Chow ring

In algebraic geometry, the Chow ring (named after W. L. Chow) of an algebraic variety is an algebraic-geometric analogue of the cohomology ring of the variety considered as a topological space: its elements are formed out of actual subvarieties (so-called algebraic cycles) and its multiplicative structure is derived from the intersection of subvarieties. In fact, there is a natural map from one to the other which preserves the geometric notions which are common to the two (for example, Chern classes, intersection pairing, and a form of Poincaré duality). The advantage of the Chow ring is that its geometric definition allows it to be defined without reference to non-algebraic concepts; in addition, using algebraic techniques that are not available in the purely topological case, certain constructions that exist for both rings are simpler in the Chow ring.

There is also a bivariant version of the Chow theory (often referred to as the "operational Chow theory") introduced by William Fulton and Robert MacPherson.

Rational equivalence
Further information: Equivalence relations on algebraic cycles

Before defining the Chow ring, we must define the notion of "rational equivalence", which as the name indicates, is an equivalence relation on a certain set. If X is an algebraic variety and Y, Z are two subvarieties, we say that Y and Z are rationally equivalent if there is a flat family parameterized by P1, contained in the product family P1 × X, two of whose fibers are Y and Z. In more classical language, we want a subvariety V of the product family two of whose fibers are Y and Z, and all of whose fibers are subvarieties of X with the same Hilbert polynomial. If we think of P1 as a line, then this notion is an algebraic analogue of cobordism.
Definition of the Chow ring

It is part of the definition of rational equivalence that it only holds between subvarieties of equal dimension. For the purposes of constructing the Chow ring, we are interested in the codimension of the subvariety (that is, the difference between its dimension and that of X) since it makes the product work properly, so we define the groups Ak(X), for integers k satisfying 0 \leq k \leq \dim{X}, to be the abelian group of formal sums of subvarieties of X of codimension k modulo rational equivalence. The Chow ring itself is the direct sum of these, namely,

\( A^*(X) = \bigoplus_{k = 0}^{\dim{X}} A^k(X). \)

The ring structure is given by intersection of varieties: that is, if we have two classes [Y], [Z] in Ak(X) and Al(X) respectively, we define their product to be

\( [Y] \cdot [Z] = [Y \cap Z]. \)

This definition has a number of technicalities that will be discussed below; here it suffices to say that in the best case, which can be shown always to hold up to rational equivalence, this intersection has codimension k + l, hence lies in Ak + l(X). This makes the Chow ring into a graded ring. As a matter of notation, an element of the Chow ring is often called a "cycle".

Geometric interpretation

The geometric content of the Chow ring is the combination of rational equivalence and intersection product, which results in the seemingly formal numerical coefficients acquiring an interpretation in terms of the degree of a subvariety. For example, the Chow ring of projective space Pn can be shown to be:

\( A^*(\mathbb{P}^n) = \mathbb{Z}[\omega]/(\omega^{n + 1}) \)

where \omega is the rational equivalence class of a hyperplane (the zero locus of a single linear functional). Furthermore, any subvariety Y of degree d and codimension k is rationally equivalent to d \omega^k, which means, for example, if we have two subvarieties Y and Z of complementary dimension (meaning their dimensions sum to n) and degrees d, e respectively, we get that their product is simply

\( [Y] \cdot [Z] = de \; \omega^n \)

where \omega^n is the class of a point. This says, at least in the case when Y and Z intersect transversely (see below), that there are exactly de points of intersection; this is Bézout's theorem. Observations such as this, vastly generalized, give rise to the methods of enumerative geometry.
Functoriality

The functoriality of cycles, i.e. flat pullback and proper pushforward defined on the level of groups of algebraic cycles Z*(X) extend to Chow groups and give homomorphism of groups

\( f^* \colon A^k(X') \to A^k(X) \,\! and f_* \colon A_k(X) \to A_k(X'). \,\! \)

