Analogs of homology groups for algebraic varieties
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by Claude Chevalley (1958)) 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-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.
Rational equivalence and Chow groups.
For what follows, define a variety over a field formula_1 to be an integral scheme of finite type over formula_1. For any scheme formula_3 of finite type over formula_1, an algebraic cycle on formula_3 means a finite linear combination of subvarieties of formula_3 with integer coefficients. (Here and below, subvarieties are understood to be closed in formula_3, unless stated otherwise.) For a natural number formula_8, the group formula_9 of formula_8-dimensional cycles (or formula_8-cycles, for short) on formula_3 is the free abelian group on the set of formula_8-dimensional subvarieties of formula_3.
For a variety formula_15 of dimension formula_16 and any rational function formula_17 on formula_15 which is not identically zero, the divisor of formula_17 is the formula_8-cycle
formula_21
where the sum runs over all formula_8-dimensional subvarieties formula_23 of formula_15 and the integer formula_25 denotes the order of vanishing of formula_17 along formula_23. (Thus formula_25 is negative if formula_17 has a pole along formula_23.) The definition of the order of vanishing requires some care for formula_15 singular.
For a scheme formula_3 of finite type over formula_1, the group of formula_8-cycles rationally equivalent to zero is the subgroup of formula_9 generated by the cycles formula_36 for all formula_37-dimensional subvarieties formula_15 of formula_3 and all nonzero rational functions formula_17 on formula_15. The Chow group formula_42 of formula_8-dimensional cycles on formula_3 is the quotient group of formula_9 by the subgroup of cycles rationally equivalent to zero. Sometimes one writes formula_46 for the class of a subvariety formula_23 in the Chow group, and if two subvarieties formula_23 and formula_15 have formula_50, then formula_23 and formula_15 are said to be rationally equivalent.
For example, when formula_3 is a variety of dimension formula_54, the Chow group formula_55 is the divisor class group of formula_3. When formula_3 is smooth over formula_1 (or more generally, a locally Noetherian normal factorial scheme ), this is isomorphic to the Picard group of line bundles on formula_3.
Examples of Rational Equivalence.
Rational Equivalence on Projective Space.
Rationally equivalent cycles defined by hypersurfaces are easy to construct on projective space because they can all be constructed as the vanishing loci of the same vector bundle. For example, given two homogeneous polynomials of degree formula_60, so formula_61, we can construct a family of hypersurfaces defined as the vanishing locus of formula_62. Schematically, this can be constructed as
formula_63
using the projection formula_64 we can see the fiber over a point formula_65 is the projective hypersurface defined by formula_66. This can be used to show that the cycle class of every hypersurface of degree formula_60 is rationally equivalent to formula_68, since formula_69 can be used to establish a rational equivalence. Notice that the locus of formula_70 is formula_71 and it has multiplicity formula_60, which is the coefficient of its cycle class.
Rational Equivalence of Cycles on a Curve.
If we take two distinct line bundles formula_73 of a smooth projective curve formula_74, then the vanishing loci of a generic section of both line bundles defines non-equivalent cycle classes in formula_75. This is because formula_76 for smooth varieties, so the divisor classes of formula_77 and formula_78 define inequivalent classes.
The Chow ring.
When the scheme formula_3 is smooth over a field formula_1, the Chow groups form a ring, not just a graded abelian group. Namely, when formula_3 is smooth over formula_1, define formula_83 to be the Chow group of codimension-formula_8 cycles on formula_3. (When formula_3 is a variety of dimension formula_54, this just means that formula_88.) Then the groups formula_89 form a commutative graded ring with the product:
formula_90
The product arises from intersecting algebraic cycles. For example, if formula_91 and formula_23 are smooth subvarieties of formula_3 of codimension formula_8 and formula_95 respectively, and if formula_91 and formula_23 intersect transversely, then the product formula_98 in formula_99 is the sum of the irreducible components of the intersection formula_100, which all have codimension formula_101.
