\(K\)-Theory: Foundations¶
Job: define algebraic \(K_0,K_1,K_2\) and topological \(K\)-theory; record Bott periodicity; one-line Quillen constructions for higher \(K\).
Core¶
Algebraic \(K_0\)¶
For a ring \(R\) (unital), let \(\mathbf{P}(R)\) be f.g.\ projective left \(R\)-modules. The Grothendieck group \(K_0(R)\) is the abelian group generated by isomorphism classes \([P]\) with relations \([P\oplus Q]=[P]+[Q]\).
| Example | \(K_0\) |
|---|---|
| Field \(k\) / PID / local ring | \(\mathbb{Z}\) via rank |
| \(\mathbb{Z}/n\) | \(\mathbb{Z}\) (projectives free) |
| Dedekind domain \(\mathcal{O}_K\) | \(\mathbb{Z}\oplus\mathrm{Cl}_K\) (rank + Steinitz class) |
| \(C(X)\) commutative \(C^*\)-alg | topological \(K^0(X)\) (Swan) |
Schemes. \(K_0(X)=\) Grothendieck group of vector bundles (or of perfect complexes when \(X\) is singular — careful). Resolution theorem: for regular \(X\), coherent sheaves give the same \(K_0\) via finite locally free resolutions.
Functoriality. Ring homs \(R\to S\) induce \(P\mapsto S\otimes_R P\) on \(K_0\). For rings, Morita equivalences induce isomorphisms.
Algebraic \(K_1\)¶
$$ K_1(R)=\mathrm{GL}(R)/E(R),\qquad \mathrm{GL}(R)=\varinjlim_n\mathrm{GL}_n(R), $$ where \(E(R)\) is the subgroup generated by elementary matrices \(e_{ij}(a)=I+aE_{ij}\) (\(i\neq j\)). Whitehead lemma: \(E(R)=[\mathrm{GL}(R),\mathrm{GL}(R)]\) for many rings (e.g.\ Euclidean); in general \(E(R)=[\mathrm{GL},\mathrm{GL}]\) still holds (Whitehead).
| Example | \(K_1\) |
|---|---|
| Field \(k\) | \(k^\times\) via det (\(E=\mathrm{SL}\)) |
| Euclidean domain | \(R^\times\) |
| General commutative \(R\) | \(R^\times\oplus\mathrm{SK}_1(R)\) with \(\mathrm{SK}_1=\mathrm{SL}/E\) |
Determinants. \(\det:K_1(R)\to R^\times_{\mathrm{ab}}\) (abelianization of units for noncommutative \(R\)).
Algebraic \(K_2\)¶
Steinberg group \(\mathrm{St}(R)\): generators \(x_{ij}(a)\) with Steinberg relations mirroring elementary matrices. Natural surjection \(\mathrm{St}(R)\twoheadrightarrow E(R)\); define $$ K_2(R)=\ker\bigl(\mathrm{St}(R)\to E(R)\bigr)=H_2(E(R),\mathbb{Z}) $$ (central extension universal among certain stem extensions).
Matsumoto (fields). For a field \(F\), $$ K_2(F)\cong F^\times\otimes_{\mathbb{Z}}F^\times\big/\bigl\langle a\otimes(1-a):a\neq 0,1\bigr\rangle $$ (Steinberg symbols \(\{a,b\}\)). Relates to Brauer group via Galois symbols / Merkurjev–Suslin (\(K_2(F)/n\cong {}_n\mathrm{Br}(F)\) for \(n\) invertible).
| Symbol identity | Relation |
|---|---|
| \(\{a,1-a\}=1\) | Steinberg |
| \(\{a,b\}\{a,c\}=\{a,bc\}\) | bilinearity |
| \(\{a,b\}\{b,a\}=1\) | anticommutativity (with care at \(2\)) |
Topological \(K\)-theory¶
For compact Hausdorff \(X\), \(\mathbf{Vect}_\mathbb{C}(X)\) under \(\oplus\) yields $$ KU^0(X)=K^0(X)=\text{Grothendieck group of complex vector bundles}. $$ Reduced \(\widetilde{K}^0(X)\) for pointed \(X\). Representability: \(K^0(X)=[X,\mathbb{Z}\times BU]\). Higher groups \(K^{-n}(X)=\widetilde{K}^0(\Sigma^n X_+)\) extend to a cohomology theory.
