\(K\)-Theory

Job: extract stable algebraic invariants of rings, schemes, and spaces — from \(K_0\) projective modules / vector bundles through higher algebraic \(K\) and topological \(K\)-theory with Bott periodicity.

Core

\(K\)-theory measures projective modules (algebra) or vector bundles (topology) up to stable equivalence, then lifts to a spectrum-valued invariant \(K(R)\) or \(KU^*(X)\) with long exact sequences under localization and fibrations.

Layer Content
Algebraic \(K_0,K_1,K_2\) projectives, \(\mathrm{GL}\), Steinberg / symbols
Topological \(K\) complex/real \(K\)-theory; Bott
Higher algebraic \(K\) Quillen \(Q\)-construction / \(+\) construction
Localization localization sequences; dévissage

Reading map

Page Focus
Foundations \(K_0\)–\(K_2\); topological \(K\); Bott; Quillen one-liner
Algebraic \(K\)-theory higher \(K\); localization sequences pointer

Dependency order

  1. Abstract algebra (modules, \(\mathrm{GL}_n\)).
  2. Algebraic topology (for topological \(K\)).
  3. Homological algebra (exact categories; resolutions).
  4. Foundations \(\to\) higher algebraic \(K\) \(\to\) applications (motives, DAG perfect complexes).

Interfaces

Area Contact
Algebraic geometry \(K_0(X)\), Grothendieck–Riemann–Roch
Derived AG \(K(\mathrm{Perf}(X))\)
Algebraic topology \(KU\), \(KO\), Adams operations
Homological algebra Quillen exact categories
Number theory \(K_*(\mathcal{O}_F)\), regulators, zeta values
Representation theory representation rings as \(K_0\)
Prismatic cohomology prismatic / syntomic cohomology adjacent to trace methods (TC/THH)

Notation (paper-standard)

  • \(K_0(R)\), \(K_1(R)=\mathrm{GL}(R)/E(R)\), \(K_2(R)=H_2(E(R),\mathbb{Z})\)
  • \(K_n(R)=\pi_n(BQ\mathrm{P}(R))\) or \(\pi_n(B\mathrm{GL}(R)^+)\)
  • \(KU^0(X)=\mathbb{Z}\times BU(X)\); \(\beta\in KU^{-2}(\mathrm{pt})\) Bott element

See it

Grothendieck–Riemann–Roch. For proper \(f:X\to Y\) of nonsingular varieties, \(\mathrm{ch}(f_!E)\mathrm{td}(T_Y)=f_*(\mathrm{ch}(E)\mathrm{td}(T_X))\) in Chow / cohomology after chern character from \(K_0\). This is the prototype of comparing \(K\)-theory to cycle theory — higher GRR and motivic refinements continue the story.

Check

Why is \(K_0(k)\cong\mathbb{Z}\) for a field \(k\)?

F.g.\ projective \(k\)-modules \(=\) finite-dimensional vector spaces, classified by dimension; \([V]-[W]\mapsto\dim V-\dim W\) gives the isomorphism \(K_0(k)\cong\mathbb{Z}\).

What does Bott periodicity assert for complex topological \(K\)-theory?

\(KU^{*+2}(X)\cong KU^*(X)\) naturally; equivalently \(\Omega^2BU\simeq\mathbb{Z}\times BU\) (weak equivalence after group completion package).

Pitfalls

  • Algebraic \(\neq\) topological \(K\) of \(\operatorname{Spec}\mathbb{R}\) / spaces: related by comparison maps; not isomorphisms in general.
  • \(K_0\) of schemes vs rings: \(K_0(X)\) uses vector bundles / perfect complexes — for singular \(X\) use perfect complexes (\(G\)-theory vs \(K\)-theory).
  • Stabilization: \(K_1\) uses \(\mathrm{GL}=\varinjlim\mathrm{GL}_n\); finite \(n\) gives unstable \(K\)-theory.

Refs

  • Quillen, Higher algebraic \(K\)-theory I
  • Milnor, Introduction to Algebraic \(K\)-Theory
  • Weibel, The \(K\)-book
  • Atiyah, \(K\)-Theory

Next

Foundations · Algebraic \(K\) · Abstract algebra · Algebraic topology · Algebraic geometry · Homological algebra