More generally, in various cases, intersection theory constructs an explicit cycle that represents the product formula_98 in the Chow ring. For example, if formula_91 and formula_23 are subvarieties of complementary dimension (meaning that their dimensions sum to the dimension of formula_3) whose intersection has dimension zero, then formula_98 is equal to the sum of the points of the intersection with coefficients called intersection numbers. For any subvarieties formula_91 and formula_23 of a smooth scheme formula_3 over formula_1, with no assumption on the dimension of the intersection, William Fulton and Robert MacPherson's intersection theory constructs a canonical element of the Chow groups of formula_100 whose image in the Chow groups of formula_3 is the product formula_98.
Examples.
Projective space.
The Chow ring of projective space formula_114 over any field formula_1 is the ring
 formula_116
where formula_117 is the class of a hyperplane (the zero locus of a single linear function). Furthermore, any subvariety formula_91 of degree formula_60 and codimension formula_120 in projective space is rationally equivalent to formula_121. It follows that for any two subvarieties formula_91 and formula_23 of complementary dimension in formula_114 and degrees formula_120, formula_126, respectively, their product in the Chow ring is simply
 formula_127
where formula_128 is the class of a formula_1-rational point in formula_114. For example, if formula_91 and formula_23 intersect transversely, it follows that formula_100 is a zero-cycle of degree formula_134. If the base field formula_1 is algebraically closed, this means that there are exactly formula_134 points of intersection; this is a version of Bézout's theorem, a classic result of enumerative geometry.
Projective bundle formula.
Given a vector bundle formula_137 of rank formula_138 over a smooth proper scheme formula_3 over a field, the Chow ring of the associated projective bundle formula_140 can be computed using the Chow ring of formula_3 and the Chern classes of formula_142. If we let formula_143 and formula_144 the Chern classes of formula_142, then there is an isomorphism of rings
formula_146
Hirzebruch surfaces.
For example, the Chow ring of a Hirzebruch surface can be readily computed using the projective bundle formula. Recall that it is constructed as formula_147 over formula_148. Then, the only non-trivial Chern class of this vector bundle is formula_149. This implies that the Chow ring is isomorphic to
formula_150
Remarks.
For other algebraic varieties, Chow groups can have richer behavior. For example, let formula_3 be an elliptic curve over a field formula_1. Then the Chow group of zero-cycles on formula_3 fits into an exact sequence
formula_154
Thus the Chow group of an elliptic curve formula_3 is closely related to the group formula_156 of formula_1-rational points of formula_3. When formula_1 is a number field, formula_156 is called the Mordell–Weil group of formula_3, and some of the deepest problems in number theory are attempts to understand this group. When formula_1 is the complex numbers, the example of an elliptic curve shows that Chow groups can be uncountable abelian groups.
Functoriality.
For a proper morphism formula_163 of schemes over formula_1, there is a pushforward homomorphism formula_165 for each integer formula_8. For example, for a proper scheme formula_3 over formula_1, this gives a homomorphism formula_169, which takes a closed point in formula_3 to its degree over formula_1. (A closed point in formula_3 has the form formula_173 for a finite extension field formula_142 of formula_1, and its degree means the degree of the field formula_142 over formula_1.)
For a flat morphism formula_163 of schemes over formula_1 with fibers of dimension formula_138 (possibly empty), there is a homomorphism formula_181.
A key computational tool for Chow groups is the localization sequence, as follows. For a scheme formula_3 over a field formula_1 and a closed subscheme formula_23 of formula_3, there is an exact sequence
formula_186
where the first homomorphism is the pushforward associated to the proper morphism formula_187, and the second homomorphism is pullback with respect to the flat morphism formula_188. The localization sequence can be extended to the left using a generalization of Chow groups, (Borel–Moore) motivic homology groups, also known as higher Chow groups.
For any morphism formula_163 of smooth schemes over formula_1, there is a pullback homomorphism formula_191, which is in fact a ring homomorphism formula_192.
Examples of flat pullbacks.
