Intersection Theory

Frontmatter

Introduction

Abstract

A useful intersection theory requires more than the construction of rings of cycle classes on non-singular varieties. For example, if A and B are subvarieties of a non-singular variety X, the intersection product A·B should be an equivalence class of algebraic cycles closely related to the geometry of how A∩B, A and B are situated in X. Two extreme cases have been most familiar. If the intersection is proper, i.e., dim (A∩B) = dim A + dim B - dim X,then A · B is a linear combination of the irreducible components of A∩B, with coefficients the intersection multiplicities. At the other extreme, if A = B is a non-singular subvariety, the self-intersection formula says that A·B is represented by the top Chern class of the normal bundle of A in X. In each case A·B is represented by a cycle on A∩B, well-defined up to rational equivalence on A∩B. One consequence of the theory developed here is a construction of, and formulas for, the intersection product A·B as a rational equivalence class of cycles on A∩B, regardless of the dimensions of the components of A∩B. We call such classes refined intersection products. Similarly other intersection formulas such as the Giambelli-Thom-Porteous formulas for the degeneracy loci of a vector bundle homomorphism, are constructed on and related to the geometry of these loci, including the cases where the loci have excess dimensions.

William Fulton

Chapter 1. Rational Equivalence

Abstract

A cycle on an arbitrary algebraic variety (or scheme) X is a finite formal sum Σn _v [V] of (irreducible) subvarieties of X, with integer coefficients. A rational function r on any subvariety of X determines a cycle [div (r)]. Cycles differing by a sum of such cycles are defined to be rationally equivalent. Alternatively, rational equivalence is generated by cycles of the form [V(0)] — [V(∞)] for subvarieties V of \( X \times {\mathbb{P}^1} \) which project dominantly to \( {\mathbb{P}^1} \). The group of rational equivalence classes on X is denoted A _* X.

For a proper morphism f:X → Y, there is an induced push-forward of cycles. The fundamental theorem of this chapter states that rational equivalence pushes forward, so there is an induced homomorphism f _* from A _* X to A _* Y, making A _* a covariant functor for proper morphisms.

For flat morphisms f:X → Y (of constant relative dimension) there are contravariant pull-back homomorphisms f* from A _* Y to A _* X. There is a useful exact sequence

$$ {A_ * }Y \to {A_ * }X \to {A_ * }\left( {X - Y} \right) \to 0 $$

for a closed subscheme Y of X, and exterior products

$$ {A_ * }X \otimes {A_ * }Y\xrightarrow{X}{A_ * }\left( {X \times Y} \right) $$

The groups A _* X will play a role analogous to homology groups in topology. In succeeding chapters it will be shown how geometric objects (vector bundles, regularly imbedded subschemes,…) give rise to operations on these groups (Chern classes, intersection products,…). Eventually corresponding contravariant, ring-valued functors A* will be constructed, with cap-products from \( {A^ * }X \otimes {A_ * }X \) To \( {A_ * }X \) and other properties familiar from topology. When X is non-singular, \( {A_ * }X \cong {A_ * }X\) in the non-singular case, but not in general, A _* X will have a ring structure. The actual relation of these groups to homology groups is discussed in Chapter 19.

William Fulton

Chapter 2. Divisors

Abstract

If D is a Cartier divisor on a scheme X, and αis a k-cycle on X, we construct an intersection class

$$ D \cdot \alpha \in {A_{k - 1}}\left( {\left| D \right| \cap \left| \alpha \right|} \right) $$

where \( {\left| D \right|^1} \), \( \left| \alpha \right| \) are the supports of D and α. For α = [V], V a subvariety, D · [V] is defined by one of two procedures: (i) if \( V \not\subset \left| D \right| \), D restricts to a Cartier divisor on V, and D · [V] is defined to be the associated Weil divisor of this restriction; (ii) if \( V \subset \left| D \right| \), the restriction of the line bundle \( {\theta _x}\left( D \right) \) to V is the line bundle of a well-defined linear equivalence class of Cartier divisors on V, and D · [V] is represented by the associated Weil divisor of any such Cartier divisor.

