Grothendieck Group - Examples

Examples

• The easiest example of the Grothendieck group construction is the construction of the integers from the natural numbers. First one observes that the natural numbers together with the usual addition indeed form a commutative monoid (N,+).

Now when we use the Grothendieck group construction we obtain the formal differences between natural numbers as elements n - m and we have the equivalence relation

.

Now define

,

for all n ∈ N. This defines the integers Z. Indeed this is the usual construction to obtain the integers from the natural numbers. See "Construction" under Integers for a more detailed explanation.

• 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 in . Moreover for an exact sequence

m = l + n, so

Thus, the Grothendieck group is isomorphic to Z and is generated by . Finally for a bounded complex of finite dimensional vector spaces V*,

where is the standard Euler characteristic defined by

• is often defined for a ring. The usual construction is as follows: For a (not necessarily commutative) ring R, one defines the category to be the category of all finitely generated projective modules over the ring. K₀(R) is then defined to be the Grothendieck group of . This gives a (contravariant) functor of R.

• A special case of the above is the case where R is the ring of (say complex-valued) smooth functions on a compact manifold X. In this case the projective R-modules are dual to vector bundles over X (by the Serre-Swan theorem). The above construction thus reconstructs the zeroth topological K-theory group K₀(X), i.e. the Grothendieck group of the commutative monoid of (isomorphism classes) vector bundles over X with addition being the direct sum. (This time K₀(X) is covariant functor of X because of the duality in the intermediate step).

• A ringed-space-version of the latter example works as follows: Choose to be the category of all locally free sheaves over X. K₀(X) is again defined as the Grothendieck group of this category and again this gives a functor.

• For a ringed space, one can also define the category to be the category of all coherent sheaves on X. This includes the special case (if the ringed space is an affine scheme) of being the category of finitely generated modules over a noetherian ring R. In both cases is an abelian category and a fortiori an exact category so the construction above applies.

• In the special case where R is a finite dimensional algebra over some field this reduces to the Grothendieck group G₀(R) mentioned above. If the ring is additionally ℤ-graded, the Grothendieck group is naturally a ℤ-module, where q correspond to the grading shift by 1.

• There is another Grothendieck group G₀ of a ring or a ringed space which is sometimes useful. The category in the case is chosen to be the category of all quasicoherent sheaves on the ringed space which reduces to the category of all modules over some ring R in case of affine schemes. G₀ is not a functor, but nevertheless it carries important information.

• Since the (bounded) derived category is triangulated, there is a Grothendieck group for derived categories too. This has applications in representation theory for example. For the unbounded category the Grothendieck group however vanishes. For a derived category of some complex finite dimensional positively graded algebra there is a subcategory in the unbounded derived category containing the abelian category A of finite dimensional graded modules whose Grothendieck group is the q-adic completion of the Grothendieck group of A.