Chow ring
The ring of rational equivalence classes of algebraic cycles (cf. Algebraic cycle) on a non-singular quasi-projective algebraic variety. Multiplication in this ring is defined in terms of intersections of cycles (cf. Intersection theory).
The Chow ring of a variety
is a graded commutative ring, where
denotes the group of classes of cycles of codimension
. For a morphism
the inverse-image homomorphism
is a homomorphism of rings, and the direct-image homomorphism
is (for proper
) a homomorphism of
-modules. This means that there is a projection formula:
![]() |
The Chow ring is the domain of values for the theory of Chern classes of vector bundles (cf. [1]). More precisely, if is a locally trivial sheaf of rank
over a variety
, if
is its projectivization, if
is the canonical projection, and if
is the class of divisors corresponding to the invertible sheaf
, then
is an imbedding and the Chow ring
may be identified with the quotient ring of the polynomial ring
by the ideal generated by the polynomial
![]() |
The coefficient is called the
-th Chern class of the sheaf
.
In the case of a variety over the field of complex numbers, there is a homomorphism into the singular cohomology ring that preserves the degree and commutes with the inverse-image and direct-image homomorphisms.
If is a singular quasi-projective variety, then its Chow ring
is defined as the direct limit of rings
over all morphisms
, where
is non-singular. One obtains a contravariant functor into the category of graded rings, satisfying the projection formula (cf. [3]).
References
[1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
[2] | "Anneaux de Chow et applications" , Sem. Chevalley (1958) Zbl 0098.13101 |
[3] | W. Fulton, "Rational equivalence on singular varieties" Publ. Math. IHES , 45 (1975) pp. 147–167 MR0404257 Zbl 0332.14002 |
Comments
For a Noetherian scheme (or ring), let
denote the
-groups of (the category of) finitely-generated projective modules over
; cf. Algebraic
-theory. Let
denote the sheaf obtained by sheafifying (in the Zariski topology) the pre-sheaf
where
runs through the open (affine) subschemes of
. One then has the Bloch formula [a1]
![]() |
providing a link between the Chow groups of and the cohomology of
with values in the
-sheaves of
. Using results on the algebraic
-theory of fields, [a2], this can be used to obtain results on the
, in particular
, [a3]. Another often used notation for the Chow group is
instead of
.
Cf. Sheaf theory for the notions of sheafification, pre-sheaf, sheaf, and cohomology with values in a sheaf.
References
[a1] | S. Bloch, "Lectures on algebraic cycles" , Dept. Math. Duke Univ. (1980) MR0558224 Zbl 0436.14003 |
[a2] | A.S. Merkur'ev, A.A. Suslin, "![]() |
[a3] | J.-L. Colliot-Thélène, "Hilbert's theorem 90 for ![]() |
[a4] | W. Fulton, "Intersection theory" , Springer (1984) MR0735435 MR0732620 Zbl 0541.14005 |
Chow ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chow_ring&oldid=25193