We prove that if a is rationally equivalent to zero on X, then D·α is zero in \( {A_{k - 1}}\left( {\left| D \right|} \right) \) ; there are therefore induced homomorphisms

$$ {A_k}X \to {A_{k - 1}}\left( {\left| D \right|} \right) $$

In the special but important case where D is the inverse image of a point for a morphism from X to a smooth curve, D·α is the specialization of α; in this case (or whenever D is principal) D·α can be well-defined as a cycle, setting \( D \cdot \left[ V \right] = 0 \) if \( V \subset D \) The above fact therefore includes the assertion that rational equivalence is preserved under specialization.

If D and D’ are Cartier divisors on a scheme X, and a is α k-cycle on X,a crucial property is the commutative law

$$ D \cdot \alpha \in {{A}_{{k - 1}}}\left( {\left| D \right| \cap \left| \alpha \right|} \right)$$

in \( {A_{k - 2}}\left( {\left| D \right| \cap \left| {D'} \right| \cap \left| \alpha \right|} \right) \) Consider, for example, the case where \( f:X \to {\mathbb{A}^2} \) is a morphism, and D and D’ are the inverse images of the two axes. One may specialize a cycle first to the part of X over the x-axis, and then specialize the resulting cycle to f ^-1 (0); or one may first specialize over the y-axis, then over the origin. The resulting cycles one arrives at by these two routes may well be different², but the above says they are rationally equivalent.

Both of the above facts follow from the identity (Theorem 2.4):

$$ D \cdot \left[ {D'} \right] = D' \cdot \left[ D \right] in {{A}_{{n - 2}}}\left( {\left| D \right| \cap \left| {D'} \right|} \right)$$

for Cartier divisors D, D’ on an n-dimensional variety X, with [D], [D’] their associated Weil divisors.

A Cartier divisor D on a scheme X determines a line bundle \( L = {\copyright_X}(D)\) and a trivialization of L over X—|D| . Only the line bundle, the support, and the trivialization are needed to carry out the above intersection construction’. These concepts are formalized in the notation of a pseudo-divisor (§ 2.2); there is the added advantage that a pseudo-divisor, unlike the stricter notion of a Cartier divisor, pulls back under arbitrary morphisms

Intersecting with divisors is used to construct homomorphisms

$$ {A_k}X \to {A_{k - 1}}X,\alpha \to {c_1}(L) \cap \alpha , $$

for a line bundle L on X, and to construct Gysin homomorphisms

$$ {i^*}:{A_k}X \to {A_{k - 1}}D$$

when i is the inclusion of an effective Cartier divisor D in X. These operations will be generalized to higher codimension in subsequent chapters.

William Fulton

Chapter 3. Vector Bundles and Chern Classes

Abstract

We will construct, for any vector bundle E on a scheme X, Chern class operations

$$ {c_i}\left( E \right) \cap \_:{A_k}X \to {A_{k - 1}}X $$

satisfying properties expected from topology. From the special case of line bundles done in § 2.5, we first construct inverse Chern classes, or Segre classes, which are then inverted to produce Chern classes. The first Chern class operations are also used to describe A _* E and A _* P(E) in terms of A _* X

Chern classes will be used later for one of the constructions of general intersection products. Although Chern classes are not absolutely needed for intersection theory, they are used in most applications. For the quickest route to intersection theory proper, the reader will need only Proposition 3.1(a) and Theorem 3.3.

For vector bundles, Chern classes and Segre classes determine each other; Chern classes are preferred since they vanish beyond the rank of the bundle. We will see in the next chapter that for cones — “singular vector bundles” — there is a natural analogue of Segre classes, but not Chern classes. Segre classes for normal cones have other remarkable properties not shared by Chern classes (cf. § 4.2).

