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Grothendieck group (Definition)

Let $ S$ be an abelian semigroup. The Grothendieck group of $ S$ is $ K(S) = S\times S/\mathord{\sim}$, where $ \sim$ is the equivalence relation: $ (s,t) \sim (u,v)$ if there exists $ r \in S$ such that $ s+v+r = t+u+r$. This is indeed an abelian group with zero element $ (s,s)$ (any $ s \in S$) and inverse $ -(s,t) = (t,s)$. It is common to use the suggestive notation $ t-s$ for $ (t,s)$.

The Grothendieck group construction is a functor from the category of abelian semigroups to the category of abelian groups. A morphism $ f\colon S \to T$ induces a morphism $ K(f)\colon K(S) \to K(T)$ which sends an element $ (s^+,s^-) \in K(S)$ to $ (f(s^+),f(s^-)) \in K(T)$.

Example 1
Let $ (\mathbb{N},+)$ be the semigroup of natural numbers with composition given by addition. Then, $ K(\mathbb{N},+) = \mathbb{Z}$.
Example 2
Let $ (\mathbb{Z},\times)$ be the semigroup of non-zero integers with composition given by multiplication. Then, $ K(\mathbb{Z},\times) = \mathbb{Q}$.
Example 3
Let $ G$ be an abelian group, then $ K(G) \cong G$ via the identification $ (g,h) \leftrightarrow g-h$ (or $ (g,h) \leftrightarrow gh^{-1}$ if $ G$ is multiplicative).

Let $ C$ be a symmetric monoidal category. Its Grothendieck group is $ K([C])$, i.e. the Grothendieck group of the isomorphism classes of objects of $ C$.



"Grothendieck group" is owned by mhale.
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See Also: Algebraic K-theory, K-theory

Other names:  group completion
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Cross-references: objects, classes, isomorphism, monoidal category, symmetric, multiplicative, multiplication, integers, addition, composition, natural numbers, semigroup, induces, morphism, category, functor, inverse, zero element, abelian group, equivalence relation, Abelian semigroup
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This is version 6 of Grothendieck group, born on 2003-05-21, modified 2006-12-04.
Object id is 4290, canonical name is GrothendieckGroup.
Accessed 5506 times total.

Classification:
AMS MSC: 18F30 (Category theory; homological algebra :: Categories and geometry :: Grothendieck groups)
 13D15 (Commutative rings and algebras :: Homological methods :: Grothendieck groups, $K$-theory)
 16E20 (Associative rings and algebras :: Homological methods :: Grothendieck groups, $K$-theory, etc.)
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