Supplied projective resolution data

Depends on

• Projective resolutions in an abelian category
• Abelian category

Used by

• Central extensions of perfect groups Corollary
• Kernel of the universal central extension Corollary
• Multiplier of a finitely presented group Corollary
• Nonuniqueness of Schur covers Counterexample
• Ext via a projective resolution of the first variable Definition
• Group homology as a derived functor Definition
• Left derived objects relative to supplied projective resolution data Definition
• Binary icosahedral cover of A5 Example
• Multiplier of a cyclic group Example
• Multiplier of a finite abelian group Example
• All finite Schur covers are unique False statement
• FALSE: enough projectives imply a canonical resolution for every object False statement
• FALSE: universality removes the need for supplied resolution data False statement
• Universal coefficients split naturally False statement
• A morphism has a comparison lift between the supplied projective resolutions Lemma
• Multiplier of a finite group is finite Lemma
• Objectwise comparison of two projective resolution data induces an isomorphism on derived objects Lemma
• Chevalley–Eilenberg cohomology computes Ext of the trivial module Proposition
• Multiplier of a cyclic group Proposition
• Universal central extension groups are superperfect Proposition
• Existence of Schur covering groups Theorem
• Hopf formula for the Schur multiplier Theorem
• Multiplier of an abelian group Theorem
• Universal coefficients in degree two 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

• Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)