Grothendieck group

The Grothendieck group is a mathematical construction that associates a group of a commutative semigroup. This by Alexander Grothendieck named construction of the localization of the ring theory is modeled and how these can be described by a universal property.

Universal property

We have the following sentence:

If a commutative semigroup, then there is a commutative group and a semigroup homomorphism with the following property: there exists a group homomorphism for each group and each semigroup homomorphism with.

Construction

A proof follows from the following construction, which is modeled on the localization of the ring theory. Let be a commutative semigroup. On the Cartesian product we define an equivalence relation by

.

It now shows that this is indeed an equivalence relation defines the equivalence class of is denoted by. It is now and further shows that is defined by a group link to. In this case, the neutral element is (whatever ), the inverses is given by the formula. If, finally, as one can show that and satisfy the condition from the universal property.

Properties

• As usual, one shows using the universal property that the group is uniquely determined up to isomorphism. Hence they are called the Grothendieck group of.
• The semigroup homomorphism from the above universal property is injective if the semigroup has the Kürzbarkeitseigenschaft.

Examples

• For the semigroup the formation of the Grothendieck group coincides with the usual construction of the integers. It has therefore, where the isomorphism is given by. If we identify the Grothendieck group of with, as is the inclusion. It does not matter whether one understands the natural numbers with or without neutral.
• Very similar considerations to the multiplicative semigroup cause, and in this identification again coincides with the inclusion.
• In the multiplicative semigroup (the index 0 signaling that zero belongs to ) is no reduction property before. In this case two pairs, and equivalent, as it is. Therefore and for all.

Grothendieck group as a functor

The construction described above assigns to each commutative semigroup to a commutative group. If a semigroup homomorphism in the category of commutative semigroups, one can construct a group homomorphism as follows. Agent is initially obtained with a semi- group homomorphism and therefrom by the universal property of a group homomorphism.

By this definition becomes a covariant functor from the category in the category of abelian groups.

If we consider an Abelian group only as a semigroup, one can form. It turns out that, the isomorphism is given by. In fact, linksadjungiert for forgetful.

Application

In addition to the above-described construction of the integers from the natural numbers, the formation of a ring group K0 is an important application. For each ring we consider the quantity (!) Of isomorphism classes of finitely generated projective left -modules with the direct sum as semigroup link. The K0 - group of the ring is then defined as the Grothendieck group of.