William Fulton

Chapter 4. Cones and Segre Classes

Abstract

If X is a proper subvariety of a variety Y, the Segre class (s X, Y) of X in Y is the class in A _* X defined as follows: let C = C _x Y be the normal cone to X in Y, P (C) the projectivized normal cone, p the projection from P (C) to X. Then

$$ s\left( {X,Y} \right) = {\sum\limits_{i \geqslant 0} {{p_*}\left({{c_1}\left( \theta \right)\left( 1 \right)} \right)} ^i} \cap \left[ {p\left( C \right)} \right] $$

When Xis regularly imbedded in Y, C = N is a vector bundle, and

$$ s\left( {X,Y} \right) = c{\left( N \right)^{ - 1}} \cap \left[ X \right] $$

These Segre classes have a fundamental birational invariance: if f: Y’ → Y is a birational proper morphism, and X’= f ^-1 (X), then s (X’, Y’) pushes forward to s (X, Y).

The coefficient of [X] in s (X, Y) is the multiplicity of Y along X.

Segre classes will be used in one of our later constructions of intersection products, and in several intersection formulas.

This chapter contains the construction of Segre classes for general cones, and for general closed subschemes of a scheme. The birational invariance is a special case of a general proposition describing the behavior of Segre classes under proper push-forward and flat pull-back.

Segre classes arise naturally in many areas of algebraic geometry. Some of these occurrences are discussed in the examples and in the last two sections, which are not required for later chapters.

William Fulton

Chapter 5. Deformation to the Normal Cone

Abstract

If X is a closed subscheme of Y, there is a family of imbeddings X S _Y , parametrized by S such that for t = 0 (in fact for t ≠ ∞) the imbedding is the given imbedding of X in Y, and for t = ∞ one has the zero section imbedding of X in the normal cone C _x Y. The existence of such a deformation, together with the “principle of continuity” that intersection products should vary nicely in families, explains the prominent role to be played by the normal cone in constructing intersection products.

William Fulton

Chapter 6. Intersection Products

Abstract

Given a regular imbedding i: X → Y of codimension d, a k-dimensional variety V, and a morphism f: V→Y, an intersection product X·V is constructed in A _k-d (W), W=f ^-1 (X). Although the case of primary interest is when f is a closed imbedding, so W = X∩V, there is significant benefit in allowing general morphisms f. Let g: W → X be the induced morphism. The normal cone C_W Vto W in V is a closed subcone of g* N _X Y, of pure dimension k. We define X·V to be the result of intersecting the k-cycle [C _W V] by the zero-section of g^*N _XY:

$$ X \cdot V = {s^ * }\left[ {{C_W}V} \right] $$

where s: W → g ^* N _X Y is the zero-section, and s* is the Gysin map constructed in Chapter 3. Alternatively X·V is the (k -d)-dimensional component of the class

$$ c\left( {{g^ * }{N_X}Y} \right) \cap s\left( {W,V} \right) $$

where s (W, V) is the Segre class of W in V.

If the k-cycle [C _w V] is written out as a sum Σm _i [C _i ],with C _i irreducible, one has a corresponding decomposition X·V =Σm _i α _i , with α_i a well-defined cycle-class on the support of C_i.

If the imbedding of W in V is regular of codimension d’, then E = g ^* N _X Y/N _w V is the quotient bundle, there is an excess intersection formula

$$ X\cdot V = {{c}_{{d - d}}}\left( E \right) \cap \left[ W \right] $$

Given i: X → Y as above, and a morphism f: Y’→Y, form the fibre square

$$ \begin{array}{*{20}{c}} {X'\mathop{{ \to Y'}}\limits^{j} } \hfill \\ {g \downarrow {{ \downarrow }^{f}}} \hfill \\ {X\mathop{{ \to Y}}\limits_{i} } \hfill \\ \end{array} $$

There are refined Gysin homomorphisms

$${{i}^{!}}:{{A}_{k}}Y\prime \to {{A}_{{k - d}}}X\prime$$

determined by the formula i^![V] = X·V for subvarieties Vof Y’.

In this chapter the fundamental properties of these intersection operations are proved. After proving that i! is well-defined on rational equivalence classes, the most important of these properties are:

(i)

Compatibility with flat pull-back (§ 6.2)

 

(ii)

(i)Compatibility with proper push-forward (§ 6.2)

 

(iii)

Commutativity (§ 6.4)

 

(iv)

Functoriality (§ 6.5).

