Grothendieck group

Definition

Let $M$ be a commutative monoid, written additively. A Grothendieck group of $M$ is an abelian group $K(M)$ with a monoid homomorphism $i \colon M \to K(M)$ with this universal property. For every abelian group $A$ and monoid homomorphism $f \colon M \to A$ , there is exactly one group homomorphism $g \colon K(M) \to A$ with $g \circ i = f$ .

Remarks

It exists and is unique up to unique isomorphism. One construction: on $M \times M$ put $(m, n) \sim (m', n')$ if $m + n' + p = m' + n + p$ for some $p \in M$ . The classes form an abelian group under componentwise addition, $i(m)$ is the class of $(m, 0)$ , and the class of $(m, n)$ is $i(m) - i(n)$ . Example: for $M = \N$ under addition, the inclusion $\N \to \Z$ is a Grothendieck group, since a homomorphism $f$ from $\N$ extends uniquely by $g(k) = k f(1)$ . Non-example: the inclusion $\N \to \Q$ is not, because the identity map $\N \to \N \subseteq \Z$ has no extension to a group homomorphism $\Q \to \Z$ . The map $i$ is injective exactly when $M$ is cancellative; for $M = \N \cup \set{\infty}$ with $n + \infty = \infty$ , one gets $K(M) = 0$ .

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Abelian group concept
Monoid concept
Grothendieck group definition

Tag	`JEWP2J397B`
URI	`m2://JEWP2J397B`
Kind	definition
Writing	W0 stub
MSC	19A49, 18F30
Defines	Grothendieck group
Uses	Monoid, Abelian group

Grothendieck group

Definition

Remarks

Reading path

Further reading

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