Homotopy Invariance and the Fundamental Theorem

Homotopy invariance is the statement that polynomial extensions do not change K-theory for regular rings. This is the algebraic analogue of the topological fact that contractible deformations do not change K-theory. Combined with the fundamental theorem, it provides the key structural link between K-groups of a ring and its polynomial/Laurent extensions.

Homotopy invariance for regular rings

Theorem3.3Homotopy Invariance

For a regular Noetherian ring $R$ , the inclusion $R \hookrightarrow R[t]$ induces isomorphisms

$K_n(R) \xrightarrow{\;\sim\;} K_n(R[t]) \quad \text{for all } n \geq 0.$

More generally, $K_n(R) \xrightarrow{\sim} K_n(R[t_1, \ldots, t_m])$ for all $m \geq 1$ .

RemarkFailure for non-regular rings

Homotopy invariance fails badly for singular or non-reduced rings. Define the nil-groups:

$NK_n(R) = \ker(K_n(R[t]) \xrightarrow{t \mapsto 0} K_n(R)).$

Then $K_n(R[t]) \cong K_n(R) \oplus NK_n(R)$ and:

For $R$ regular: $NK_n(R) = 0$ (homotopy invariance).
For $R = k[\epsilon]/(\epsilon^2)$ : $NK_0(R) \cong \Omega^1_{k/\mathbb{Z}} / dR$ , involving Kahler differentials.
For $R = \mathbb{Z}/4\mathbb{Z}$ : $NK_1(R) \neq 0$ (nontrivial nilpotent elements).

The nil-groups $NK_n(R)$ are uniquely $p$ -divisible for all primes $p$ that are units in $R$ , and are modules over the "big Witt vectors" $W(R)$ .

The fundamental theorem

Definition3.4The fundamental theorem of algebraic K-theory

For any ring $R$ , the fundamental theorem provides a splitting:

$K_n(R[t, t^{-1}]) \cong K_n(R) \oplus K_{n-1}(R) \oplus NK_n(R) \oplus NK_n(R)$

for all $n \in \mathbb{Z}$ (using negative K-theory for $n < 0$ ). The four summands come from:

$K_n(R) \to K_n(R[t, t^{-1}])$ : the inclusion.
$K_{n-1}(R) \to K_n(R[t, t^{-1}])$ : the "multiplication by $t$ " map (Bass's construction).
Two copies of $NK_n(R)$ : from $K_n(R[t])$ and $K_n(R[t^{-1}])$ .

For regular $R$ , the nil-groups vanish and $K_n(R[t, t^{-1}]) \cong K_n(R) \oplus K_{n-1}(R)$ .

ExampleGroup rings of free abelian groups

For $R$ regular and $G = \mathbb{Z}^m$ , iterating the fundamental theorem:

$K_n(R[\mathbb{Z}^m]) = K_n(R[t_1^{\pm 1}, \ldots, t_m^{\pm 1}]) \cong \bigoplus_{i=0}^{m} \binom{m}{i} K_{n-i}(R).$

For example, with $R = \mathbb{Z}$ and $m = 2$ :

$K_n(\mathbb{Z}[\mathbb{Z}^2]) \cong K_n(\mathbb{Z}) \oplus K_{n-1}(\mathbb{Z})^2 \oplus K_{n-2}(\mathbb{Z}).$

In degree $n = 1$ : $K_1(\mathbb{Z}[s^{\pm 1}, t^{\pm 1}]) \cong K_1(\mathbb{Z}) \oplus K_0(\mathbb{Z})^2 \oplus K_{-1}(\mathbb{Z}) = \mathbb{Z}/2 \oplus \mathbb{Z}^2 \oplus 0 = \mathbb{Z}/2 \oplus \mathbb{Z}^2$ .

Homotopy invariance for G-theory

Theorem3.4G-theory homotopy invariance

For any Noetherian ring $R$ (not necessarily regular), the inclusion $R \hookrightarrow R[t]$ induces isomorphisms

$G_n(R) \xrightarrow{\;\sim\;} G_n(R[t]) \quad \text{for all } n \geq 0.$

This is much stronger than the K-theory version since it holds without regularity.

ExampleG-theory of singular rings

For $R = k[x]/(x^2)$ (dual numbers):

$K_0(R) = \mathbb{Z}$ and $K_0(R[t]) = \mathbb{Z}$ : homotopy invariance holds for $K_0$ here by accident (projective modules over local rings are free).
But $K_1(R[t]) \supsetneq K_1(R)$ : the nil-group $NK_1(R) \neq 0$ detects nilpotent endomorphisms.
Meanwhile $G_n(R) = G_n(R[t])$ for all $n$ by the theorem.
In fact $G_n(k[x]/(x^2)) \cong K_n(k)$ by devissage, so $G$ -theory "sees through" the nilpotent thickening.

The Karoubi--Villamayor K-theory

Definition3.5Karoubi--Villamayor groups

The Karoubi--Villamayor K-groups $KV_n(R)$ for $n \geq 1$ are defined as

$KV_n(R) = \pi_n(K_0(R[\Delta^\bullet]) \times BGL(R[\Delta^\bullet])^+)$

where $R[\Delta^p] = R[t_0, \ldots, t_p] / (t_0 + \cdots + t_p - 1)$ is the ring of polynomial functions on the standard $p$ -simplex. This forms a simplicial ring, and $KV_n$ is the homotopy group of the associated simplicial K-theory space.

By construction, $KV_n$ is homotopy invariant for all rings:

$KV_n(R) \cong KV_n(R[t]).$

For regular rings, $KV_n(R) = K_n(R)$ . For non-regular rings, $KV_n(R)$ is a "regularized" version of $K_n(R)$ that kills the nil-groups.

RemarkComparison of theories

The different flavors of K-theory are related by:

$K_n(R) \twoheadrightarrow KV_n(R) \quad \text{(surjection for } n \geq 1\text{)}$

with kernel related to nil-groups. There is an exact sequence:

$NK_{n+1}(R) \to KV_{n+1}(R) \to K_n(R) \to KV_n(R) \to 0$

For regular rings, $NK_n = 0$ and all three theories ( $K_n$ , $G_n$ , $KV_n$ ) agree. For non-regular rings:

$K_n(R)$ sees nilpotent elements (via nil-groups).
$G_n(R)$ is homotopy invariant and sees singularities differently.
$KV_n(R)$ is homotopy invariant by construction and provides a "smooth" approximation.