 

For example, to calculate X • V, by (i) it suffices to calculate X • V' for any V' mapping properly and birationally to V; one may blow up V along V ∩f W to reduce to a case where the excess intersection formula applies. A particular case of (ii) is the assertion that the intersection products restrict to open subschemes: one may often compute intersection products locally. An important case of commutativity asserts that intersections may be carried out before or after specialization in a family; this will include a strong version of the “principle of continuity” in Chapter 10.

When Y' = Y, i ^! determines the (ordinary) Gysin homomorphisms

$$ {i^*}:{A_k}Y \to {A_{k - d}}X$$

Functoriality (iv) refines the statement that (j i)* = i* j* for i: X → Y, j: Y→Z regular embeddings.

More generally, if f: X→Y is a local complete intersection morphism, there are Gysin homomorphisms f*, and refined homomorphisms f ^! .These Gysin homomorphisms are used to describe the group A _* \( \widetilde Y\) ,when \( \widetilde Y\) is the blow-up of a scheme Y along a regularly imbedded subscheme. A new blowup formula describes the Gysin map from A _* Y to A * \( \widetilde Y\) explicitly.

The rest of this book is based on this intersection product and the fundamental properties proved in § 6.1— § 6.5. As in Chap. 2, the formal properties can be motivated from topology. As we shall see in Chap. 19, a regular imbedding X SY of codimension d determines an orientation, or generalized Thom class, in H ^2d (Y, Y—X). The Gysin maps are the algebraic geometry versions of cap product by this orientation class, or with its pull-back to Y', if Y' maps to Y.

William Fulton

Chapter 7. Intersection Multiplicities

Abstract

As in Chap. 6, consider a fibre square

$$ \begin{array}{*{20}{c}} W \to V \\ \downarrow {}{ \downarrow {}^f} \\ X{\mathop \to \limits_i }Y \end{array} $$

with i a regular imbedding of codimension d, V a k-dimensional variety. If Z is an irreducible component of W of dimension k — d, the intersection multiplicity i (Z, X·V; Y) is defined to be the coefficient of Z in the intersection class X·V ∈ A _k-d (W). The intersection multiplicity is a positive integer, satisfying

$$ i\left({Z,X \cdot V;Y} \right) \leqq length\left({{\upsilon _{z,w}}} \right) $$

Examples show that this inequality may be strict; equality holds, however, if \( {\mathcal{O}_{z,v}} \) is a Cohen-Macaulay ring.

On the other hand, the criterion of multiplicity one asserts that i (Z, X·V;Y) is one precisely when \( {\mathcal{O}_{z,v}} \) is a regular local ring with maximal ideal generated by the ideal of X in Y.

The standard properties of intersection multiplicities, worked out in the examples, follow from the basic properties of the general intersection product which were proved in Chapter 6.

William Fulton

Chapter 8. Intersections on Non-singular Varieties

Abstract

If Y is a non-singular variety, the diagonal imbedding δ of Y in Y x Y is a regular imbedding. For x, y ∈ A _* Y, the product x·y ∈ A _* Y is defined by the formula

$$ x \cdot y = {\delta ^ * }\left( {x \times y} \right) $$

Setting \( {A^P}Y = {A_{N - P}}Y \), n = dim Y, this product makes A _* Y into a commutative, graded, ring, with unit [Y].

If f: X → Y is a morphism, with Y non-singular, the graph morphism γ _f from X to X x Y is a regular imbedding. For x ∈ A _* X, y ∈ A _* Y, define

$$ \chi { \cdot _f}y = \gamma \frac{ * }{f}\left( {x \times y} \right) \in {A_ * }{\rm X} $$

This product makes A _* X into a graded module over A*Y. If X is also nonsingular, setting

$$ {f^ * }\left( y \right) = \left[ {\rm X} \right]{ \cdot _f}y $$

defines a homomorphism f*: A* Y → A*X of graded rings.

Using the refined operation \( y\frac{!}{f} \) in place of \( y\frac{ * }{f},x{ \cdot _f}y \) has a canonical refinement in \( {A_ * }\left( {\left| x \right| \cap {f^{ - 1}}\left( {\left| y \right|} \right)} \right) \) In particular, if V and W are subvarieties of a non-singular variety Y, the intersection class V·W is defined in A_m, (V∩W)is defined in \( {A_m}\left( {V \cap W} \right) \) m = dim V +dim W - dim Y. Any m-dimensional irreducible component Z of V∩W W has a coefficient in V·W, called the intersection multiplicity, and denoted i(Z, V·W; Y). The expected properties of these intersection products and multiplicities follow readily from the general properties proved in Chaps. 6 and 7.

