This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.
Given a commutative monoid M, "the most general" abelian group K that arises from M is to be constructed by introducing inverse elements to all elements of M. Such an abelian group K always exists; it is called the Grothendieck group of M. It is characterized by a certain universal property and can also be concretely constructed from M. If M does not have the cancellation property (that is, there exists a, b and c in M such that
), then the Grothendieck group K cannot contain M. In particular, in the case of a monoid operation denoted multiplicatively that has a zero element satisfying
Alternatively, the Grothendieck group K of M can also be constructed using generators and relations: denoting by
while + denotes the addition in the monoid M.) This construction has the advantage that it can be performed for any semigroup M and yields a group which satisfies the corresponding universal properties for semigroups, i.e. the "most general and smallest group containing a homomorphic image of M".
For a commutative monoid M, the map i : M → K is injective if and only if M has the cancellation property, and it is bijective if and only if M is already a group.
The easiest example of a Grothendieck group is the construction of the integers
First one observes that the natural numbers (including 0) together with the usual addition indeed form a commutative monoid
Indeed, this is the usual construction to obtain the integers from the natural numbers.
of a compact manifold M is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of vector bundles of finite rank on M with the monoid operation given by direct sum.
of a (not necessarily commutative) ring R is the Grothendieck group of the monoid consisting of isomorphism classes of finitely generated projective modules over R, with the monoid operation given by the direct sum.
is the ring of complex-valued smooth functions on a compact manifold M. In this case the projective R-modules are dual to vector bundles over M (by the Serre–Swan theorem).
Another construction that carries the name Grothendieck group is the following: Let R be a finite-dimensional algebra over some field k or more generally an artinian ring.
Since any short exact sequence of vector spaces splits, it holds that
In fact, for any two finite-dimensional vector spaces V and W the following holds: The above equality hence satisfies the condition of the symbol
Note that any two isomorphic finite-dimensional K-vector spaces have the same dimension.
In fact, every finite n-dimensional K-vector space V is isomorphic to
This in turn implies that every finite abelian group G satisfies
Indeed, the observation made from the previous paragraph shows that every abelian group A has its symbol
Furthermore, the rank of the abelian group satisfies the conditions of the symbol
Suppose one has the following short exact sequence of abelian groups: Then tensoring with the rational numbers
On the other hand, one also has the following relation; for more information, see Rank of an abelian group.
By choosing a suitable basis and writing the corresponding matrices in block triangular form one easily sees that character functions are additive in the above sense.
one has a canonical element In fact the Grothendieck group was originally introduced for the study of Euler characteristics.
A common generalization of these two concepts is given by the Grothendieck group of an exact category
The precise axioms for this distinguished class do not matter for the construction of the Grothendieck group.
and one relation for each exact sequence Alternatively and equivalently, one can define the Grothendieck group using a universal property: A map
This gives the notion of a Grothendieck group in the previous section if one chooses
This procedure produces the Grothendieck group of the commutative monoid
means the "set" [ignoring all foundational issues] of isomorphism classes in