Alphabeta MathSupporting prerequisites · 01 — Derived category of an abelian category
Presents 21 supporting prerequisites, covering published library prerequisites and dependency levels 0–22.
1 · Prerequisites
- The calculus of fractions constructs the localization
- Quasi isomorphisms admit the roof calculus in the homotopy category
- Cochain complex in an abelian category
- Bounded, bounded below, and bounded above complexes
- Partial order and partially ordered set
- Finite, countably infinite, countable, uncountable
- The cardinality | A| of a finite set
- A subset of a finite set is finite, with | B| ≤ | A|, and equality holds if and only if B = A
- Real Coxeter representation and contragredient notation
- Finite parabolic longest elements and disconnected factors
- Root signs, inversions and descent translation
- Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives
- Connected component notation [I,a]
- Coxeter diagrams: edges, labels, components and finite type
- Sign facets of a cone arrangement
- Facet incidence and localized chamber signs
- Linear symmetries act on the Salvetti model
- Coxeter-system, roots, length and weak-order conventions
- Standard parabolic subgroups and parabolic roots
- The incident facet and chamber poset
- The Salvetti order complex and its finite CW case
- Complexified complements in an open convex cone
- Nullhomotopic maps and contractible spaces
- A nonempty space is contractible if and only if its identity map is nullhomotopic
- An abstract simplicial complex
- The geometric realization of an abstract simplicial complex
- Luis Paris, K(pi,1) conjecture for Artin groups (2014) · §3.1, proof of Theorem 3.1: chain extensions and Delta on printed p.387, omega on p.388, Claims 1–7 and 9–15 on pp.388–392; arrangement hypotheses on p.364.
- A free group action has no nonidentity element fixing a point
- Root-permuting normaliser complement
- Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order
- Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink–Howlett groupoids · §1.4, PDF p.4; Proposition 2.10(b),(d) and Proposition 2.11(c), PDF p.23
- Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink–Howlett groupoids · §1.4, PDF p.4; Proposition 2.10(b),(c) and Proposition 2.11(c), PDF p.23
- Standard expressions and their lengths
- Opposite-category convention for Brink–Howlett
- Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank
- The longest element as the opposition of the chamber, and longest elements of finite parabolics
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · Lemma 1.25, opening formula and (d); section 1.27, generator definition and rank-one relation; section 1.4, rank convention.
- Brink–Howlett groupoid
- Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · §§1.4–1.5, PDF p.4; §1.19, PDF p.8; Proposition 3.14(a),(b) and proof, PDF pp.43–44
- Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · Sections 1.3–1.4, especially (1.4.1) and (1.4.4); section 1.21; sections 3.1, 3.3–3.4.
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.4, equations (1.4.3)–(1.4.4), PDF p. 4
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · Proposition 3.13, PDF p. 43; Theorem 1.22(b), PDF p. 9; definition of simple in §1.19, PDF p. 8; §1.4, PDF p. 4
- Every standard expression counts the same parallel classes
- Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.19, PDF p. 8; §§1.20–1.22, PDF pp. 8–9; Lemma 1.25(d) and §1.27, PDF p. 10
- Two-generator minimal common left multiple
- Atomic braid relations
- Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · Sections 1.3–1.5 (especially (1.4.1), (1.4.4) and the first equivalence in 1.5); sections 1.7 and 1.9; Lemma 1.25, opening formula and (a); Lemma 3.12(a) and its proof.
- Brink–Howlett generators and invariance of standard length
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
Definition
Derived category of an abelian category
Definition
Intervals in a poset; locally finite, lower-finite and upper-finite posets
Lemma
Roots of the infinite dihedral geometric representation
Lemma
Component opposition transports the retained simple roots
Lemma
Invariant choices of facet points for finite linear symmetry groups
Lemma
Outside parabolic roots form an invariant set
Lemma
Natural finite Salvetti good covers
Lemma
Connected propagation spans the component
Lemma
Free complement actions have trivial incident-pair stabilizers
Lemma
Disjoint translates of invariant Salvetti cover members
Lemma
Minimum representatives of normaliser cosets preserve the simple roots
Lemma
Root-permuting elements are minimum coset representatives
Lemma
Transfer to the root-action Brink–Howlett groupoid
Lemma
Appending an elementary factor is additive or removes its inverse
Lemma
Parallel classes of roots and finite inversion sets
Lemma
Compressed inversion count is additive for Coxeter-reduced composites
Lemma
Independent arbitrary-rank factorization and simple-class inputs
Lemma
Standard expressions are minimum generator words
Lemma
Two elementary prefixes force a finite rank-two relative root set
Lemma
Standard expressions are connected by atomic braid moves
Lemma