Tag
Extensions
7 resources tagged “Extensions” across the knowledge library.
7 items
Guide6 Aug 2026Computing Ext GroupsTechniques for computing Ext: choice of variable, use of the long exact sequences, standard computations over the integers, and Ext for cyclic and finitely generated modules.4 min readRead MoreGuide6 Aug 2026Extensions of Modules and the Baer SumExtensions of modules, equivalence of extensions, the Baer sum defined by pullback and pushout, and the resulting abelian group structure on Ext.4 min readRead MoreGuide6 Aug 2026The Ext FunctorExt defined via extensions, via projective resolutions and via injective resolutions, the agreement of the three definitions, and the basic vanishing criteria.4 min readRead MoreGuide6 Aug 2026The Stein–Serre Theorem for Abelian GroupsThe Stein-Serre theorem characterising countable torsion-free abelian groups that are free, via vanishing of Ext, and related Whitehead-type problems.3 min readRead MoreGuide6 Aug 2026The Tensor Product of ModulesThe tensor product by universal property and construction, right exactness, flat modules, and the tensor-hom adjunction.4 min readRead MoreGuide6 Aug 2026The Tor FunctorTor defined by projective resolutions, symmetry in its two variables, vanishing criteria, and the standard computations over the integers.4 min readRead MoreGuide6 Aug 2026The Two Long Exact Sequences of ExtThe long exact sequences of Ext in the first and second variables, their connecting homomorphisms, naturality, and the direction reversal caused by contravariance.3 min readRead More