The Grothendieck Group K₀

The Grothendieck group $K_0$ is the most fundamental invariant in algebraic K-theory. Given a ring $R$ , the group $K_0(R)$ encodes information about finitely generated projective modules over $R$ , modulo a relation that identifies modules related by direct sum decomposition.

Construction of K₀

Definition1.1Grothendieck group of a commutative monoid

Let $(M, +, 0)$ be a commutative monoid. The Grothendieck group (or group completion) of $M$ is an abelian group $K(M)$ together with a monoid homomorphism $\iota: M \to K(M)$ satisfying the universal property: for every monoid homomorphism $f: M \to A$ with $A$ an abelian group, there exists a unique group homomorphism $\bar{f}: K(M) \to A$ such that $\bar{f} \circ \iota = f$ .

Concretely, $K(M)$ can be constructed as the quotient of the free abelian group $\mathbb{Z}[M]$ by the subgroup generated by elements $[m + n] - [m] - [n]$ for all $m, n \in M$ . Equivalently, $K(M) = (M \times M) / \sim$ where $(a, b) \sim (c, d)$ if there exists $e \in M$ with $a + d + e = b + c + e$ .

Definition1.2K₀ of a ring

Let $R$ be a ring (associative, unital). The group $K_0(R)$ is the Grothendieck group of the commutative monoid $(\operatorname{Proj}(R), \oplus)$ , where $\operatorname{Proj}(R)$ denotes the set of isomorphism classes of finitely generated projective $R$ -modules under direct sum.

Explicitly, elements of $K_0(R)$ are formal differences $[P] - [Q]$ of finitely generated projective modules, with the relation $[P] - [Q] = [P'] - [Q']$ if and only if there exists a finitely generated projective module $S$ such that

$P \oplus Q' \oplus S \cong P' \oplus Q \oplus S.$

ExampleK₀ of a field

If $F$ is a field, every finitely generated projective $F$ -module is free, so $\operatorname{Proj}(F) \cong \mathbb{N}$ via $[F^n] \mapsto n$ . The Grothendieck group is

$K_0(F) \cong \mathbb{Z},$

generated by $[F]$ . The rank homomorphism $\operatorname{rk}: K_0(F) \to \mathbb{Z}$ sending $[P] - [Q] \mapsto \dim P - \dim Q$ is an isomorphism.

ExampleK₀ of a PID

If $R$ is a principal ideal domain (PID), then every finitely generated projective module is free (since submodules of free modules over PIDs are free). Thus $K_0(R) \cong \mathbb{Z}$ , generated by $[R]$ . This applies to $\mathbb{Z}$ , $k[x]$ , and all DVRs.

The rank and reduced K₀

Definition1.3Rank homomorphism and reduced K₀

For a ring $R$ , the rank homomorphism is the map

$\operatorname{rk}: K_0(R) \to \mathbb{Z}, \quad [P] \mapsto \operatorname{rk}(P)$

where the rank is computed at any prime ideal. The reduced Grothendieck group is

$\widetilde{K}_0(R) = \ker(\operatorname{rk}: K_0(R) \to \mathbb{Z}).$

Elements of $\widetilde{K}_0(R)$ measure the failure of finitely generated projective modules to be stably free. If $\widetilde{K}_0(R) = 0$ , then every finitely generated projective module is stably free (i.e., $P \oplus R^m \cong R^n$ for some $m, n$ ).

ExampleK₀ of a Dedekind domain

For a Dedekind domain $R$ with fraction field $F$ , the structure theorem gives $K_0(R) \cong \mathbb{Z} \oplus \operatorname{Cl}(R)$ , where $\operatorname{Cl}(R)$ is the ideal class group. The isomorphism sends $[P]$ to $(\operatorname{rk}(P), [\det P])$ where $\det P = \bigwedge^{\operatorname{rk} P} P$ .

For $R = \mathbb{Z}[\sqrt{-5}]$ , we have $\operatorname{Cl}(R) \cong \mathbb{Z}/2\mathbb{Z}$ , so $K_0(\mathbb{Z}[\sqrt{-5}]) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ . The nontrivial element of $\widetilde{K}_0$ is represented by the ideal $(2, 1 + \sqrt{-5})$ .

Functorial properties

RemarkFunctoriality of K₀

$K_0$ is a covariant functor from rings to abelian groups. A ring homomorphism $f: R \to S$ induces

$f_*: K_0(R) \to K_0(S), \quad [P] \mapsto [P \otimes_R S]$

via extension of scalars. This is well-defined because base change preserves finitely generated projective modules.

For commutative rings, $K_0(R)$ is a commutative ring with multiplication $[P] \cdot [Q] = [P \otimes_R Q]$ and identity $[R]$ . The rank map $\operatorname{rk}: K_0(R) \to \mathbb{Z}$ is then a ring homomorphism, and $\widetilde{K}_0(R)$ is an ideal.

If $R$ is a commutative Noetherian ring, there is also the group $G_0(R) = K_0(\operatorname{mod}\text{-}R)$ , the Grothendieck group of finitely generated (not necessarily projective) modules. The natural map $K_0(R) \to G_0(R)$ is an isomorphism when $R$ is regular (the resolution theorem).

ExampleMorita invariance

For the matrix ring $M_n(R)$ , Morita equivalence gives $K_0(M_n(R)) \cong K_0(R)$ for all $n \geq 1$ . The equivalence sends a projective $M_n(R)$ -module $P$ to the projective $R$ -module $eP$ where $e = e_{11}$ is the matrix unit. This is a special case of the general principle: $K_0$ is an invariant of the Morita equivalence class of a ring.