Bott periodicity (complex). $$ K^{n+2}(X)\cong K^n(X); $$ Bott element \(\beta\in K^{-2}(\mathrm{pt})\cong\widetilde{K}^0(S^2)\) generates and product with \(\beta\) is the isomorphism. Real \(KO\)-theory has period \(8\).
| Computation | Result |
|---|---|
| \(K^0(\mathrm{pt})\) | \(\mathbb{Z}\) |
| \(K^1(\mathrm{pt})\) | \(0\) |
| \(K^0(S^2)\) | \(\mathbb{Z}\oplus\mathbb{Z}\) |
| \(K^0(\mathbb{CP}^n)\) | \(\mathbb{Z}[H]/(H-1)^{n+1}\) |
Chern character. \(\mathrm{ch}:K^0(X)\otimes\mathbb{Q}\to \prod_i H^{2i}(X;\mathbb{Q})\) rational isomorphism (with even cohomology).
Atiyah–Bott–Shapiro / Clifford: periodic \(K\)-homology and index theory — topological \(K\) is the natural home of the Atiyah–Singer index (pointer to mathematical physics / geometry).
Quillen’s constructions (one-liner)¶
Plus construction. \(K_n(R)=\pi_n(B\mathrm{GL}(R)^+)\) for \(n\ge 1\), where \((-)^+\) kills \(E(R)=\mathrm{GL}'\) perfect radical on \(\pi_1\) while preserving homology.
\(Q\)-construction. For an exact category \(\mathcal{M}\) (e.g.\ \(\mathbf{P}(R)\)), Quillen’s category \(Q\mathcal{M}\) has morphisms admissible mono/epi zigzags; \(K_n(\mathcal{M})=\pi_{n+1}(BQ\mathcal{M})\). Recovers \(K_0\) as \(\pi_1\) and agrees with \(+\) for \(\mathbf{P}(R)\).
Higher groups \(K_n(R)\) for \(n\ge 3\) are defined this way; see algebraic.md.
See it¶
Dirichlet unit theorem via \(K_1\). \(K_1(\mathcal{O}_K)\cong\mathcal{O}_K^\times\) (for number rings, \(\mathrm{SK}_1=0\) often); Dirichlet describes \(\mathcal{O}_K^\times\cong\mu\times\mathbb{Z}^{r_1+r_2-1}\). Higher \(K_n(\mathcal{O}_K)\) (Borel) relate to special values of \(\zeta_K\) via regulators — Beilinson conjectures package.
Swan theorem. For compact Hausdorff \(X\), \(K_0(C(X))\cong K^0_{\mathrm{top}}(X)\): f.g.\ projective modules over \(C(X)\) \(\leftrightarrow\) complex vector bundles. Algebraic and topological \(K_0\) meet on this class of rings.
Check¶
Show that a short exact sequence \(0\to P'\to P\to P''\to 0\) of projectives splits, hence \([P]=[P']+[P'']\) is forced by \(\oplus\) relations alone for \(K_0(R)\).
Projectives: \(\mathrm{id}_{P''}\) lifts through \(P\twoheadrightarrow P''\) \(\Rightarrow\) section \(\Rightarrow\) \(P\cong P'\oplus P''\). Thus additivity for SES is automatic in \(\mathbf{P}(R)\); for coherent sheaves on schemes one needs the full relation \([M]=[M']+[M'']\) for SES in the definition of \(G_0\).
Why is \(\{a,-a\}=1\) in \(K_2(F)\) for a field \(F\)?
From Steinberg and bilinearity: \(\{a,-a\}=\{a,(-1)a\}=\{a,-1\}\{a,a\}\); \(\{a,a\}=\{a,-1\}\) because \(\{a,a\}=\{a,(-1)a\cdot(-1)\}=\ldots\) standard identity \(\{a,a\}=\{a,-1\}\) and \(\{a,-1\}^2=1\) yields \(\{a,-a\}=1\). (Write full expansion in Matsumoto presentation.)
Pitfalls¶
- \(G_0\) vs \(K_0\): \(G_0\) uses all f.g.\ modules / coherent sheaves; equals \(K_0\) for regular rings/schemes (resolution theorem), not in general.
- Noncommutative rings: det less powerful; \(K_1\) abelianization story changes.
- Topological \(K^1\) vs algebraic \(K_1\) of \(C(X)\): related by comparison but algebraic \(K_1(C(X))\) is subtler than \([X,U]\).
- Bott period \(2\) is complex: real \(KO\) is period \(8\) — do not mix.
Refs¶
- Milnor, Introduction to Algebraic \(K\)-Theory
- Quillen, Higher algebraic \(K\)-theory I (Lecture Notes in Math.\ 341)
- Atiyah, \(K\)-Theory; Karoubi, \(K\)-Theory
- Weibel, The \(K\)-book
Next¶
K-theory index · Algebraic \(K\) · Modules · Algebraic topology · Algebraic NT · AG foundations · NCG / Baum–Connes — the Baum–Connes assembly maps to \(K_\bullet(C^*_r(G))\)