Higher Algebraic $K$-Theory¶

Job: define $K_n$ for $n\ge 0$ via Quillen; record fundamental localization / dévissage / resolution theorems used in geometry and number theory.

Core¶

Higher $K$-groups¶

For a ring $R$, $$ K_n(R)=\pi_n\bigl(B\mathrm{GL}(R)^+\bigr)\quad(n\ge 1),\qquad K_0(R)=\text{as before}, $$ or equivalently $K_n(R)=\pi_{n+1}(BQ\mathbf{P}(R))$ via the $Q$-construction. For an exact category $\mathcal{M}$, $K_n(\mathcal{M})=\pi_{n+1}(BQ\mathcal{M})$.

Waldhausen $S_\bullet$-construction. For categories with cofibrations and weak equivalences (e.g.\ perfect complexes with q.i.), $K$ is defined as loops on $|wS_\bullet\mathcal{C}|$; agrees with Quillen on exact categories and extends to schemes via $\mathrm{Perf}(X)$.

Input	$K$-theory spectrum
Ring $R$	$K(R)$ with $\pi_n=K_n(R)$
Scheme $X$	$K(X)=K(\mathrm{Perf}(X))$
Exact category	$K(\mathcal{M})$
dg-category / stable $\infty$-cat	noncommutative motives package (BGT, …)

Fundamental theorems (Quillen)¶

Resolution theorem. If $\mathcal{M}\subset\mathcal{P}$ exact categories and every object of $\mathcal{P}$ has a finite resolution by objects of $\mathcal{M}$, then $K(\mathcal{M})\xrightarrow{\sim}K(\mathcal{P})$ under standard hyp. $\Rightarrow$ for regular Noetherian $R$, $G_n(R)\cong K_n(R)$ where $G_n=K_n(\mathbf{M}(R))$ (f.g.\ modules).

Dévissage. If $\mathcal{A}\subset\mathcal{B}$ abelian, every object of $\mathcal{B}$ has a finite filtration with quotients in $\mathcal{A}$, then $K(\mathcal{A})\xrightarrow{\sim}K(\mathcal{B})$ (e.g.\ finite length modules over Artinian rings: reduce to residue fields).

Additivity. Exact sequences of functors induce relations $[F]=[F']+[F'']$ on $K$-theory spectra (Waldhausen).

Localization sequences (pointer)¶

Quillen localization (rings). For multiplicative $S\subset R$ central Ore (or commutative), there is a long exact sequence relating $K_*(R)$, $K_*(S^{-1}R)$, and $K$-theory of modules with $S$-torsion — schematically, for regular rings and $S=R\setminus\mathfrak{p}$ etc., one obtains localization sequences in $G$-theory: $$ \cdots\to G_{n+1}(R_S)\to G_n(R\text{ on }V(S))\to G_n(R)\to G_n(R_S)\to\cdots. $$

Schemes (Thomason–Trobaugh). For quasi-compact quasi-separated $X$ and open $U=X\setminus Z$, $$ \cdots\to K_{n+1}(U)\to K_n(X\text{ on }Z)\to K_n(X)\to K_n(U)\to\cdots $$ with $K(X\text{ on }Z)=$ perfect complexes acyclic on $U$. This is the workhorse for computing $K(X)$ by stratifications and for Mayer–Vietoris.

Projective bundle formula / Bass fundamental theorem. $$ K_n(R[t,t^{-1}])\cong K_n(R)\oplus K_{n-1}(R) $$ (under regularity / Nil-vanishing hyp.\ for the full Bass story); defines negative $K$-groups by descent $K_{-n}(R)=\operatorname{coker}(K_{1-n}(R[t])\to K_{1-n}(R[t,t^{-1}]))$ etc.

Computations and structures¶

Object	Known features
$K_n(\mathbb{F}_q)$	Quillen: periodic, order related to $q^i-1$
$K_n(\mathbb{Z})$	partial (Borel ranks; Rognes–Weibel; Voevodsky for $K_n^M$)
$K_n(k)$ field	motivic cohomology via Bloch–Lichtenbaum / Voevodsky FS
$K_*^M(F)$ Milnor	symbols; VMST theorem identifies with motivic cohomology

Adams operations $\psi^k$ act on $K_*(R)_{\mathbb{Q}}$; eigenspaces relate to weight filtrations and regulators.

