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.
Grothendieck group of a commutative monoid
Motivation Given a commutative monoid M {\displaystyle M} , "the most general" abelian group K {\displaystyle K} that arises from M {\displaystyle M} is to be constructed by introducing inverse elements to all elements of M {\displaystyle M} . Such an abelian group K {\displaystyle K} always exists; it is called the Grothendieck group of M {\displaystyle M} . It is characterized by a certain universal property and can also be concretely constructed from M {\displaystyle M} . If M {\displaystyle M} does not have the cancellation property (that is, there exists a , b {\displaystyle a,b} and c {\displaystyle c} in M {\displaystyle M} such that a ≠ b {\displaystyle a\neq b} and a c = b c {\displaystyle ac=bc} ), then the Grothendieck group K {\displaystyle K} cannot contain M {\displaystyle M} . In particular, in the case of a monoid operation denoted multiplicatively that has a zero element satisfying 0 ⋅ x = 0 {\displaystyle 0\cdot x=0} for every x ∈ M , {\displaystyle x\in M,} the Grothendieck group must be the trivial group (group with only one element), since one must have
x = 1 ⋅ x = ( 0 − 1 ⋅ 0 ) ⋅ x = 0 − 1 ⋅ ( 0 ⋅ x ) = 0 − 1 ⋅ 0 = 0 {\displaystyle x=1\cdot x=(0^{-1}\cdot 0)\cdot x=0^{-1}\cdot (0\cdot x)=0^{-1}\cdot 0=0}
for every x {\displaystyle x} .
Universal property Let M be a commutative monoid. Its Grothendieck group is an abelian group K with a monoid homomorphism i : M → K {\displaystyle i\colon M\to K} satisfying the following universal property: for any monoid homomorphism f : M → A {\displaystyle f\colon M\to A} from M to an abelian group A, there is a unique group homomorphism g : K → A {\displaystyle g\colon K\to A} such that f = g ∘ i . {\displaystyle f=g\circ i.}
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.
Explicit constructions
To construct the Grothendieck group K of a commutative monoid M, one forms the Cartesian product M × M {\displaystyle M\times M} . The two coordinates are meant to represent a positive part and a negative part, so ( m 1 , m 2 ) {\displaystyle (m_{1},m_{2})} corresponds to m 1 − m 2 {\displaystyle m_{1}-m_{2}} in K. Addition on M × M {\displaystyle M\times M} is defined coordinate-wise:
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