Mayer--Vietoris Sequences

The Mayer--Vietoris sequence in algebraic K-theory allows computation of K-groups of a ring (or scheme) from knowledge of K-groups of simpler pieces. It is the algebraic analogue of the classical Mayer--Vietoris sequence in singular homology.

Milnor's Mayer--Vietoris for K₀ and K₁

Definition3.3Cartesian square of rings

A Cartesian square (or pullback square) of rings is a commutative diagram

$\begin{array}{ccc} R & \xrightarrow{f_1} & R_1 \\ \downarrow^{f_2} & & \downarrow^{g_1} \\ R_2 & \xrightarrow{g_2} & R_{12} \end{array}$

such that $R \cong R_1 \times_{R_{12}} R_2 = \{(r_1, r_2) \in R_1 \times R_2 : g_1(r_1) = g_2(r_2)\}$ . The square is called surjective if at least one of $g_1$ or $g_2$ is surjective.

Theorem3.2Milnor's Mayer--Vietoris

For a surjective Cartesian square of rings, there is an exact sequence:

$K_1(R) \to K_1(R_1) \oplus K_1(R_2) \to K_1(R_{12}) \xrightarrow{\partial} K_0(R) \to K_0(R_1) \oplus K_0(R_2) \to K_0(R_{12}).$

The boundary map $\partial$ sends a unit $u \in GL_n(R_{12})$ to the class $[P_u] - [R^n]$ where $P_u$ is the projective $R$ -module obtained by "patching" via $u$ :

$P_u = \{(v_1, v_2) \in R_1^n \times R_2^n : g_1(v_1) = u \cdot g_2(v_2)\}.$

ExampleConductor square

Let $R = \mathbb{Z}[\sqrt{-5}]$ , $\mathfrak{f} = (2, 1+\sqrt{-5})$ its conductor in the normalization. The conductor square is:

$\begin{array}{ccc} \mathbb{Z}[\sqrt{-5}] & \to & \mathbb{Z}[\sqrt{-5}]/(2) \\ \downarrow & & \downarrow \\ \mathbb{Z}[\sqrt{-5}]_\mathfrak{f} & \to & (\mathbb{Z}[\sqrt{-5}]/(2))_\mathfrak{f} \end{array}$

More generally, for a seminormal ring $R$ with normalization $\tilde{R}$ and conductor $\mathfrak{c}$ :

$\begin{array}{ccc} R & \to & R/\mathfrak{c} \\ \downarrow & & \downarrow \\ \tilde{R} & \to & \tilde{R}/\mathfrak{c} \end{array}$

The Mayer--Vietoris sequence becomes:

$K_1(\tilde{R}) \oplus K_1(R/\mathfrak{c}) \to K_1(\tilde{R}/\mathfrak{c}) \xrightarrow{\partial} K_0(R) \to K_0(\tilde{R}) \oplus K_0(R/\mathfrak{c}) \to K_0(\tilde{R}/\mathfrak{c})$

This computes $K_0$ and $K_1$ of singular curves in terms of their normalization.

The node and the cusp

ExampleK₀ of the nodal curve

For the coordinate ring of the node $R = k[x, y]/(xy)$ (two lines crossing at the origin), the normalization is $\tilde{R} = k[x] \times k[y]$ with conductor $\mathfrak{c} = (x) \times (y)$ . The conductor square:

$\begin{array}{ccc} k[x,y]/(xy) & \to & k \\ \downarrow & & \downarrow^{\Delta} \\ k[x] \times k[y] & \to & k \times k \end{array}$

The Mayer--Vietoris sequence gives:

$K_1(k[x] \times k[y]) \oplus K_1(k) \to K_1(k \times k) \xrightarrow{\partial} K_0(R) \to K_0(k[x] \times k[y]) \oplus K_0(k)$

Since $K_1(k[t]) = k^{\times}$ and $K_1(k) = k^{\times}$ , the boundary map analysis yields:

$K_0(k[x,y]/(xy)) \cong \mathbb{Z}^2$

with one rank for each irreducible component. Meanwhile $SK_1(R) = 0$ and $K_1(R) \cong k^{\times} \times k^{\times}$ .

ExampleK₁ of the cuspidal curve

For the cusp $R = k[t^2, t^3] \subseteq k[t]$ (with $\operatorname{char}(k) \neq 2, 3$ ), the normalization is $\tilde{R} = k[t]$ and the conductor is $\mathfrak{c} = t^2 k[t]$ . The conductor square:

$\begin{array}{ccc} k[t^2, t^3] & \to & k[t^2, t^3]/(t^2) \cong k \oplus k\epsilon \\ \downarrow & & \downarrow \\ k[t] & \to & k[t]/(t^2) \cong k[\epsilon] \end{array}$

where $\epsilon^2 = 0$ . The Mayer--Vietoris sequence gives:

$k[t]^{\times} \times (k \oplus k\epsilon)^{\times} \to k[\epsilon]^{\times} \xrightarrow{\partial} K_0(R) \to \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$

The group $k[\epsilon]^{\times} = \{a + b\epsilon : a \in k^{\times}\} \cong k^{\times} \times k^+$ . The map from $k[t]^{\times} \times (k^{\times} \times k^+)$ surjects onto the $k^{\times}$ part. The additive group $k^+ = (k, +)$ contributes:

$K_1(k[t^2, t^3]) / K_1(k) \cong k^+$

detecting the cuspidal singularity via the additive group of the ground field.

Higher Mayer--Vietoris

RemarkAlgebraic K-theory of pushouts

For Cartesian squares satisfying appropriate conditions (e.g., $\operatorname{GL}$ -fibration or excision conditions), the Mayer--Vietoris sequence extends to all higher K-groups:

$\cdots \to K_{n+1}(R_{12}) \xrightarrow{\partial} K_n(R) \to K_n(R_1) \oplus K_n(R_2) \to K_n(R_{12}) \to \cdots$

This holds unconditionally when one of $g_1, g_2$ is surjective and the square is a Milnor square (the kernel of the surjection is an ideal of both $R$ and the other ring).

Suslin proved that for $\operatorname{GL}$ -fibered squares, the higher Mayer--Vietoris sequence holds. This was later subsumed by the excision results of Suslin--Wodzicki and the more general framework of non-connective K-theory.

RemarkExcision in algebraic K-theory

Unlike topological K-theory, algebraic K-theory does not satisfy excision in general. For a two-sided ideal $I \trianglelefteq R$ and the map $R \to R/I$ , the relative K-groups $K_n(R, I)$ fit into:

$\cdots \to K_n(R, I) \to K_n(R) \to K_n(R/I) \to K_{n-1}(R, I) \to \cdots$

Excision would say $K_n(R, I) \cong K_n(S, I)$ for any $S$ with $I \trianglelefteq S$ and $S/I = R/I$ . This fails in general but holds when $I$ is a $K$ -excisive ideal (e.g., when $R \to R/I$ has a pro- $H$ -unital kernel). Suslin--Wodzicki gave a characterization: excision holds iff $\operatorname{Tor}^{\mathbb{Z} \ltimes I}_*(I, \mathbb{Z}) = 0$ .