In the modern language, Grothendieck defined only K0, the zeroth K-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem.
The lower K-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found.
The Riemann–Roch theorem states that the difference in dimensions between these subspaces is equal to the degree of the line bundle (a measure of twistedness) plus one minus the genus of X.
This gives two ways of determining an element in the Chow group of Y from a vector bundle on X: Starting from X, one can first compute the pushforward in K-theory and then apply the Chern character and Todd class of Y, or one can first apply the Chern character and Todd class of X and then compute the pushforward for Chow groups.
While the group K0 seemed to satisfy the necessary properties to be the beginning of a cohomology theory of algebraic varieties and of non-commutative rings, there was no clear definition of the higher Kn(X).
Even as such definitions were developed, technical issues surrounding restriction and gluing usually forced Kn to be defined only for rings, not for varieties.
Of particular note is that Bass, building on his earlier work with Murthy,[7] provided the first proof of what is now known as the fundamental theorem of algebraic K-theory.
Not only did this recover K1 and K2, the relation of K-theory to the Adams operations allowed Quillen to compute the K-groups of finite fields.
Since this sequence was fundamental to many of the facts in the subject, regularity hypotheses pervaded early work on higher K-theory.
Stephen Smale's h-cobordism theorem[25] asserted that if n ≥ 5, W is compact, and M, N, and W are simply connected, then W is isomorphic to the cylinder M × [0, 1] (in TOP, PL, or DIFF as appropriate).
The s-cobordism theorem, due independently to Mazur,[26] Stallings, and Barden,[27] explains the general situation: An h-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion M ⊂ W vanishes.
This generalizes the h-cobordism theorem because the simple connectedness hypotheses imply that the relevant Whitehead group is trivial.
In fact the s-cobordism theorem implies that there is a bijective correspondence between isomorphism classes of h-cobordisms and elements of the Whitehead group.
Jean Cerf proved that for simply connected smooth manifolds M of dimension at least 5, isotopy of h-cobordisms is the same as a weaker notion called pseudo-isotopy.
This space contains strictly more information than the Whitehead group; for example, the connected component of the trivial cobordism describes the possible cylinders on M and in particular is the obstruction to the uniqueness of a homotopy between a manifold and M × [0, 1].
Quillen suggested to his student Kenneth Brown that it might be possible to create a theory of sheaves of spectra of which K-theory would provide an example.
[32] Spencer Bloch, influenced by Gersten's work on sheaves of K-groups, proved that on a regular surface, the cohomology group
[35] Quillen's proposed spectral sequence would start from the étale cohomology of a ring R and, in high enough degrees and after completing at a prime l invertible in R, abut to the l-adic completion of the K-theory of R. In the case studied by Lichtenbaum, the spectral sequence would degenerate, yielding Lichtenbaum's conjecture.
This was done by Thomason in a lengthy monograph which he co-credited to his dead friend Thomas Trobaugh, who he said gave him a key idea in a dream.
[40] Thomason combined Waldhausen's construction of K-theory with the foundations of intersection theory described in volume six of Grothendieck's Séminaire de Géométrie Algébrique du Bois Marie.
Goodwillie, motivated by his "calculus of functors", conjectured the existence of a theory intermediate to K-theory and Hochschild homology.
In the mid-1980s, Bokstedt gave a definition of topological Hochschild homology that satisfied nearly all of Goodwillie's conjectural properties, and this made possible further computations of K-groups.
This transformation factored through the fixed points of a circle action on THH, which suggested a relationship with cyclic homology.
The functor K0 takes a ring A to the Grothendieck group of the set of isomorphism classes of its finitely generated projective modules, regarded as a monoid under direct sum.
If the ring A is commutative, we can define a subgroup of K0(A) as the set where : is the map sending every (class of a) finitely generated projective A-module M to the rank of the free
In the case of a central simple algebra A over a field F, the reduced norm provides a generalisation of the determinant giving a map K1(A) → F∗ and SK1(A) may be defined as the kernel.
[77] The above expression for K2 of a field k led Milnor to the following definition of "higher" K-groups by thus as graded parts of a quotient of the tensor algebra of the multiplicative group k× by the two-sided ideal, generated by the For n = 0,1,2 these coincide with those below, but for n ≧ 3 they differ in general.
denotes the group of m-th roots of unity in some separable extension of k. This extends to satisfying the defining relations of the Milnor K-group.
[84] One possible definition of higher algebraic K-theory of rings was given by Quillen Here πn is a homotopy group, GL(R) is the direct limit of the general linear groups over R for the size of the matrix tending to infinity, B is the classifying space construction of homotopy theory, and the + is Quillen's plus construction.
[69] Parshin's conjecture concerns the higher algebraic K-groups for smooth varieties over finite fields, and states that in this case the groups vanish up to torsion.