Intersection with geometry and number theory¶

Chern classes / Chern character from $K$ to motivic cohomology / Deligne cohomology produce Beilinson regulators.
Perfect complexes on derived schemes: $K(X)$ for $X$ derived still makes sense via $\mathrm{Perf}$ — link DAG.
Trace methods (Bökstedt, Hesselholt–Madsen, Nikolaus–Scholze): TC / THH approximate $p$-adic $K$-theory of rings of integers and syntomic cohomology.

See it¶

Localization for $\mathbb{Z}$. From the localization sequence for $G$-theory along $\mathbb{Z}\to\mathbb{Q}$ and dévissage of finite abelian groups into $\mathbb{F}_p$, $$ \cdots\to K_{n+1}(\mathbb{Q})\to \bigoplus_p K_n(\mathbb{F}_p)\to K_n(\mathbb{Z})\to K_n(\mathbb{Q})\to\cdots $$ (with care at $K_0$ ranks). Combined with Quillen’s computation of $K_*(\mathbb{F}_p)$ and Borel’s ranks of $K_n(\mathbb{Z})\otimes\mathbb{Q}$, this organizes the arithmetic of higher $K(\mathbb{Z})$.

Blow-up / projective bundle. $K_0(\mathbb{P}^n_R)\cong K_0(R)^{n+1}$ (basis $[\mathcal{O}],[\mathcal{O}(1)],\ldots,[\mathcal{O}(n)]$ after suitable identification) — classical starting point for GRR computations on projective morphisms.

Check¶

Why does the resolution theorem give $G_0(R)\cong K_0(R)$ for regular Noetherian $R$?

Every f.g.\ $R$-module has a finite resolution by f.g.\ projectives (finite global dimension locally / Serre); resolution theorem identifies $K$-theory of $\mathbf{P}(R)$ with that of $\mathbf{M}(R)$. On $K_0$, $[M]=\sum (-1)^i[P_i]$ for a projective resolution.

What is the role of the Thomason–Trobaugh localization sequence in computing $K(X)$ for a scheme with a closed stratification?

Iterate: remove open strata; each step relates $K(X)$ to $K(U)$ and $K$ supported on the closed complement. With dévissage on the closed pieces (e.g.\ regular strata $\rightsquigarrow$ $K$ of fields / regular rings), one reduces to known pieces — analogous to Mayer–Vietoris in topology.

Pitfalls¶

$K$ vs $G$: localization sequences are cleanest for $G$-theory (coherent sheaves) without regularity; $K$-theory of perfect complexes needs Thomason–Trobaugh hypotheses (qcqs).
Negative $K$-groups need not vanish (non-regular rings); Bass nil groups $NK_n$ measure failure of homotopy invariance.
Spectrum vs groups: multiplicative structure lives at the spectrum / $E_\infty$ level ($K$-theory spectrum is an $E_\infty$-ring spectrum under mild hyp.).
Milnor $K^M$ vs Quillen $K$: agree for $n\le 2$ on fields; diverge for $n\ge 3$ in general (but related by motivic spectral sequence).

Refs¶

Quillen, Higher algebraic $K$-theory I–II
Thomason–Trobaugh, Higher algebraic $K$-theory of schemes
Weibel, The $K$-book
Friedlander–Suslin, Lecture notes on $K$-theory; Srinivas, Algebraic $K$-Theory

Next¶

K-theory index · Foundations · Homological algebra · AG cohomology · Derived AG · Algebraic NT · Spectral sequences · Prismatic cohomology · Condensed mathematics — condensed TC / THH; cyclotomic trace in solid $A_{\inf}$-modules · Algebraic cobordism — $\mathrm{MGL}$ the universal oriented theory; the Conner–Floyd theorem relates cobordism to $K$-theory

Input	\(K\)-theory spectrum
Ring \(R\)	\(K(R)\) with \(\pi_n=K_n(R)\)
Scheme \(X\)	\(K(X)=K(\mathrm{Perf}(X))\)
Exact category	\(K(\mathcal{M})\)
dg-category / stable \(\infty\)-cat	noncommutative motives package (BGT, …)

Object	Known features
\(K_n(\mathbb{F}_q)\)	Quillen: periodic, order related to \(q^i-1\)
\(K_n(\mathbb{Z})\)	partial (Borel ranks; Rognes–Weibel; Voevodsky for \(K_n^M\))
\(K_n(k)\) field	motivic cohomology via Bloch–Lichtenbaum / Voevodsky FS
\(K_*^M(F)\) Milnor	symbols; VMST theorem identifies with motivic cohomology

Higher Algebraic \(K\)-Theory¶