K-Theory of Schemes

Algebraic K-theory extends naturally from rings to schemes, where it captures deep geometric and arithmetic information. For a scheme $X$ , the K-groups $K_n(X)$ are defined using vector bundles (or perfect complexes), while the G-theory groups $G_n(X)$ use coherent sheaves.

Definitions

Definition5.1K-theory of a scheme

For a scheme $X$ , define:

$K_n(X)$ (K-theory): the Quillen K-groups of the exact category $\mathcal{V}(X)$ of locally free sheaves (vector bundles) of finite rank on $X$ : $K_n(X) = K_n(\mathcal{V}(X)).$
$G_n(X)$ (G-theory): the Quillen K-groups of the abelian category $\operatorname{Coh}(X)$ of coherent sheaves on $X$ (when $X$ is Noetherian): $G_n(X) = K_n(\operatorname{Coh}(X)).$

There is a natural Cartan map $K_n(X) \to G_n(X)$ induced by the inclusion $\mathcal{V}(X) \hookrightarrow \operatorname{Coh}(X)$ . By the resolution theorem, this is an isomorphism when $X$ is regular.

RemarkFunctoriality

The two theories have complementary functoriality:

K-theory $K_n(X)$ :

Contravariant for arbitrary morphisms $f: X \to Y$ : pullback $f^*: K_n(Y) \to K_n(X)$ via $\mathcal{E} \mapsto f^*\mathcal{E}$ .
Makes $K_0(X)$ a commutative ring via tensor product: $[\mathcal{E}] \cdot [\mathcal{F}] = [\mathcal{E} \otimes \mathcal{F}]$ .

G-theory $G_n(X)$ :

Contravariant for flat morphisms: $f^*: G_n(Y) \to G_n(X)$ when $f$ is flat.
Covariant for proper morphisms: $f_*: G_n(X) \to G_n(Y)$ via $\mathcal{F} \mapsto \sum (-1)^i [R^i f_* \mathcal{F}]$ .
$G_0(X)$ is a module over $K_0(X)$ via $[\mathcal{E}] \cdot [\mathcal{F}] = [\mathcal{E} \otimes \mathcal{F}]$ .

K₀ of smooth varieties

ExampleK₀ of a smooth curve

For a smooth projective curve $C$ over a field $k$ :

$K_0(C) \cong \mathbb{Z} \oplus \operatorname{Pic}(C)$

where $\mathbb{Z}$ is the rank and $\operatorname{Pic}(C)$ is the Picard group. Every vector bundle $\mathcal{E}$ on $C$ satisfies $[\mathcal{E}] = \operatorname{rk}(\mathcal{E})[\mathcal{O}] + [\det \mathcal{E}] - [\mathcal{O}]$ in $K_0(C)$ modulo higher filtration.

For $C = \mathbb{P}^1$ : $\operatorname{Pic}(\mathbb{P}^1) \cong \mathbb{Z}$ generated by $\mathcal{O}(1)$ , so $K_0(\mathbb{P}^1) \cong \mathbb{Z}^2$ with basis $[\mathcal{O}], [\mathcal{O}(1)]$ .

For an elliptic curve $E$ : $\operatorname{Pic}^0(E) \cong E(k)$ (the group of rational points), so $K_0(E) \cong \mathbb{Z} \oplus E(k) \oplus \mathbb{Z}$ where the first $\mathbb{Z}$ is rank and the last $\mathbb{Z}$ is the degree component of $\operatorname{Pic}$ .

ExampleK₀ of smooth surfaces

For a smooth projective surface $S$ over an algebraically closed field:

$K_0(S) \otimes \mathbb{Q} \cong CH^*(S) \otimes \mathbb{Q} = \bigoplus_{i=0}^{2} CH^i(S) \otimes \mathbb{Q}$

via the Chern character. The integral structure is more subtle. The filtration $F^i K_0(S)$ by codimension of support gives:

$\operatorname{gr}^0 K_0(S) = \mathbb{Z} \quad (\text{rank})$ $\operatorname{gr}^1 K_0(S) = \operatorname{Pic}(S) \quad (\text{first Chern class})$ $\operatorname{gr}^2 K_0(S) = CH^2(S) \quad (\text{second Chern class})$

For $S = \mathbb{P}^2$ : $K_0(\mathbb{P}^2) \cong \mathbb{Z}^3$ with basis $[\mathcal{O}], [\mathcal{O}(1)], [\mathcal{O}(2)]$ and ring structure $K_0(\mathbb{P}^2) \cong \mathbb{Z}[h]/(h^3)$ where $h = [\mathcal{O}(1)] - 1$ .

