Mathematics / research
Homological algebra and derived constructions
Study exactness and its failures through chain complexes, resolutions, derived functors, and spectral sequences.
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Inside this module
Module 01 research
Homological algebra and derived constructions
Study exactness and its failures through chain complexes, resolutions, derived functors, and spectral sequences.
- 01
Chain complexes, chain maps, and homotopies
Separate differential identities, induced cohomology maps, and chain-level homotopy data.
27 min - 02
Exact sequences and connecting homomorphisms
Construct the boundary map by lifting a cycle and identify its obstruction to a closed lift.
28 min - 03
Projective, injective, and flat modules
Distinguish lifting, extension, and tensor exactness through concrete module tests.
29 min - 04
Projective and injective resolutions
Replace a module by an exact augmented complex and track the choices through homotopy.
30 min - 05
Tensor products and Tor functors
Measure lost tensor exactness using resolutions, torsion kernels, and balanced arguments.
31 min - 06
Hom and Ext functors
Compute derived Hom with the correct variance and interpret first Ext through extensions.
31 min - 07
Derived categories and quasi-isomorphisms
Localize the homotopy category while retaining the replacement data needed for derived functors.
33 min - 08
Triangulated structures and mapping cones
Build a signed cone, verify its differential, and use triangles without treating them as arbitrary exact sequences.
34 min - 09
Double complexes and spectral sequences
Totalize with a checked sign and read a convergent filtered calculation without losing extension data.
35 min - 10
Koszul complexes and local algebra
Build an exterior-algebra complex and locate the regularity hypotheses that make it a resolution.
34 min