Engineering↗
Derived Functors
Handbook guide to Derived Functors, covering core definitions, key results, formulas, reasoning methods and worked examples in graduate abstract algebra.
Tag
derived functors
Every page in the KEVOS library tagged derived functors. 8 pages.
Engineering↗
Chain Complexes and Homology
Chain and cochain complexes, cycles and boundaries, homology as a functor, and the abelian category of complexes.
Engineering↗
Chain Homotopy and the Comparison Theorem
Chain homotopy between chain maps, homotopy equivalence, the comparison theorem for projective resolutions, and independence of derived functors from the chosen resolution.
Engineering↗
Change of Rings
Restriction and extension of scalars, induced and coinduced modules, the change-of-rings theorems, and the associated spectral sequences.
Engineering↗
Ext via Projective and Injective Resolutions
Computing Ext by resolving either variable, the double complex proof that the two agree, and the practical consequences for choosing a computation.
Engineering↗
The Long Exact Homology Sequence
The long exact sequence associated to a short exact sequence of complexes, construction of the connecting homomorphism, naturality, and the standard applications.
Engineering↗
The Long Exact Sequences of Derived Functors
Derivation of the long exact sequences of derived functors via the horseshoe lemma, their naturality, dimension shifting and the standard computational patterns.
Engineering↗
Yoneda Ext and n-Fold Extensions
The Yoneda description of Ext^n by n-fold extensions, the equivalence relation, splicing as composition, and the Yoneda product.
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