In algebraic K-theory, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Bloch's formula, introduced by

Spencer Bloch Spencer Janney Bloch (born May 22, 1944; New York City) is an American mathematician known for his contributions to algebraic geometry and algebraic ''K''-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Departm ...

for

K_2

, states that the

Chow group In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-c ...

of a smooth variety ''X'' over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

is isomorphic to the cohomology of ''X'' with coefficients in the K-theory of the structure sheaf

\mathcal_X

; that is, ::

\operatorname^q(X) = \operatorname^q(X, K_q(\mathcal_X))

where the right-hand side is the sheaf cohomology;

K_q(\mathcal_X)

is the sheaf associated to the presheaf

U \mapsto K_q(U)

, ''U'' Zariski open subsets of ''X''. The general case is due to Quillen.For a sketch of the proof, besides the original paper, see http://www-bcf.usc.edu/~ericmf/lectures/zurich/zlec5.pdf For ''q'' = 1, one recovers

\operatorname(X) = H^1(X, \mathcal_X^*)

. (see also Picard group.) The formula for the

mixed characteristic In commutative algebra, a ring of mixed characteristic is a commutative ring R having characteristic zero and having an ideal I such that R/I has positive characteristic.. Examples * The integers \mathbb have characteristic zero, but for any pr ...

is still open.

References

* Daniel Quillen: Higher algebraic K-theory: I. In: H. Bass (ed.): Higher K-Theories. Lecture Notes in Mathematics, vol. 341. Springer-Verlag, Berlin 1973. Algebraic K-theory Algebraic geometry Theorems in algebraic topology {{algebraic-geometry-stub