free; this assertion is correct, but was not settled until 20 years later. (Swan's theorem is another aspect of this analogy.) Developments The other historical origin of algebraic K-theory was the work of J. H. C. Whitehead and others on what later became known as Whitehead torsion. There followed a period in which there were various partial definitions of higher K-theory functors. Finally, two useful and equivalent definitions were given by Daniel Quillen using homotopy theory in 1969 and 1972. A variant was also given by Friedhelm Waldhausen in order to study the algebraic K-theory of spaces, which is related to the study of pseudo-isotopies. Much modern research on higher K-theory is related to algebraic geometry and the study of motivic cohomology. The corresponding constructions involving an auxiliary quadratic form received the general name L-theory. It is a major tool of surgery theory. In string theory, the K-theory classification of Ramond–Ramond field strengths and the charges of stable D-branes was first proposed in 1997. Examples and properties K0 of a field The easiest example of the Grothendieck group is the Grothendieck group of a point for a field . Since a vector bundle over this space is just a finite dimensional vector space, which is a free object in the category of coherent sheaves, hence projective, the monoid of isomorphism classes is corresponding to the dimension of the vector space. It is an easy exercise to show that the Grothendieck group is then . K0 of an Artinian algebra over a field One important property of the Grothendieck group of a Noetherian scheme is that it is invariant under reduction, hence . Hence the Grothendieck group of any Artinian -algebra is a direct sum of copies of , one for each connected component of its spectrum. For example, K0 of projective space One of the most commonly used computations of the Grothendieck group is with the computation of for projective space over a field. This is because the intersection numbers of a projective can be computed by embedding and using the push pull formula . This makes it possible to do concrete calculations with elements in without having to explicitly know its structure since One technique for determining the Grothendieck group of comes from its stratification as since the Grothendieck group of coherent sheaves on affine spaces are isomorphic to , and the intersection of is generically for . K0