18Gxx Homological algebra
This subtopic studies homological algebra, including chain complexes, derived functors, and derived categories used across modern algebra and geometry.
Specific topics
18G05 Projectives and injectives in homological algebra
Overview
18G05 studies general homological algebra in categories in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
General homological algebra in categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for general homological algebra in categories
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G10 Resolutions; derived functors
Overview
18G10 studies resolutions and derived functors in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Resolutions and derived functors (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for resolutions and derived functors
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G15 Ext and Tor, generalizations, Künneth formula
Overview
18G15 studies spectral sequences and hyperhomology in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Spectral sequences and hyperhomology (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for spectral sequences and hyperhomology
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G20 Homological dimension (category-theoretic aspects)
Overview
18G20 studies homological dimension and exact categories in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Homological dimension and exact categories (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for homological dimension and exact categories
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G25 Relative homological algebra
Overview
18G25 studies derived categories and triangulated structures in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Derived categories and triangulated structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for derived categories and triangulated structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G30 Simplicial sets; simplicial objects in a category
Overview
18G30 studies cohomology theories in algebra and topology in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Cohomology theories in algebra and topology (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for cohomology theories in algebra and topology
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G35 Chain complexes
Overview
18G35 studies differential graded and model categorical homological algebra in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Differential graded and model categorical homological algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for differential graded and model categorical homological algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G40 Spectral sequences, hypercohomology
Overview
18G40 studies homotopical algebra in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Homotopical algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for homotopical algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G45 $2$-homological algebra
Overview
18G45 studies relative homological algebra in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Relative homological algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for relative homological algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G50 Nonabelian homological algebra (category-theoretic aspects)
Overview
18G50 studies computational homological algebra in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Computational homological algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for computational homological algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G55 Homotopical algebra
Overview
18G55 studies applications of homological algebra in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Applications of homological algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of homological algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks
18G60 Other (co)homology theories
Overview
18G60 studies miscellaneous homological algebra topics in homological algebra. It develops categorical tools for exactness, resolution, derived functors, and cohomological methods across algebra and topology.
Related Wikipedia Page
Miscellaneous homological algebra topics (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for miscellaneous homological algebra topics
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to measure algebraic complexity and to organize cohomological constructions.
Applications
- Resolutions and derived functors
- Spectral sequences and triangulated structures
- Homological methods in algebra and topology
References
Recommended Textbooks