# Math 677: Homological Algebra.

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### Title

Homological Algebra.

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### Description

Chain complexes, derived functors, cohomology of groups, ext and tor, spectral sequences, etc. Application to algebraic geometry and algebraic number theory.

## Desired Learning Outcomes

### Minimal learning outcomes

Outlined below are topics that all successful Math 677 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

1. Category Theory
• Definitions and Examples
• Modules
• Snake Lemma
• Opposite Category
• Morphism properties
• Monic/Epic morphism
• Initial/Terminal object
• Zero object
• Kernel/Cokernel
• Product/Coproduct
• Functors/Cofunctors
• Limits/Colimits
• Direct Limits
• Natural transformation
• Functor Categories
• Ab-categories
• Biproducts
2. Abelian Categories
• Left/Right Exact functors
• Freyd-Mitchell Embedding Theorem
• Category of Chain Complexes
• Homology Functor
• Projective/Injective Objects
• Projective/Injective Resolutions
• Comparison Theorem
• Horseshoe Lemma
• Derived Functors
• Ext/Tor
• Derived Functors of the Inverse Limit
• Acyclic Objects/Resolutions
• Univesal delta-functors
3. Spectral Sequences
• Convergence
• Spectral Sequence of a Filtration
• The Classical Convergence Theorem
• Complete Convergence Theorem
• Spectral Sequence of a Double Complex
• Balancing Ext and Tor
• Kuenneth Spectral Sequence
• Universal Coefficient Theorems
• Hyperhomology
• Cartan-Eilenberg Resolutions
• Grothendieck Spectral Sequence
• Spectral Sequence of an Exact Couple
4. Calculating Ext and Tor
• Tor for Abelian Groups
• Tor and Flatness
• Ext for Nice Rings
• Ext and Extensions
5. Dimension
• Projective/Injective Dimension
• Flat Dimension
• Change of Rings Theorems
6. Group Homology/Cohomology
• Shapiro's Lemma
• The Bar Resolution

### Textbooks

Possible textbooks for this course include (but are not limited to):

• Weibel: An introduction to homological algebra