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Article items · 04 — Perfect complexes on schemes and their support
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 4–6.
1 · Prerequisites
- External input on derived enhancements
- Local perfection implies quasi-coherent cohomology
- Restriction, stalks and cohomological support
- Stable supported perfect subcategories and retracts
- Enhanced affine perfect equivalence
- Perfect complexes over a ring and its graded version
- The Axiom of Choice
- The derived category inherits a triangulated structure
- Cohomology descends to the derived category
- Regular equals smooth over a perfect field
- regular local rings are domains and cohen macaulay
- A Noetherian space is a finite union of irreducible closed subsets
- Rational equivalence and the Chow group of cycles
- Serre splitting above the dimension
- Bloch cycle complexes and higher Chow groups
- Reindexing a chain complex cohomologically
- Cohomology after an even cochain shift
- Derived finite coefficients and the Bockstein sequence
- 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
- Finite-dimensional families of generated bundle functionals
- Kernels of surjections of vector bundles
- External finite-type dimension, rational-point and etale-fiber inputs
- Stable spectra and exact triangles
- Article scheme conventions
- Smooth density and normal codimension over a perfect field
- Functions extend across codimension two
- Line bundles are detected off codimension two
- Finite-variable polynomial algebras over fields are Noetherian by finite generators
- Decreasing spectral filtrations and associated graded
- Schemes and morphisms over a base
- Small sites and the plus construction
- The plus construction is a presheaf
- The plus presheaf is separated
- The plus construction turns a separated presheaf into a sheaf
- Universal mapping property of the plus construction
- Reindexing cochains as chains
- Cohomology of an integer shift
- Reindexing commutes with shifts
- Homotopy groups of integer spectral shifts
- Reading spectrum homotopy from an Omega tower
- Eilenberg–Mac Lane spectrum of a chain complex
- Higher-cycle localization including nonseparated schemes
- Separated raw-cycle Zariski descent input
- The kernel of restriction is the supported subcomplex
- Kernel models the homotopy fiber when the cokernel is acyclic
- Marc Levine, Techniques of localization in the theory of algebraic cycles · §2.2 p.6; Theorems1.9 and6.12; §6.11 pp.37–38; §7 pp.43–44, proving Theorem1.7
- Mazza, Voevodsky and Weibel, Lecture Notes on Motivic Cohomology (xprova author PDF) · Definition19.10, Theorem19.11, Proposition19.12 and proof, printedpp168–169/PDF179–180
- Graded pieces of a direct-sum tail tower
- Exact couple
- Derived exact couple
- Cohomological spectral sequence
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
DefinitionDependency level 4
Perfect complexes on schemes and their support
LemmaDependency level 4
Gluing upper bounds through an actual fiber sequence
LemmaDependency level 4
Smooth affine components and gluing a trivial line
LemmaDependency level 4
Cycle-complex degree translation
LemmaDependency level 4
Cohomology of equivalent derived complexes
LemmaDependency level 4
Functorial finite coefficients for derived complexes
LemmaDependency level 4
One nonempty parameter open gives smooth-locus avoidance and transversality
LemmaDependency level 4
Homotopy exactness of a spectral fiber sequence
LemmaDependency level 4
Smooth-locus restriction properties
DefinitionDependency level 5
Multiplicative filtrations and graded products
DefinitionDependency level 5
Scheme sites and sheafification
DefinitionDependency level 5
Valuative dimension and degreewise bounded towers
LemmaDependency level 5
Images of filtration stages
LemmaDependency level 5
Bockstein obstruction to prime torsion
LemmaDependency level 5
The selected zero fibers are finite etale schemes
LemmaDependency level 5
Complex shifts and spectral homotopy indices
LemmaDependency level 5
Stage projections, kernels and image groups
LemmaDependency level 5
Support fibers commute with graded triangles
LemmaDependency level 6
Actual raw simplicial cycles satisfy descent without separatedness
LemmaDependency level 6
Filtered equivalences preserve image filtrations
LemmaDependency level 6
Extending a nonnegative filtration to integer indices
LemmaDependency level 6
Naturality of image filtrations
LemmaDependency level 6
Nesting of image filtration groups
LemmaDependency level 6
Products of filtration images
LemmaDependency level 6
Image filtration for a split tower
LemmaDependency level 6
The source qcqs cdh site
LemmaDependency level 6
Products and projections on image filtrations
PropositionDependency level 6