Chow group

In algebraic geometry , a branch of mathematics , Chow groups are an important invariant of varieties.

definition

Let be a smooth, irreducible, projective variety over an algebraically closed field . ${\displaystyle X}$

The group of algebraic cycles of codimension i

${\displaystyle {\mathcal {Z}}^{i}(X)}$

is defined as the free Abelian group generated by the irreducible (not necessarily smooth) sub-varieties of the codimension . So an element is a finite sum ${\displaystyle W\subset X}$ ${\displaystyle i}$ ${\displaystyle Z\in {\mathcal {Z}}^{i}(X)}$

${\displaystyle Z=\sum _{\alpha }n_{\alpha }W_{\alpha }}$

with and irreducible sub-variety of the codimension . ${\displaystyle n_{\alpha }\in \mathbb {Z} }$ ${\displaystyle W_{\alpha }\subset X}$ ${\displaystyle i}$

Two sub-varieties

${\displaystyle Y,Z\subset X}$

are called rational equivalent if there is a sub-variety

${\displaystyle V\subset X\times P^{1}}$which is flat over , ${\displaystyle P^{1}}$

as well as with ${\displaystyle a,b\in P^{1}}$

${\displaystyle pr_{X}(V\cap X\times \left\{a\right\})=Y,pr_{X}(V\cap X\times \left\{b\right\})=Z}$

gives. Rational equivalence defines an equivalence relation on the cycle group . ${\displaystyle {\mathcal {Z}}^{i}(X)}$

The Chow group is defined as the quotient of the Zykel group modulo rational equivalence: ${\displaystyle CH^{i}(X)}$

${\displaystyle CH^{i}(X)={\mathcal {Z}}^{i}(X)/\sim }$.

Chow ring

The intersection product of sub-varieties (clearly: modulo rational equivalence, one puts sub-varieties in a general position and then takes their average) defines a mapping ${\displaystyle \times }$

${\displaystyle CH^{i}(X)\otimes CH^{j}(X)\rightarrow CH^{i+j}(X)}$

for everyone . The chow ring is the direct sum of the chow groups ${\displaystyle i,j}$

${\displaystyle CH(X)=\bigoplus _{i=0}^{\infty }CH^{i}(X)}$

with the multiplication defined by the cut product.

By means of the cleavage product to define the global average product by ${\displaystyle \times \colon CH^{k}(X)\otimes CH^{l}(X)\to CH^{k+l}(X)}$ ${\displaystyle \cdot \colon CH^{k}(X)\otimes CH^{l}(X)\to CH^{k+l-dim(X)}(X)}$

${\displaystyle x\cdot y:=\Delta ^{*}(x\times y)}$

for diagonal embedding . ${\displaystyle \Delta \colon X\to X\times X}$

Examples

• For every smooth, irreducible variety is

${\displaystyle CH^{0}(X)=\mathbb {Z} }$.

• ${\displaystyle CH^{1}(X)}$is the Picard group

${\displaystyle CH^{1}(X)=Pic(X)}$.

• For -dimensional affine space holds ${\displaystyle n}$ ${\displaystyle A^{n}}$

${\displaystyle CH^{k}(A^{n})=0}$for , ${\displaystyle k\not =0,n}$

${\displaystyle CH^{n}(A^{n})=\mathbb {Z} }$.

• For the -dimensional projective space applies ${\displaystyle n}$ ${\displaystyle P^{n}}$

${\displaystyle CH^{k}(P^{n})=\mathbb {Z} }$ For ${\displaystyle 0\leq k\leq n+1}$

${\displaystyle CH^{k}(P^{n})=0}$ For ${\displaystyle k>n+1}$

Relationship to the algebraic K-theory

Let be the function field of the variety and the Milnor's K-theory of this field. Then ${\displaystyle K(X)}$ ${\displaystyle X}$ ${\displaystyle K_{*}^{M}(K(X))}$

${\displaystyle CH^{k}(X)=Kokern(\bigcup _{x\in X_{(p+1)}}K_{1}^{M}(K(X))\to \bigcup _{x\in X_{(p)}}K_{0}^{M}(K(X))),}$

where is the set of all points of the dimension . ${\displaystyle X_{(p)}}$ ${\displaystyle X}$ ${\displaystyle p}$

literature

• Wei-Liang Chow : On Equivalence Classes of Cycles in an Algebraic Variety , Annals of Mathematics, Volume 64, 1956, pp. 450-479, ISSN  0003-486X
• William Fulton : Intersection theory , results of mathematics and their border areas. 3rd episode. A Series of Modern Surveys in Mathematics 2, Berlin, New York: Springer-Verlag 1998, ISBN 978-0-387-98549-7 , MR 1644323