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Concept
Grothendieck group
Summary
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 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 , 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 of an , 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.
K-theory#Grothendieck completion
To construct the Grothendieck group K of a commutative monoid M, one forms the Cartesian product . The two coordinates are meant to represent a positive part and a negative part, so corresponds to in K.
Official source
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