Thomason--Trobaugh K-theory

Definition5.2K-theory of perfect complexes

For a general (possibly singular or non-Noetherian) scheme $X$ , the Thomason--Trobaugh K-theory is defined using the Waldhausen construction applied to the category of perfect complexes $\operatorname{Perf}(X)$ :

$K_n(X) = K_n(\operatorname{Perf}(X))$

where $\operatorname{Perf}(X)$ is the full subcategory of the derived category $D(\mathcal{O}_X\text{-}\operatorname{Mod})$ consisting of complexes locally quasi-isomorphic to bounded complexes of locally free sheaves of finite rank.

For a Noetherian regular scheme, $\operatorname{Perf}(X) = D^b(\operatorname{Coh}(X))$ and this agrees with the Quillen definition. For singular schemes, $\operatorname{Perf}(X) \subsetneq D^b(\operatorname{Coh}(X))$ .

ExampleK-theory of singular schemes

For the singular scheme $X = \operatorname{Spec} k[x, y]/(xy)$ (the node):

Using Thomason--Trobaugh K-theory:

$K_0(X) \cong \mathbb{Z}^2$ (one for each irreducible component, since every perfect complex splits at the generic points).
$K_1(X) \cong k^{\times} \times k^{\times}$ (units from each branch).
$K_{-1}(X) \cong \mathbb{Z}$ (detecting the singularity via Bass's negative K-theory).

The negative K-group $K_{-1}(X)$ vanishes for regular schemes and provides a measure of the severity of the singularity.

The Brown--Gersten--Quillen spectral sequence

Definition5.3BGQ spectral sequence

For a regular Noetherian scheme $X$ of finite Krull dimension, there is a convergent spectral sequence:

$E_1^{p,q} = \bigoplus_{x \in X^{(p)}} K_{-p-q}(k(x)) \implies K_{-p-q}(X)$

where $X^{(p)}$ denotes the set of points of codimension $p$ and $k(x)$ is the residue field at $x$ .

The $E_1$ -page is the Gersten complex:

$\bigoplus_{x \in X^{(0)}} K_n(k(x)) \to \bigoplus_{x \in X^{(1)}} K_{n-1}(k(x)) \to \cdots \to \bigoplus_{x \in X^{(n)}} K_0(k(x))$

and the $E_2$ -terms are the Zariski cohomology of the K-theory sheaf:

$E_2^{p,q} = H^p_{\text{Zar}}(X, \mathcal{K}_{-q}).$

By Gersten's conjecture (proved for smooth varieties over a field), the Gersten complex is exact, so $E_2^{p,q} = H^p(X, \mathcal{K}_{-q})$ where $\mathcal{K}_n$ is the Zariski sheaf associated to the presheaf $U \mapsto K_n(U)$ .

RemarkRelation to Chow groups

From the BGQ spectral sequence, $E_2^{p,-p} = H^p_{\text{Zar}}(X, \mathcal{K}_p) = CH^p(X)$ (Chow groups). The edge map gives the cycle class map:

$CH^p(X) = E_2^{p,-p} \to E_\infty^{p,-p} \hookrightarrow \operatorname{gr}^p K_0(X)$

which is Grothendieck's Chern character in the appropriate degree. After tensoring with $\mathbb{Q}$ , the spectral sequence degenerates at $E_2$ and gives:

$K_n(X) \otimes \mathbb{Q} \cong \bigoplus_{p} H^p_{\text{Zar}}(X, \mathcal{K}_{n+p}) \otimes \mathbb{Q}$

relating K-groups to motivic cohomology.