Grothendieck Group

The Grothendieck group is a mathematical construction that assigns a group to a commutative semigroup . This construction, named after Alexander Grothendieck , is based on the localization from ring theory and, like this, can be described by a universal property .

Universal property

The following sentence applies:

Is a commutative semigroup, so there is a commutative group and a Halbgruppen- homomorphism with the following property: For every group and every semigroup homomorphism there exists a group homomorphism with . ${\displaystyle H}$ ${\displaystyle {\mathcal {G}}(H)}$ ${\displaystyle \phi _{H}\colon H\rightarrow {\mathcal {G}}(H)}$ ${\displaystyle G}$ ${\displaystyle \phi \colon H\rightarrow G}$ ${\displaystyle \psi \colon {\mathcal {G}}(H)\rightarrow G}$ ${\displaystyle \phi =\psi \circ \phi _{H}}$

construction

A proof results from the following construction, which is based on the localization from the ring theory. Be a commutative semigroup. On the Cartesian product , define an equivalence relation by ${\displaystyle H}$ ${\displaystyle H^{2}=H\times H}$

${\displaystyle (x,y)\sim (x',y')\quad :\Leftrightarrow \quad \exists t\in H:x+y'+t=x'+y+t}$.

One now shows that this actually defines an equivalence relation, the equivalence class of is denoted by. You now set and continue to show that a group link defines on. Here, the neutral element (independently of ) the inverse form is represented by the formula given. If one finally posits , one can show that and meet the condition from the universal property. ${\displaystyle (x,y)}$ ${\displaystyle [(x,y)]}$ ${\displaystyle {\mathcal {G}}(H):=H^{2}/\sim }$ ${\displaystyle [(x,y)]+[(x',y')]:=[(x+x',y+y')]}$ ${\displaystyle {\mathcal {G}}(H)}$ ${\displaystyle [(x,x)]}$ ${\displaystyle x\in H}$ ${\displaystyle -[(x,y)]=[(y,x)]}$ ${\displaystyle \phi _{H}\colon H\rightarrow {\mathcal {G}}(H),x\mapsto [(x+x,x)]}$ ${\displaystyle {\mathcal {G}}(H)}$ ${\displaystyle \phi _{H}}$

properties

• As usual, one shows with the help of the universal property that the group is uniquely determined except for isomorphism. Therefore they called the Grothendieck group of . ${\displaystyle {\mathcal {G}}(H)}$ ${\displaystyle {\mathcal {G}}(H)}$ ${\displaystyle H}$
• The semigroup homomorphism from the above universal property is injective if and only if the semigroup has the property that can be shortened . ${\displaystyle \phi _{H}}$

Examples

• For the semigroup , the formation of the Grothendieck group coincides with the usual construction of whole numbers . One therefore has , where the isomorphism is given by . If one identifies the Grothendieck group of with , then there is inclusion . It does not matter whether one understands natural numbers with or without zero. ${\displaystyle H=(\mathbb {N} ,+)}$ ${\displaystyle {\mathcal {G}}(\mathbb {N} ,+)\cong \mathbb {Z} }$ ${\displaystyle [(n,m)]\mapsto n-m}$ ${\displaystyle (\mathbb {N} ,+)}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \phi _{H}}$ ${\displaystyle \mathbb {N} \subset \mathbb {Z} }$ ${\displaystyle \mathbb {N} }$
• Very similar considerations on the multiplicative half-group lead to , and this identification coincides again with inclusion . ${\displaystyle H=(\mathbb {N} \setminus \{0\},\cdot )}$ ${\displaystyle {\mathcal {G}}(\mathbb {N} \setminus \{0\},\cdot )\cong (\mathbb {Q} ^{+},\cdot )}$ ${\displaystyle \phi _{H}}$ ${\displaystyle \mathbb {N} \setminus \{0\}\subset \mathbb {Q} ^{+}}$
• The multiplicative semigroup (the index 0 indicates that the zero belongs to) is not shortened. In this case two pairs and are equivalent because it holds . Hence, and for everyone . ${\displaystyle H=(\mathbb {N} _{0},\cdot )}$ ${\displaystyle N_{0}}$ ${\displaystyle (m,n)}$ ${\displaystyle (m',n')}$ ${\displaystyle m\cdot n'\cdot 0=m'\cdot n\cdot 0}$ ${\displaystyle {\mathcal {G}}(\mathbb {N} _{0},\cdot )\cong \{0\}}$ ${\displaystyle \phi _{H}(n)=0}$ ${\displaystyle n\in \mathbb {N} _{0}}$

Grothendieck group as functor

The construction described above assigns a commutative group to each commutative semigroup. If a semigroup homomorphism is in the category of commutative semigroups, a group homomorphism can be constructed as follows . Means one initially obtains a half group homomorphism and therefrom by means of the universal property of a group homomorphism with . ${\displaystyle \phi \colon H\rightarrow K}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {G}}(\phi )\colon {\mathcal {G}}(H)\rightarrow {\mathcal {G}}(K)}$ ${\displaystyle \phi _{K}\colon K\rightarrow {\mathcal {G}}(K)}$ ${\displaystyle \phi _{K}\circ \phi \colon H\rightarrow {\mathcal {G}}(K)}$ ${\displaystyle {\mathcal {G}}(\phi )\colon {\mathcal {G}}(H)\rightarrow {\mathcal {G}}(K)}$ ${\displaystyle \phi _{K}\circ \phi ={\mathcal {G}}(\phi )\circ \phi _{H}}$

This definition makes a covariant functor from the category to the category of the Abelian groups. ${\displaystyle {\mathcal {G}}}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {{\mathcal {A}}b}}$

If one regards an Abelian group only as a semigroup, one can form. It turns out that where the isomorphism is given by . Indeed, it is left adjoint to the forget function . ${\displaystyle G}$ ${\displaystyle {\mathcal {G}}(G)}$ ${\displaystyle {\mathcal {G}}(G)\cong G}$ ${\displaystyle [(x,y)]\mapsto x-y}$ ${\displaystyle {\mathcal {G}}\colon {\mathcal {H}}\rightarrow {{\mathcal {A}}b}}$ ${\displaystyle {{\mathcal {A}}b}\rightarrow {\mathcal {H}}}$

application

In addition to the above-described construction of the whole numbers from the natural numbers, the formation of the K ₀ group of a ring is an important application. For each ring one considers the set (!) Of the isomorphism classes of finitely generated projective -left- modules with the direct sum as a semigroup link . The K ₀ group of the ring is then defined as the Grothendieck group of . ${\displaystyle R}$ ${\displaystyle \mathrm {Proj} (R)}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle \mathrm {Proj} (R)}$

literature

• Jonathan Rosenberg : Algebraic K-Theory and Its Applications (= Graduate Texts in Mathematics. Vol. 147). Springer, New York NY et al. 1994, ISBN 3-540-94248-3 .