Non-Connective K-Theory and the Bass Construction

Non-connective algebraic K-theory extends the K-groups to negative degrees, detecting singularities and non-regularity of rings and schemes. The Bass construction and the Schlichting--Thomason approach provide systematic definitions that satisfy excision and descent properties.

Negative K-groups

Definition7.5Bass's negative K-groups

For a ring $R$ , Bass defines the negative K-groups $K_{-n}(R)$ for $n \geq 1$ by iterating the fundamental theorem. The Bass--Heller--Swan decomposition:

$K_m(R[t, t^{-1}]) \cong K_m(R) \oplus K_{m-1}(R) \oplus NK_m(R) \oplus NK_m(R)$

allows one to define $K_{m-1}(R)$ as a summand of $K_m(R[t, t^{-1}])$ , bootstrapping from $K_0$ and $K_1$ downward:

$K_{-1}(R) = \operatorname{coker}\left(K_0(R[t]) \oplus K_0(R[t^{-1}]) \to K_0(R[t, t^{-1}])\right)$

$K_{-n}(R) = K_{-n+1}(R[t, t^{-1}]) / (K_{-n+1}(R) \oplus K_{-n}(R) \oplus \text{Nil terms})$

For regular rings: $K_{-n}(R) = 0$ for all $n \geq 1$ .

ExampleNon-vanishing negative K-groups

For the coordinate ring of the node $R = k[x, y]/(xy)$ : $K_{-1}(R) \cong \mathbb{Z}$ This detects the singularity: the two branches meeting at a point contribute a $\mathbb{Z}$ from the "gluing data" that fails to be captured by $K_0$ .
For the cone over a smooth projective variety $X$ , the ring $R = k[C(X)]$ (homogeneous coordinate ring) satisfies: $K_{-n}(R) \cong \widetilde{K}_{-n}(X)$ for $n \geq 1$ , where $\widetilde{K}$ is reduced K-theory.
For a singular surface $S$ with isolated singularity at $p$ , $K_{-1}(\mathcal{O}_{S,p})$ captures the local geometry of the singularity.

The non-connective K-theory spectrum

Definition7.6Non-connective delooping

The non-connective K-theory spectrum $\mathbf{K}^B(R)$ (Bass K-theory) is defined as the homotopy colimit:

$\mathbf{K}^B(R) = \operatorname{hocolim}\left(\mathbf{K}(R) \to \Omega^{-1}\mathbf{K}(R[t, t^{-1}]) / \mathbf{K}(R[t]) \vee \mathbf{K}(R[t^{-1}]) \to \cdots \right)$

where each step applies the "Bass delooping." This produces a spectrum with:

$\pi_n(\mathbf{K}^B(R)) = \begin{cases} K_n(R) & n \geq 0 \\ K_n^{\text{Bass}}(R) & n < 0 \end{cases}$

The non-connective spectrum $\mathbf{K}^B$ satisfies:

Localization: For every localization $R \to S^{-1}R$ , there is a fiber sequence of spectra.
Excision: The Mayer--Vietoris sequence extends to all integers.
Descent: $\mathbf{K}^B$ satisfies Nisnevich descent on schemes (Thomason--Trobaugh).

RemarkConnective vs non-connective

The connective K-theory spectrum $\mathbf{K}(R)$ has $\pi_n = 0$ for $n < 0$ by definition. The non-connective $\mathbf{K}^B(R)$ can have nonzero negative homotopy groups.

The connective cover gives a map $\mathbf{K}(R) \to \mathbf{K}^B(R)$ inducing isomorphisms on $\pi_n$ for $n \geq 0$ . The difference matters for:

Localization sequences: The connecting maps $K_{n+1}(U) \to K_n(Z)$ in the Thomason--Trobaugh sequence work best with $\mathbf{K}^B$ , since $K_n(Z)$ can be negative.
Descent: $\mathbf{K}^B$ satisfies Nisnevich descent while $\mathbf{K}$ generally does not (the obstruction is measured by negative K-groups).
Regularity detection: $R$ is regular iff $K_{-n}^B(R) = 0$ for all $n > 0$ (assuming $R$ is Noetherian of finite dimension).

Schlichting's approach

Definition7.7Schlichting's non-connective K-theory

Schlichting provides an intrinsic construction of non-connective K-theory for exact categories, avoiding the Bass delooping machine. For an exact category $\mathcal{E}$ , define:

$\mathbf{K}^{\text{Sch}}(\mathcal{E}) = \operatorname{hofib}\left(K(\mathcal{E}) \to K(\mathcal{E}[\mathcal{E}])\right)[-1]$

where $\mathcal{E}[\mathcal{E}]$ is the "idempotent completion" or "Karoubi envelope," and the homotopy fiber captures the "missing" negative K-theory.

For $\mathcal{E} = \mathcal{P}(R)$ : $\mathbf{K}^{\text{Sch}}(\mathcal{P}(R)) \simeq \mathbf{K}^B(R)$ .

The advantage: Schlichting's construction works for abstract exact categories and DG-categories, not just module categories.

ExampleComputing K₋₁

For a commutative Noetherian ring $R$ , Bass gives an explicit description:

$K_{-1}(R) = \operatorname{coker}\left(K_0(R[t]) \oplus K_0(R[t^{-1}]) \xrightarrow{j_+ - j_-} K_0(R[t, t^{-1}])\right)$

where $j_\pm$ are induced by the inclusions. If $R$ is reduced, this simplifies:

$K_{-1}(R) \cong \operatorname{coker}\left(\widetilde{K}_0(R) \oplus \widetilde{K}_0(R) \to \widetilde{K}_0(R[t, t^{-1}]) / \widetilde{K}_0(R)\right)$

For the node $R = k[x, y]/(xy) = k[x] \times_{k} k[y]$ : Using the Mayer--Vietoris square and the fundamental theorem, one computes $K_{-1}(R) \cong \mathbb{Z}$ , generated by the "loop" class around the singular point.

For regular $R$ : both $K_0(R) \to K_0(R[t])$ are isomorphisms by homotopy invariance, so $K_{-1}(R) = 0$ .

Properties of negative K-theory

RemarkVanishing and regularity

The following are equivalent for a Noetherian ring $R$ of finite Krull dimension $d$ :

$R$ is regular (all localizations have finite global dimension).
$K_{-n}(R) = 0$ for all $n \geq 1$ .
$K_{-1}(R) = 0$ and $R$ is locally regular in codimension $\leq d - 1$ (under mild hypotheses).

Moreover, $K_{-n}(R) = 0$ for $n > d$ (Weibel's vanishing conjecture, proved by Kerz--Strunk--Tamme): the negative K-groups are bounded below by the Krull dimension.

This gives a "K-theoretic characterization of regularity": a Noetherian ring is regular if and only if its non-connective K-theory spectrum is connective.