In fact, \( f^* gives a ring homomorphism on the entire Chow ring (meaning it respects the intersection product, which is clear at least on the set-theoretic level), but f_* does not (since it fails even on the set-theoretic level: we do not always have \( f(Y \cap Z) = f(Y) \cap f(Z)). However, we get the so-called projection formula: for Y a subvariety of X and Y′ a subvariety of X′,

\( f_*([Y] \cdot f^*([Y'])) = f_*([Y]) \cdot [Y']. \)

Cohomological connections

The Chow ring is very similar to the integer-valued cohomology on X. In fact, there is an obvious map

\( f \colon A^*(X) \to H^{2*}(X)\,\! \)

(by abuse of notation, the above denotes the subring of the cohomology ring generated in the even dimensions) which sends each rational equivalence class [Y] first to the homology class determined by the closed subvariety Y, and then to its Poincaré dual (this explains the even dimensionality: a complex algebraic variety always has even real dimension, hence determines a homology class in even degree). This can be shown to respect rational equivalence. Furthermore, part of Poincaré duality is that the intersection product of homology classes corresponds to the cup product of cohomology classes, so the map is actually a ring homomorphism.

There exist a number of facts that take identical form when stated either for the Chow ring or the cohomology ring. For example, the push-pull formula is true in homology and cohomology as well. More seriously, it is a basic result that the cohomology ring of Pn is the same as that given above for its Chow ring, even up to the interpretation of \omega (this says, in fact, that the map f defined in the previous paragraph is an isomorphism for projective space). However, the cohomological proof is quite technical. By contrast, we can give a simple geometric proof of the formula for the Chow ring:

First, let H be a hyperplane, which is isomorphic to a copy of Pn − 1. Any other hyperplane J is rationally equivalent, since if the two are defined by linear forms L and M, we can think of these forms as points on Pn (via their coefficients), which therefore define a unique line between them. The points of this line are themselves linear forms which define a family of hyperplanes, among which are, by construction, H and J. The intersection H \cap J is a hyperplane in H, and by definition its class is also equal to \omega^2. In this way we can produce a nested family of hyperplanes, each isomorphic to successive projective spaces and equivalent to powers of \omega.

Using these observations, we examine an arbitrary subvariety Y of codimension k and degree d. If k = 0 then Y is necessarily equal to Pn itself, since projective space is irreducible. If not, assume for simplicity that H is defined by the vanishing of the last coordinate and that the point P = [0 : \dots : 0 : 1] does not lie on Y, and define for each \( t = [t_0 : t_1] \) in P1 other than [0 : 1] the map

\( f_t \colon \mathbb{P}^n \setminus \{P\} \to \mathbb{P}^n \setminus \{P\}, f_t([a_0 : \dots : a_{n - 1}: a_n]) = [t_0 a_0 : \dots : t_0 a_{n - 1} : t_1 a_n]. \)

The images under these maps of Y form a family of varieties over all of P1 except a single point. We take the closure of this family within the product family P1 × Pn to obtain a rational equivalence of Y (that it is a rational equivalence follows from the fact that forming this closure corresponds to taking the "flat limit", a nontrivial but standard fact). Furthermore, the fiber over the point at infinity is the projection of Y onto the hyperplane H, hence has the same degree and dimension as Y. Since H is itself a projective space we iterate the construction until Y has too large a dimension to proceed. This shows that Y is rationally equivalent to \( d \omega^k, and we have already found the product structure.

A similar proof establishes a generalization of this theorem, known in cohomology as the Leray–Hirsch theorem, which computes the Chow ring of a projective space bundle in terms of the Chern classes of the corresponding vector bundle and the Chow ring of the base space. The cohomological proof requires the use of spectral sequences.

