isomorphism classes of algebraic vector bundles on . Then, as before, the direct sum of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid . Then, the Grothendieck group is defined by the application of the Grothendieck construction on this abelian monoid. Grothendieck group of coherent sheaves in algebraic geometry In algebraic geometry, the same construction can be applied to algebraic vector bundles over a smooth scheme. But, there is an alternative construction for any Noetherian scheme . If we look at the isomorphism classes of coherent sheaves we can mod out by the relation if there is a short exact sequence This gives the Grothendieck-group which is isomorphic to if is smooth. The group is special because there is also a ring structure: we define it as Using the Grothendieck–Riemann–Roch theorem, we have that is an isomorphism of rings. Hence we can use for intersection theory. Early history The subject can be said to begin with Alexander Grothendieck (1957), who used it to formulate his Grothendieck–Riemann–Roch theorem. It takes its name from the German Klasse, meaning "class". Grothendieck needed to work with coherent sheaves on an algebraic variety X. Rather than working directly with the sheaves, he defined a group using isomorphism classes of sheaves as generators of the group, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is called K(X) when only locally free sheaves are used, or G(X) when all are coherent sheaves. Either of these two constructions is referred to as the Grothendieck group; K(X) has cohomological behavior and G(X) has homological behavior. If X is a smooth variety, the two groups are the same. If it is a smooth affine variety, then all extensions of locally free sheaves split, so the group has an alternative definition. In topology, by applying the same construction to vector bundles, Michael Atiyah and Friedrich Hirzebruch defined K(X) for a topological space X in 1959, and using the Bott periodicity theorem they made it the basis of an extraordinary cohomology theory. It played a major role in the second proof of the Atiyah–Singer index theorem (circa 1962). Furthermore, this approach led to a noncommutative K-theory for C*-algebras. Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that every finitely generated projective module over a polynomial ring is