Bézout’s theorem, in its simplest form, states that \( {A^ * }\left( {{\mathbb{P}^n}} \right) \cong \mathbb{Z}\left[ h \right]/\left( {{h^{n + 1}}} \right) \) where h is the class of a hyperplane. A deeper analysis of intersections on projective space will be given in Chap. 12.

William Fulton

Chapter 9. Excess and Residual Intersections

Abstract

If X Y is a regular imbedding, V⊂Y a subvariety, we have constructed (§ 6.1) an intersection product X·V in A _m (X∩V), where m = dim V - codim (X, Y). If a closed subscheme Z of X∩V is given, the basic problem of residual intersections is to write X·Vas the sum of a class on Z and a class on a “residual set” R. There is a canonical choice for the class on Z, namely

$$ {\left\{ {c\left( N \right) \cap s\left( {Z,V} \right)} \right\}_m} $$

where N is the restriction to Z of N _X Y, and s(Z,V) is the Segre class. Our problem is therefore to compute this class on Z, and to construct and compute a residual intersection class \( \mathbb{R} \) in A _m (R), for an appropriate closed set R such that Z∪R= X∩V, with

$$ X \cdot V = {\left\{ {c\left( N \right) \cap s\left( {Z,V} \right)} \right\}_m} + \mathbb{R} $$

If m = 0, and R is a finite set, knowing X·V and \( {\left\{ {c\left( N \right) \cap s\left( {Z,V} \right)} \right\}_0} \) gives a formula for the weighted number of points of R. This is the basis for applications of the excess intersection formula to enumerative geometry.

In case Z is a (scheme-theoretic) connected component of X∩V, R is the union of the other connected components; since \( {A_ * }\left( {X \cap V} \right) = {A_ * }\left( Z \right) \oplus {A_ * }\left( R \right) \) the above decomposition is part of the construction of Chap. 6. Computations, applications, and a few of the many classical examples are considered in § 9.1.

The general case is considered in § 9.2. In the main theorem Z is assumed to be a Cartier divisor on V; in this case there is a natural scheme structure on the residual set, which can be used to construct \( \mathbb{R} \) If Z is arbitrary, one blows up V along Z to reduce to the divisor case.

An important and typical application of the residual intersection theorem is to the formula for the double point cycle class of a morphism, which is given in § 9.3.

William Fulton

Chapter 10. Families of Algebraic Cycles

Abstract

If T is a non-singular curve, and p: S → is a morphism, any (k+ 1)-cycle \( \alpha = \sum {n_i}\left[ {{\mathfrak{F}_i}} \right] \) On S determines an algebraic family of k-cycles α_t, on the fibres Y _{t =P} ^-1 (t):

$$ {\alpha _t} = \begin{array}{*{20}{c}} \sum \\ {{\gamma _i}} \end{array}{n_i}\left[ {{{\left( {{V_i}} \right)}_t}} \right] $$

Rationally equivalent (k + 1)-cycles on S determine rationally equivalent k-cycles in each fibre. The basic operations of intersection theory preserve algebraic families. For example, if S is smooth over T, and {α_t} and {β_t} are algebraic families of cycles, then the intersection products α _t · β _t , also vary in an algebraic family. These facts are consequences of the general theorems of Chap. 6, and the recognition of α _t , as the image of α by the refined Gysin homomorphism constructed from the diagram

$$ \begin{array}{*{20}{c}} {{Y_t}}& \to &\gamma \\ \downarrow &{}&{{ \downarrow ^p}} \\ {\left\{ t \right\}}& \to &T \end{array} $$

In this formulation, T may be replaced by any variety of arbitrary dimension, with t a regular point of T. This provides a simple method for studying algebraic equivalence.

