Коммутативті моноидты кеңейтуші абелдік топ

Abelian group extending a commutative monoid
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K theory. 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 and ), 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 for every the Grothendieck group must be the trivial group (group with only one element), since one must have

for every x.

Let M be a commutative monoid. Its Grothendieck group is an abelian group K with a monoid homomorphism satisfying the following universal property: for any monoid homomorphism from M to an abelian group A, there is a unique group homomorphism such that

This expresses the fact that any abelian group A that contains a homomorphic image of M will also contain a homomorphic image of K, K being the "most general" abelian group containing a homomorphic image of M.

In the language of category theory, any universal construction gives rise to a functor; one thus obtains a functor from the category of commutative monoids to the category of abelian groups which sends the commutative monoid M to its Grothendieck group K. This functor is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids. 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 Grothendieck group is the fundamental construction of K theory. The group 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. This gives a contravariant functor from manifolds to abelian groups. This functor is studied and extended in topological K theory.

The zeroth algebraic K group 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. Then is a covariant functor from rings to abelian groups. The two previous examples are related: consider the case where 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). Thus and are the same group.

Generalizing even further it is also possible to define the Grothendieck group for triangulated categories. The construction is essentially similar but uses the relations [X] − [Y] + [Z] = 0 whenever there is a distinguished triangle X → Y → Z → X[1].