There are certain facts that do not hold of the Chow ring, but do hold of cohomology. Notably, the Künneth formula fails, though the Leray–Hirsch theorem reestablishes it for the product of projective spaces. Furthermore, although the Chow ring is contravariantly functorial on varieties, it does not form a cohomology theory in the sense of algebraic topology because no notion of relative Chow groups exists; indeed, no concept of boundary exists for algebraic varieties, so a direct attack on the analogy is hopeless.
Details of the construction

The definition of Ak(X) given above requires some clarification regarding "modulo rational equivalence". The relevant technical detail is that, as in the computation of the Chow ring of projective space, it is sometimes (in fact, usually) the case that two cycles which are not the cycles associated to a variety may be rationally equivalent, yet rational equivalence as stated appears to take notice only of the set structure. The solution is via scheme theory, namely, that a subvariety Y defined by a sheaf of ideals I can be considered to have a multiplicity d if we replace I with I^d. Then the classical statement of rational equivalence is inadequate, and we must pay close attention to the details of flat families. Finally, a formal sum of classes, such as aY + bZ, should be considered as the disjoint union of the varieties-with-degrees aY and bZ. Once these conventions are established, we may impose rational equivalence as a relation on the free abelian group of cycles to get the Chow ring.

The definition of the intersection product is somewhat more complex. The main problem is that of maintaining the correct dimension in the intersection. If Y and Z are two subvarieties of codimensions k and l, it is not always the case that their intersection has codimension k + l; for a trivial example, they could be equal. To handle this difficulty, the "moving lemma" is proved, which states that in any two rational equivalence classes we can always find representatives that intersect "generically transversely", in which case their intersection behaves well. Transversality of subvarieties is defined similarly as for manifolds: one defines the Zariski tangent spaces to the subvarieties, which are naturally subspaces of that of X, and if these subspaces span, then the intersection is transverse. It is generically transverse if transversality holds on an open, dense subset of the intersection.

In a sense it is disingenuous to claim that the Chow ring yields simpler proofs for facts that can be proved for cohomology as well. The machinery of scheme theory, flat families and flat limits in particular, and the moving lemma all furnish a great deal of technical difficulty underlying the Chow ring. However, these technical details for the most part underlie the theory, and once they are established the geometric advantage becomes clear.
Variants

Chow groups have been extended to higher Chow groups; paralleling the extension of K0 (zeroth algebraic K-theory) by higher algebraic K-theory.[1]

Arithmetic Chow groups are an amalgamation of Chow groups of varieties over Q together with a component encoding Arakelov-theoretical information, that is to say, information related to the associated complex manifold.[2]

History

The rational equivalence and the ring A* was defined by the Italian school of the algebraic geometry in the early 20th century and was used by Severi and his school. (See, for example, Severi's papers,[3][4] where Severi essentially studies the group A0(S) for an algebraic surface S, and remarks in the beginning of the Mumford's paper.[5] Segre uses a subtle study of the group A0 of a singular curve in his 1930 paper to describe branch curves of algebraic surfaces in P2.[6] The ring A* was called Chow ring after W.-L. Chow wrote an important review in 1956.[7]

References

(Bloch 1986).
(H. Gillet & C. Soulé 1992)
F. Severi, "La base per le varietà algebriche di dimensione qualunque contenute in una data", Mem. della R. Accad. d'Italia, 5, (1934), p. 239
F. Severi, "The series of sets of points on an algebraic surface", Proc. Imp. Acad. Volume 12, Supplement (1936), 1-7
D.Mumford, "Rational equivalence of 0-cycles on surfaces", J. Math. Kyoto Univ. Volume 9, Number 2 (1969), 195-204
B. Segre, "Sulla Caratterizzazione delle curve di diramazione", Mem. R. Acc. d'Italia, I 4 (1930)

W.L. Chow, "On equivalence classes of cycles in an algebraic variety", Annals of Mathematics, 1956

Bloch, Spencer (1986), "Algebraic cycles and higher K-theory", Advances in Mathematics 61 (3): 267–304, doi:10.1016/0001-8708(86)90081-2, ISSN 0001-8708, MR 852815
Chow, Wei-Liang (1956), "On Equivalence Classes of Cycles in an Algebraic Variety", Annals of Mathematics 64: 450–479, doi:10.2307/1969596, ISSN 0003-486X
Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 2, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323
Gillet, H.; Soulé, C. (1992), "An arithmetic Riemann–Roch Theorem", Invent. Math. 110: 473–543, doi:10.1007/BF01231343

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