Note that non-examples can be constructed using blowups; for example, if we take the blowup of the origin in formula_193 then the fiber over the origin is isomorphic to formula_148.
Branched coverings of curves.
Consider the branched covering of curves
formula_195
Since the morphism ramifies whenever formula_196 we get a factorization
formula_197
where one of the formula_198. This implies that the points formula_199 have multiplicities formula_200 respectively. The flat pullback of the point formula_201 is then
formula_202
Flat family of varieties.
Consider a flat family of varieties
formula_203
and a subvariety formula_204. Then, using the cartesian square
formula_205
we see that the image of formula_206 is a subvariety of formula_3. Therefore, we have
formula_208
Cycle maps.
There are several homomorphisms (known as cycle maps) from Chow groups to more computable theories.
First, for a scheme "X" over the complex numbers, there is a homomorphism from Chow groups to Borel–Moore homology:
formula_209
The factor of 2 appears because an "i"-dimensional subvariety of "X" has real dimension 2"i". When "X" is smooth over the complex numbers, this cycle map can be rewritten using Poincaré duality as a homomorphism
formula_210
In this case ("X" smooth over C), these homomorphisms form a ring homomorphism from the Chow ring to the cohomology ring. Intuitively, this is because the products in both the Chow ring and the cohomology ring describe the intersection of cycles.
For a smooth complex projective variety, the cycle map from the Chow ring to ordinary cohomology factors through a richer theory, Deligne cohomology. This incorporates the Abel–Jacobi map from cycles homologically equivalent to zero to the intermediate Jacobian. The exponential sequence shows that "CH"1("X") maps isomorphically to Deligne cohomology, but that fails for "CH""j"("X") with "j" ; 1.
For a scheme "X" over an arbitrary field "k", there is an analogous cycle map from Chow groups to (Borel–Moore) etale homology. When "X" is smooth over "k", this homomorphism can be identified with a ring homomorphism from the Chow ring to etale cohomology.
Relation to K-theory.
An (algebraic) vector bundle "E" on a smooth scheme "X" over a field has Chern classes "c""i"("E") in "CH""i"("X"), with the same formal properties as in topology. The Chern classes give a close connection between vector bundles and Chow groups. Namely, let "K"0("X") be the Grothendieck group of vector bundles on "X". As part of the Grothendieck–Riemann–Roch theorem, Grothendieck showed that the Chern character gives an isomorphism
formula_211
This isomorphism shows the importance of rational equivalence, compared to any other adequate equivalence relation on algebraic cycles.
Conjectures.
Some of the deepest conjectures in algebraic geometry and number theory are attempts to understand Chow groups. For example:
For example, let "X" be a smooth complex projective surface. The Chow group of zero-cycles on "X" maps onto the integers by the degree homomorphism; let "K" be the kernel. If the geometric genus "h"0("X", Ω2) is not zero, Mumford showed that "K" is "infinite-dimensional" (not the image of any finite-dimensional family of zero-cycles on "X"). The Bloch–Beilinson conjecture would imply a satisfying converse, Bloch's conjecture on zero-cycles: for a smooth complex projective surface "X" with geometric genus zero, "K" should be finite-dimensional; more precisely, it should map isomorphically to the group of complex points of the Albanese variety of "X".
Variants.
Bivariant theory.
Fulton and MacPherson extended the Chow ring to singular varieties by defining the "operational Chow ring" and more generally a bivariant theory associated to any morphism of schemes. A bivariant theory is a pair of covariant and contravariant functors that assign to a map a group and a ring respectively. It generalizes a cohomology theory, which is a contravariant functor that assigns to a space a ring, namely a cohomology ring. The name "bivariant" refers to the fact that the theory contains both covariant and contravariant functors.
This is in a sense the most elementary extension of the Chow ring to singular varieties; other theories such as motivic cohomology map to the operational Chow ring.
Other variants.
Arithmetic Chow groups are an amalgamation of Chow groups of varieties over Q together with a component encoding Arakelov-theoretical information, that is, differential forms on the associated complex manifold.
The theory