nLab algebraic cycle

Definition

Let $X$ be a noetherian scheme. One defines a $k$-dimensional algebraic cycle as an element of the free abelian group $Z_k(X)$ generated by the closed integral subschemes of dimension $k$, and dually a $k$-codimensional cycle is an element of the free group $Z^k(X)$ generated by the closed integral subschemes of codimension $k$ in $X$. (An important special case is the group $Z^1(X)$ of 1-codimensional cycles, better known as the group of Weil divisors.) One usually writes a cycle as a formal sum

$$C = \sum_{Z \subset X} n_Z.[Z]$$

Direct image and inverse image of cycles

For proper morphisms $f : X \to Y$, one defines the direct image of a $k$-cycle by assigning

$$f_*([Z]) = \deg_{R(f(Z))}(R(Z)) . [f(Z)]$$

when $\dim(f(Z)) = \dim(Z)$ and 0 otherwise. Here $f(Z)$ is considered as an integral subscheme of $Y$ with the reduced subscheme structure induced from $Y$. $R(Z)$ denotes the field of rational functions on $Z$ and $\deg$ denotes the degree of the field extension. One gets homomorphisms $f_* : Z_k(X) \to Z_k(Y)$ for each $k$.

For flat morphisms of relative dimension $n$, one defines the inverse image of a $k$-cycle by assigning, for a closed integral subscheme $Z \subset Y$ of dimension $k$,

$$f^*([Z]) = \sum_{Z_\alpha \subset f^{-1}(Z)} \length_{O_{X,z_\alpha}}(O_{f^{-1}(Z),z_\alpha}) [Z_\alpha]$$

where the sum is taken over the irreducible components $Z_\alpha$ of $f^{-1}(Z)$, $\length$ denotes length of modules, and $z_\alpha$ are the generic points of $Z_\alpha$. Hence one gets homomorphisms $f^* : Z_k(Y) \to Z_{k+n}(X)$.

Weil divisors and rational functions

A Weil divisor on $X$ is a 1-codimensional cycle.

A rational function $r \in R(X)$ on an integral scheme $X$ corresponds via canonical isomorphisms $R(X) \to \Frac(O_{X,x})$, for every $x \in X$, to elements $a_x/b_x \in \Frac(O_{X,x})$, and one defines the order of vanishing of $r$ at $x$ as

$$\ord_x(r) = \length_{O_{X,x}}(O_{X,x}/(a_x)) - \length_{O_{X,x}}(O_{X,x}/(b_x))$$

where $\length$ denotes length of modules.

Then one defines the Weil divisor associated to the rational function $r$ as

$$div(r) = \sum_{Z \subset X} \ord_z(r).[Z]$$

where the sum goes over closed integral subschemes $Z$ of codimension 1 and with generic point $z \in Z$.

Related concepts

• Chow groups

• Cartier divisor

• Abel-Jacobi map

• standard conjectures

References

Standard references are

• EGA IV, section 21
• Stacks Project, 02QQ

On the relation with Weil cohomology theories, algebraic K-theory, Beilinson-Lichtenbaum conjectures?, and motivic cohomology:

• Marc Levine, Algebraic cycle complexes, talk notes, 2008, pdf.

A relation to iterated integrals and diffeological spaces is discussed in

• Richard Hain, Iterated Integrals and Algebraic Cycles: Examples and Prospects (arXiv:math/0109204)