Bass--Heller--Swan Theorem

The Bass--Heller--Swan theorem (also known as the fundamental theorem of algebraic K-theory for $K_1$ ) describes how K-groups behave under Laurent polynomial extensions. It provides the first example of a "splitting" phenomenon relating K-groups of a ring to those of polynomial and Laurent extensions.

Statement

Theorem1.1Bass--Heller--Swan

For any ring $R$ , there is a natural isomorphism

$K_1(R[t, t^{-1}]) \cong K_1(R) \oplus K_0(R) \oplus NK_1(R) \oplus NK_1(R)$

where:

The map $K_1(R) \to K_1(R[t, t^{-1}])$ is induced by the inclusion $R \hookrightarrow R[t, t^{-1}]$ .
The factor $K_0(R)$ is the image of the map $K_0(R) \to K_1(R[t, t^{-1}])$ sending $[P]$ to the class of the automorphism "multiplication by $t$ " on $P \otimes_R R[t, t^{-1}]$ .
The nil-groups $NK_1(R) = \ker(K_1(R[t]) \to K_1(R))$ where the map is evaluation $t \mapsto 0$ .

The nil-groups

Definition1.11Nil-groups NK₁

The nil-group $NK_1(R)$ is defined as

$NK_1(R) = \operatorname{coker}\bigl(K_1(R) \to K_1(R[t])\bigr) = K_1(R[t]) / K_1(R).$

An element of $NK_1(R[t])$ is represented by a pair $(P, \nu)$ where $P$ is a finitely generated projective $R$ -module and $\nu: P \to P$ is a nilpotent $R$ -module endomorphism. The group $NK_1(R)$ is the Grothendieck group of such pairs modulo exact sequences.

When $R$ is a regular ring (Noetherian, finite global dimension), $NK_1(R) = 0$ . In particular, for regular rings:

$K_1(R[t, t^{-1}]) \cong K_1(R) \oplus K_0(R).$

ExampleApplication to ℤ[t, t⁻¹]

Since $\mathbb{Z}$ is regular, $NK_1(\mathbb{Z}) = 0$ , so

$K_1(\mathbb{Z}[t, t^{-1}]) \cong K_1(\mathbb{Z}) \oplus K_0(\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}.$

The factor $K_1(\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} = \{+1, -1\}$ comes from the units of $\mathbb{Z}$ . The factor $K_0(\mathbb{Z}) \cong \mathbb{Z}$ is generated by the class of multiplication by $t$ on the free module of rank 1.

More generally, iterating for $\mathbb{Z}[t_1^{\pm 1}, \ldots, t_n^{\pm 1}]$ (the group ring $\mathbb{Z}[\mathbb{Z}^n]$ ):

$K_1(\mathbb{Z}[\mathbb{Z}^n]) \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}^n.$

Proof outline

Proof

The proof proceeds via the Mayer--Vietoris sequence applied to a suitable localization.

Step 1. Consider the ring $R[t]$ and its localization $R[t, t^{-1}]$ . The evaluation maps $\varepsilon_0: R[t] \to R$ (setting $t = 0$ ) and $\varepsilon_1: R[t] \to R$ (setting $t = 1$ ) provide splittings.

Step 2. For the polynomial extension, there is an exact sequence:

$K_1(R) \xrightarrow{i_*} K_1(R[t]) \xrightarrow{\varepsilon_{0*}} K_1(R)$

where $\varepsilon_{0*} \circ i_* = \operatorname{id}$ , giving $K_1(R[t]) \cong K_1(R) \oplus NK_1(R)$ .

Step 3. The localization sequence for $R[t] \to R[t, t^{-1}]$ with respect to the multiplicative set $\{1, t, t^2, \ldots\}$ gives a long exact sequence involving relative K-groups.

Step 4. Bass constructs a map $\partial: K_1(R[t, t^{-1}]) \to K_0(R)$ as follows: represent $\alpha \in K_1(R[t, t^{-1}])$ by an automorphism $\phi$ of $R[t, t^{-1}]^n$ . Choose a lattice $L \subseteq R[t, t^{-1}]^n$ (a finitely generated $R[t]$ -submodule spanning the $R[t,t^{-1}]$ -module). Then

$\partial(\alpha) = [L / \phi(L)] - [\text{correction term}] \in K_0(R).$

Step 5. The map $K_0(R) \to K_1(R[t, t^{-1}])$ sending $[P]$ to the class of multiplication by $t$ on $P[t, t^{-1}]$ provides a splitting of $\partial$ .

Step 6. The two copies of $NK_1(R)$ come from the maps $K_1(R[t]) \to K_1(R[t, t^{-1}])$ and $K_1(R[t^{-1}]) \to K_1(R[t, t^{-1}])$ via the substitutions $t \mapsto t$ and $t \mapsto t^{-1}$ .

Combining all splittings gives the stated decomposition. $\square$

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Consequences and generalizations

RemarkHigher BHS and negative K-theory

The Bass--Heller--Swan theorem generalizes to all K-groups: for regular rings,

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

for all $n \in \mathbb{Z}$ . This in fact defines negative K-groups $K_{-n}(R)$ for $n > 0$ by iterating the Laurent polynomial construction. For regular rings, $K_{-n}(R) = 0$ for all $n > 0$ .

For non-regular rings, the nil-groups $NK_n(R)$ appear. Carter showed that $NK_1(\mathbb{Z}[G])$ can be nonzero for finite groups $G$ with elements of order $p^2$ for some prime $p$ .

ExampleNegative K-theory of singular rings

For the coordinate ring of the node $R = k[x, y]/(xy)$ :

$K_{-1}(R) \cong \mathbb{Z}$

which detects the singularity. This is related to the fact that $\operatorname{Spec}(R)$ has two branches meeting at the origin. For regular rings, negative K-groups vanish, so nonzero $K_{-n}$ is an obstruction to regularity.