
LemmaDependency level 9