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.

Universal property

In its simplest form, the Grothendieck group of a commutative monoid is the universal way of making that monoid into an abelian group. Let M be a commutative monoid. Its Grothendieck group N should have the following universal property: There exists a monoid homomorphism

i:M→N

such that for any monoid homomorphism

f:M→A

from the commutative monoid M to an abelian group A, there is a unique group homomorphism

g:N→A

such that

f=gi.

In the language of category theory, the functor that sends a commutative monoid M to its Grothendieck group N is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids.

Explicit construction

To construct the Grothendieck group of a commutative monoid M, one forms the Cartesian product

M×M.

The two coordinates are meant to represent a positive part and a negative part:

(m, n)

is meant to correspond to

m − n.

Addition is defined coordinate-wise:

(m₁, m₂) + (n₁, n₂) = (m₁ + n₁, m₂ + n₂).

Next we define an equivalence relation on M×M. We say that (m₁, m₂) is equivalent to (n₁, n₂) if, for some element k of M, m₁ + n₂ + k = m₂ + n₁ + k. It is easy to check that the addition operation is compatible with the equivalence relation. The identity element is now any element of the form (m, m), and the inverse of (m₁, m₂) is (m₂, m₁).

In this form, the Grothendieck group is the fundamental construction of K-theory. The group K₀(M) of a manifold M is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of vector bundles of finite rank on M with the monoid operation given by direct sum.

The Grothendieck group can also be constructed using generators and relations: denoting by (Z(M),+') the free abelian group generated by the set M, the Grothendieck group is the quotient of Z(M) by the subgroup generated by $\{x+\text{'}y-\text{'}(x+y)mid\; x,yin\; M\}$.

Generalization

To apply the Grothendieck group to purely algebraic settings, it is useful to generalize it to the case of an essentially small abelian category. To do this, let $mathcal\; A$ be an essentially small abelian category. Let F be the free abelian group generated by isomorphism classes of objects of the category. (This is where the hypothesis of essential smallness is necessary; without it, F would not be a set.) We will impose some relations on F. Call R the subgroup of F generated as follows: For each exact sequence 0→A→B→C→0 in $mathcal\; A$, the element

[A] + [C] - [B]

is in R. Then the Grothendieck group $K\_0(\{mathcal\; A\})$ is F/R.

K₀ of an abelian category has a similar universal property to K₀ of a commutative monoid. We make a preliminary definition: A function χ from isomorphism classes of objects of an abelian category $mathcal\; A$ to an abelian group A is called additive if, for each exact sequence 0→A→B→C→0, we have χ(A) + χ(C) - χ(B) = 0. Then, for any additive function χ:$mathcal\; A$→A, there is a unique abelian group homomorphism f:$K\_0\{mathcal\; A\}$→A such that χ factors through f and the map that takes each object of $mathcal\; A$ to the element representing its isomorphism class in $K\_0(\{mathcal\; A\})$.

This universal property makes $K\_0(\{mathcal\; A\})$ the 'universal receiver' of generalized Euler characteristics. In particular, for every bounded complex of objects in $\{mathcal\; A\}$

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

we have a canonical element

$[A^*]\; =\; sum\_i\; (-1)^i\; [A^i]\; =\; sum\_i\; (-1)^i\; [H^i\; (A^*)]\; in\; K\_0.$

In fact the Grothendieck group was originally introduced for the study of Euler characteristics.

Splitting principle

The relationship between K₀ of a commutative monoid and K₀ of an abelian category comes from the splitting principle. According to the splitting principle, we can always take an exact sequence 0→A→B→C→0 and find a closely related exact sequence in which the middle term splits, that is, it is the direct sum of the other two terms. Because of this, the Grothendieck group of the commutative monoid of vector bundles on a smooth manifold is the same as the Grothendieck group of the abelian category of vector bundles on that same smooth manifold.

K₀ is often defined for a ring or for a ringed space. The usual construction is as follows: For a not necessarily commutative ring R, one lets the abelian category $mathcal\; A$ be the category of all finitely generated projective modules over the ring. For a ringed space (X,O_X), one lets the abelian category $mathcal\; A$ be the category of all coherent sheaves on X. This makes K₀ into a functor.

There is another Grothendieck group of a ring or a ringed space which is sometimes useful. The Grothendieck group G₀ of a ring is the Grothendieck group associated to the category of all finitely generated modules over a ring. Similarly, the Grothendieck group G₀ of a ringed space is the Grothendieck group associated to the category of all quasicoherent sheaves on the ringed space. G₀ is not a functor, but nevertheless it carries important information.

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 $[V]\; =\; [k^\{mbox\{dim\}(V)\}]$ in $K\_0(Vect\_\{fin\})$. 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 $[V]\; =\; dim(V)[k]$, the Grothendieck group $K\_0(Vect\_\{fin\})$ is isomorphic to $\{mathbb\; Z\}$ and is generated by [k]. Finally for a bounded complex of finite dimensional vector spaces $V^*$,

$[V^*]\; =\; chi(V^*)[k]$

where $chi$ is the standard Euler characteristic defined by

$chi(V^*)=\; sum\_i\; (-1)^i\; \{rm\; dim\; \}\; V\; =\; sum\_i\; (-1)^i\{rm\; dim\; \}\; H^i(V^*)$

References

• Michael F. Atiyah, K-Theory, (Notes taken by D.W.Anderson, Fall 1964), published in 1967, W.A. Benjamin Inc., New York.