13Dxx Homological methods
This subtopic studies homological methods in commutative algebra, including projective resolutions, depth, regularity, and the use of derived techniques.
Specific topics
13D02 Syzygies, resolutions, complexes and commutative rings
Overview
13D02 develops syzygies, resolutions, complexes and commutative rings in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Syzygies, resolutions, complexes and commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for syzygies, resolutions, complexes and commutative rings
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D03 (Co)homology of commutative rings and algebras
Overview
13D03 develops (co)homology of commutative rings and algebras in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
(Co)homology of commutative rings and algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for (co)homology of commutative rings and algebras
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D05 Homological dimension and commutative rings
Overview
13D05 develops homological dimension and commutative rings in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Homological dimension and commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for homological dimension and commutative rings
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D07 Homological functors on modules of commutative rings
Overview
13D07 develops homological functors on modules of commutative rings in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Homological functors on modules of commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for homological functors on modules of commutative rings
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D09 Derived categories and commutative rings
Overview
13D09 develops derived categories and commutative rings in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Derived categories and commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for derived categories and commutative rings
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D10 Deformations and infinitesimal methods in commutative ring theory
Overview
13D10 develops deformations and infinitesimal methods in commutative ring theory in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Deformations and infinitesimal methods in commutative ring theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for deformations and infinitesimal methods in commutative ring theory
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D15 Grothendieck groups, $K$-theory and commutative rings
Overview
13D15 develops grothendieck groups, k-theory and commutative rings in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Grothendieck groups, K-theory and commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for grothendieck groups, k-theory and commutative rings
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D22 Homological conjectures
Overview
13D22 develops homological conjectures in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Homological conjectures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for homological conjectures
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D30 Torsion theory for commutative rings
Overview
13D30 develops torsion theory for commutative rings in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Torsion theory for commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for torsion theory for commutative rings
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Overview
13D40 develops hilbert-samuel and hilbert-kunz functions; poincare series in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Hilbert-Samuel and Hilbert-Kunz functions; Poincare series (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for hilbert-samuel and hilbert-kunz functions; poincare series
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks
13D45 Local cohomology and commutative rings
Overview
13D45 develops local cohomology and commutative rings in homological methods in commutative algebra. It focuses on structural definitions, main theorem families, and methods that connect algebraic invariants with computable criteria.
Related Wikipedia Page
Local cohomology and commutative rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and benchmark examples for local cohomology and commutative rings
- How local/global and categorical viewpoints interact in commutative settings
- Standard homological, finiteness, and multiplicative tools used in proofs
Typical Uses
Used to classify ring and module behavior, transfer properties across constructions, and organize algebraic arguments in geometry and number theory applications.
Applications
- Commutative algebra foundations for algebraic geometry
- Algorithmic and symbolic computations in ring theory
- Interfaces with number theory, logic, and deformation methods
References
Recommended Textbooks