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algebraic cycle

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Definition

Let XX be a noetherian scheme. One defines a kk-dimensional algebraic cycle as an element of the free abelian group Z k(X)Z_k(X) generated by the closed integral subschemes of dimension kk, and dually a kk-codimensional cycle is an element of the free group Z k(X)Z^k(X) generated by the closed integral subschemes of codimension kk in XX. (An important special case is the group Z 1(X)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= ZXn Z.[Z] C = \sum_{Z \subset X} n_Z.[Z]

Direct image and inverse image of cycles

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

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

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

For flat morphisms of relative dimension nn, one defines the inverse image of a kk-cycle by assigning, for a closed integral subscheme ZYZ \subset Y of dimension kk,

f *([Z])= Z αf 1(Z)length O X,z α(O f 1(Z),z α)[Z α] 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 αZ_\alpha of f 1(Z)f^{-1}(Z), length\length denotes length of modules, and z αz_\alpha are the generic points of Z αZ_\alpha. Hence one gets homomorphisms f *:Z k(Y)Z k+n(X)f^* : Z_k(Y) \to Z_{k+n}(X).

Weil divisors and rational functions

A Weil divisor on XX is a 1-codimensional cycle.

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

ord x(r)=length O X,x(O X,x/(a x))length O X,x(O X,x/(b x)) \ord_x(r) = \length_{O_{X,x}}(O_{X,x}/(a_x)) - \length_{O_{X,x}}(O_{X,x}/(b_x))

where length\length denotes length of modules.

Then one defines the Weil divisor associated to the rational function rr as

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

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

References

Standard references are

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

A relation to iterated integrals and diffeological spaces is discussed in

Last revised on April 2, 2016 at 04:39:23. See the history of this page for a list of all contributions to it.