19Dxx Higher algebraic K-theory
This subtopic studies higher algebraic K-theory, extending low-dimensional K-groups to capture deeper structural and homotopical information.
Specific topics
19D06 $Q$- and plus-constructions
Overview
19D06 studies quillen’s higher k-theory in higher algebraic K-theory. It develops higher K-groups using homotopical and categorical constructions that extend K0, K1, and K2.
Related Wikipedia Page
Quillen’s higher K-theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for quillen’s higher k-theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to study deep invariants of rings, schemes, and exact categories through homotopy-theoretic tools.
Applications
- Higher K-groups and exact categories
- Localization and devissage
- Homotopical methods in algebraic K-theory
References
Recommended Textbooks
19D10 Algebraic $K$-theory of spaces
Overview
19D10 studies exact categories and q-construction in higher algebraic K-theory. It develops higher K-groups using homotopical and categorical constructions that extend K0, K1, and K2.
Related Wikipedia Page
Exact categories and Q-construction (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for exact categories and q-construction
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to study deep invariants of rings, schemes, and exact categories through homotopy-theoretic tools.
Applications
- Higher K-groups and exact categories
- Localization and devissage
- Homotopical methods in algebraic K-theory
References
Recommended Textbooks
19D23 Symmetric monoidal categories
Overview
19D23 studies localization in k-theory in higher algebraic K-theory. It develops higher K-groups using homotopical and categorical constructions that extend K0, K1, and K2.
Related Wikipedia Page
Localization in K-theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for localization in k-theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to study deep invariants of rings, schemes, and exact categories through homotopy-theoretic tools.
Applications
- Higher K-groups and exact categories
- Localization and devissage
- Homotopical methods in algebraic K-theory
References
Recommended Textbooks
19D25 Karoubi-Villamayor-Gersten $K$-theory
Overview
19D25 studies devissage and filtration theorems in k-theory in higher algebraic K-theory. It develops higher K-groups using homotopical and categorical constructions that extend K0, K1, and K2.
Related Wikipedia Page
Devissage and filtration theorems in K-theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for devissage and filtration theorems in k-theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to study deep invariants of rings, schemes, and exact categories through homotopy-theoretic tools.
Applications
- Higher K-groups and exact categories
- Localization and devissage
- Homotopical methods in algebraic K-theory
References
Recommended Textbooks
19D35 Negative $K$-theory, NK and Nil
Overview
19D35 studies higher k-groups of rings and schemes in higher algebraic K-theory. It develops higher K-groups using homotopical and categorical constructions that extend K0, K1, and K2.
Related Wikipedia Page
Higher K-groups of rings and schemes (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for higher k-groups of rings and schemes
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to study deep invariants of rings, schemes, and exact categories through homotopy-theoretic tools.
Applications
- Higher K-groups and exact categories
- Localization and devissage
- Homotopical methods in algebraic K-theory
References
Recommended Textbooks
19D45 Higher symbols, Milnor $K$-theory
Overview
19D45 studies homotopy-theoretic methods in k-theory in higher algebraic K-theory. It develops higher K-groups using homotopical and categorical constructions that extend K0, K1, and K2.
Related Wikipedia Page
Homotopy-theoretic methods in K-theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for homotopy-theoretic methods in k-theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to study deep invariants of rings, schemes, and exact categories through homotopy-theoretic tools.
Applications
- Higher K-groups and exact categories
- Localization and devissage
- Homotopical methods in algebraic K-theory
References
Recommended Textbooks
19D50 Computations of higher $K$-theory of rings
Overview
19D50 studies bivariant and relative k-theory in higher algebraic K-theory. It develops higher K-groups using homotopical and categorical constructions that extend K0, K1, and K2.
Related Wikipedia Page
Bivariant and relative K-theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for bivariant and relative k-theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to study deep invariants of rings, schemes, and exact categories through homotopy-theoretic tools.
Applications
- Higher K-groups and exact categories
- Localization and devissage
- Homotopical methods in algebraic K-theory
References
Recommended Textbooks
19D55 $K$-theory and homology; cyclic homology and cohomology
Overview
19D55 studies applications of higher k-theory in higher algebraic K-theory. It develops higher K-groups using homotopical and categorical constructions that extend K0, K1, and K2.
Related Wikipedia Page
Applications of higher K-theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of higher k-theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to study deep invariants of rings, schemes, and exact categories through homotopy-theoretic tools.
Applications
- Higher K-groups and exact categories
- Localization and devissage
- Homotopical methods in algebraic K-theory
References
Recommended Textbooks