be thinking of the equivalence classes as formal differences Another useful observation is the invariance of equivalence classes under scaling: for any The Grothendieck completion can be viewed as a functor and it has the property that it is left adjoint to the corresponding forgetful functor That means that, given a morphism of an abelian monoid to the underlying abelian monoid of an abelian group there exists a unique abelian group morphism Example for natural numbers An illustrative example to look at is the Grothendieck completion of . We can see that For any pair we can find a minimal representative by using the invariance under scaling. For example, we can see from the scaling invariance that In general, if then which is of the form or This shows that we should think of the as positive integers and the as negative integers. Definitions There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry. Grothendieck group for compact Hausdorff spaces Given a compact Hausdorff space consider the set of isomorphism classes of finite-dimensional vector bundles over , denoted and let the isomorphism class of a vector bundle be denoted . Since isomorphism classes of vector bundles behave well with respect to direct sums, we can write these operations on isomorphism classes by It should be clear that is an abelian monoid where the unit is given by the trivial vector bundle . We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of and is denoted . We can use the Serre–Swan theorem and some algebra to get an alternative description of vector bundles over the ring of continuous complex-valued functions as projective modules. Then, these can be identified with idempotent matrices in some ring of matrices . We can define equivalence classes of idempotent matrices and form an abelian monoid . Its Grothendieck completion is also called . One of the main techniques for computing the Grothendieck group for topological spaces comes from the Atiyah–Hirzebruch spectral sequence, which makes it very accessible. The only required computations for understanding the spectral sequences are computing the group for the spheres .pg 51-110 Grothendieck group of vector bundles in algebraic geometry There is an analogous construction by considering vector bundles in algebraic geometry. For a Noetherian scheme there is a set of all