Alphabeta MathMotivic Hall Algebras and Integration Maps
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
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Linear Algebra
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- Topology of ℝ
Representation Theory of Groups
Scheme Theory
- Affine Group Schemes, Hopf Algebras, and Rational Representations
- Affine Schemes and the Structure Sheaf
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- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
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- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
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Topology
- Compactness
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- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Relative motivic stack-function groups for quiver representation stacks
Remark
Derived and cohomological Hall algebras are distinct constructions
Definition
The motivic Hall algebra of a finite quiver
Definition
Finite-field groupoid cardinality for quiver stack specializations
Lemma
The motive of a quiver extension stack
Theorem
Quiver motivic Hall convolution is associative and unital
Definition
The Euler-twisted dimension-vector algebra of a quiver
Lemma
Finite-field groupoid cardinality respects the motivic stack relations
Remark
Motivic stack Hall and untwisted finitary Hall have different carriers
Theorem
Quiver stack integration is multiplicative with the Euler twist
Sources
- Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, arXiv:math/0503029v2, §§2–3
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §§2.1–3.5
- Bertrand Toën, Derived Hall Algebras, arXiv:math/0501343v4, §§3–4
- Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, arXiv:1006.2706v2, §§2 and 8
- Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, arXiv:math/0503029v2, §5.1
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §§4.1–4.2
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §§2–3
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §4.1 and §7.1
- Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, arXiv:math/0503029v2, §§4–5
- Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, arXiv:math/0503029v2, §5.1, Theorem 5.2
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §4.2, Theorem 4.3
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §§4.3 and 5.2
- Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §§1.2–1.3
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §3
- Yakov Varshavsky, Moduli spaces of principal F-bundles, Lemma 3.3(a) and its proof
- Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §§1.1–1.3
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §5.2 and §§7.1–7.2
- Dominic Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, arXiv:math/0503029v4, §6.1, Theorem 6.1 and proof