sociology - Grothendieck group

In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid, in the best possible way. It takes its name from the more general construction in category theory, introduced by Alexander Grothendieck in his fundamental work of the mid-1950s that resulted in the development of K-theory.

The Grothendieck group

$K_0({\mathcal A})$

of a (essentially small ) abelian category ${\mathcal A}$ is the abelian group generated by the objects of the category, subject to the following relations : for each exact sequence

$0 \to A \to B \to C \to 0$ ,

we have

[B] = [A] + [C].

In particular, if objects are isomorphic in ${\mathcal A}$ they define the same element in $K_0({\mathcal A})$ .

The Grothendieck group has the following universal property: every function $χ$ from (isomorphism classes of) ${\mathcal A}$ to an abelian group $R$ , such that

χ(B) = χ(A) + χ(C)

for an exact sequence as above, factors over $K_0({\mathcal A})$ .

In other words $K_0({\mathcal A})$ is the 'universal receiver' of generalized Euler characteristics; and in particular every bounded complex of objects in ${\mathcal A}$

$\cdots \to 0 \to 0 \to A^n \to A^{n+1} \to \cdots \to A^{m-1} \to A^m \to 0 \to 0 \to \cdots$

defines a canonical element

$[A^*] = \sum_i (-1)^i [A^i] = \sum_i (-1)^i [H^i (A^*)] \in K_0.$

Example

In the abelian category of finite dimensional vector spaces over a field $k$ , two vector spaces are isomorphic if and only if they have the same dimension. Thus, for a vector space V the class $[V] = [k dim(V)]$ in $K 0 (V e c t f i n)$ . Moreover for an exact sequence

$0 \to k^l \to k^m \to k^n \to 0$

m = l + n, so

[k l + n] = [k l] + [k n] = (l + n)[k].

Thus $[V] = d i m (V)[k]$ , the Grothendieck group $K 0 (V e c t f i n)$ is isomorphic to ${\mathbb Z}$ and is generated by [k]. Finally for a bounded complex of finite dimensional vector spaces $V *$ ,

[V *] = χ(V *)[k]

where $χ$ is the standard Euler characteristic defined by

χ(V *) = \sum ( - 1) i dim(V) = \sum ( - 1) i dim(H i (V *)) i i

Categories: Abstract algebra | Homological algebra