Free abelian group on a set

Depends on

• Group and abelian group
• Monoid homomorphism and group homomorphism

Used by

• Algebraic cycles and the cycle group of a scheme of finite type over a field Definition
• Grothendieck group of an essentially small abelian category Definition
• Grothendieck group of an essentially small triangulated category Definition
• Split Grothendieck group of an additive category Definition
• Symmetrizable Cartan data for quantum groups Definition
• System of fundamental units Definition
• The Cayley graph of ℤ for the generating set {1} is a line and its word metric is |m-n| Example
• The Cayley graph of ℤⁿ for the standard basis is the integer lattice, and its word metric is the sum of coordinate differences Example
• The free-abelian-group monad sends a set to its finite formal integer combinations Example
• The subgroup 2ℤ×ℤ has index two in ℤ² and its inclusion is a quasi-isometry Example
• Finite-support regularization of quantum label-weight modules Lemma
• Rational-local quantum halves and torus specialize at q=1 Lemma
• Shift signs and exact-functor maps on triangulated K0 Lemma
• The first potentially nonzero homotopy group of a wedge of higher spheres has its cell basis Lemma
• The unmodified integral torus has a quantum-binomial basis Lemma
• Ultra-tight multisets admit integer column potentials Lemma
• Free abelian groups of rank at least two are not hyperbolic Proposition
• Aut(ℤⁿ)≅ GLₙ(ℤ) for every finite rank n Theorem
• G0 of an abelian category equals triangle K0 of its bounded derived category Theorem
• Shift-orbit bases for graded simple and projective classes Theorem
• Simple classes freely generate the Grothendieck group of a length category Theorem
• The abelianisation of a free group on X is a free abelian group on X Theorem
• Triangle K0 of perfect complexes equals split K0 of finite projectives Theorem
• Universal properties and functoriality of G0 and split K0 Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• John McKernan, Presentations and Groups of Small Order, Lecture 12 (standard reference, not scraped)