14Axx Foundations
This subtopic studies foundations of algebraic geometry, including the basic language and principles that organize the subject across examples and general theory.
Specific topics
14A10 Varieties and morphisms
Overview
14A10 studies varieties and morphisms in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.
Related Wikipedia Page
Varieties and morphisms (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for varieties and morphisms
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Foundational setup for advanced algebraic geometry
- Comparison of geometric categories and functors
- Interfaces with arithmetic geometry and moduli theory
References
Recommended Textbooks
14A15 Schemes and morphisms
Overview
14A15 studies schemes and morphisms in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.
Related Wikipedia Page
Schemes and morphisms (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for schemes and morphisms
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Foundational setup for advanced algebraic geometry
- Comparison of geometric categories and functors
- Interfaces with arithmetic geometry and moduli theory
References
Recommended Textbooks
14A20 Generalizations (algebraic spaces, stacks)
Overview
14A20 studies generalizations (algebraic spaces, stacks) in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.
Related Wikipedia Page
Generalizations (algebraic spaces, stacks) (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for generalizations (algebraic spaces, stacks)
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Foundational setup for advanced algebraic geometry
- Comparison of geometric categories and functors
- Interfaces with arithmetic geometry and moduli theory
References
Recommended Textbooks
14A21 Logarithmic algebraic geometry
Overview
14A21 studies logarithmic algebraic geometry in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.
Related Wikipedia Page
Logarithmic algebraic geometry (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for logarithmic algebraic geometry
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Foundational setup for advanced algebraic geometry
- Comparison of geometric categories and functors
- Interfaces with arithmetic geometry and moduli theory
References
Recommended Textbooks
14A22 Noncommutative algebraic geometry
Overview
14A22 studies noncommutative algebraic geometry in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.
Related Wikipedia Page
Noncommutative algebraic geometry (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for noncommutative algebraic geometry
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Foundational setup for advanced algebraic geometry
- Comparison of geometric categories and functors
- Interfaces with arithmetic geometry and moduli theory
References
Recommended Textbooks
14A25 Elementary questions
Overview
14A25 studies elementary questions in foundations of algebraic geometry. It emphasizes basic objects, morphisms, and categorical frameworks that organize modern algebraic geometry from local models to global constructions.
Related Wikipedia Page
Elementary questions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for elementary questions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Foundational setup for advanced algebraic geometry
- Comparison of geometric categories and functors
- Interfaces with arithmetic geometry and moduli theory
References
Recommended Textbooks