The principle of continuity, or conservation of number, has two parts. First, in an algebraic family of zero-cycles, on a scheme which is proper over the parameter space, all the cycles have the same degree. Second, as mentioned above, the operations of intersection theory preserve algebraic families.

Refined intersection theory yields an improvement over classical formulations of this principle. For example, the ambient variety need not be complete; all that is necessary is that the locus of intersections is proper over the parameter space. This is useful for applications to enumerative geometry, when the ambient space is a space of non-degenerate geometric figures. In the last section an example of this kind is worked out: the formula for the number of curves in an r-dimensional family of plane curves which are tangent to r given plane curves in general position, in terms of the characteristics of the family, and the degrees and classes of the given curves.

where n _i , is the degree of the i ^th surface, m_i its first class (i.e. the number of points in a general plane section at which the tangent plane passes through a fixed general point), and the characteristic \( {v^i}{\rho ^{r - i}} \) is the number of curves in the family tangent to i general lines and r - i general planes. For the family of all (plane) conics in \( {\mathbb{P}^3} \) these characteristics were found by Chasles, cf. Schubert (1)§ 20:

$$ {v^8} = 92,{v^7}\rho = 116,{v^6}{\rho ^2} = 128,{v^5}{\rho ^3} = 104,{v^4}{\rho ^4} = 64,{v^3}{\rho ^5} = 32,{v^2}{\rho ^6} = 16,v{\rho ^7} = 8,{\rho ^8} = 4 $$

Similarly, there are 666,841,088 quadric surfaces in \( {\mathbb{P}^3} \) tangent to 9 given quadrics in general position.

A method for calculating the characteristics for the family of all quadrics of dimension m in \( {\mathbb{P}^n} \) was given by Schubert (4), based on the beautiful geometry of complete quadrics. This was reconsidered by Semple (1) and Tyrell (1), and recently by Demazure, Vainsencher, De Concini and Procesi.

William Fulton

Chapter 11. Dynamic Intersections

Abstract

Let X S Y be a regular imbedding of codimension d, with normal bundle N _X Y; let V be a k-dimensional subvariety of Y, W= X∩V,N the restriction of N _X Y to W, and C⊂N the normal cone to W in V. In Chap. 6 the intersection class X·V in A _k-d (W) has been constructed to be \( s\frac{ * }{N}\left[ C \right] \) where s _N W → Nis the zero-section.

If X S Y is imbedded in a family S S Y x T of regular imbeddings, with T a non-singular curve, 0 ∈ T, X₀ = X, and S ⊂ Y x T is a deformation of V, then there is a closed set \( \left( {\begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}{X_t} \cap {V_t}} \right) \) contained in W, and a class we denote \( \begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cdot {V_t}} \right) \) in \( {A_{k - d}}\left( {\begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}{X_t} \cap {V_t}} \right) \) which refines X·V, i.e., maps to X·V in A _k-d (W)

The Kodaira-Spencer homomorphism for the deformation determines a section of N, and hence a class \( S\frac{!}{\mathfrak{F}}\left[ C \right] \) in \( S\frac{{ - 1}}{\mathfrak{F}}\left( C \right) \) which also refines X·V. In fact

$$ \begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cap {V_t}} \right) \subset s\frac{{ - 1}}{\mathfrak{X}}\left( C \right) \subset W $$

and, by these inclusions

$$ \begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cdot {V_t}} \right) \to s\frac{!}{\mathfrak{K}}\left[ C \right] \to X \cdot V \cdot $$

If X _t , meets V_t properly for generic t, then \( \begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cap {V_t}} \right) \) has dimension k - d, so \( \begin{array}{*{20}{c}}dim {\lim} \\ {t \to 0} \end{array}\left( {{X_t} \cdot {V_t}} \right) \) is a well-defined cycle representing X·V. If dim \( s\frac{{ - 1}}{\mathfrak{K}}\left( C \right) = k - d \) this limit cycle must be \( s\frac{!}{\mathfrak{K}}\left[ C \right] \) in which case the limit cycle is determined by infinitesimal data.

This allows a dynamic interpretation for the distinguished varieties and their equivalences, which can be useful for calculations. For any closed subset Z of X, let (X·V)^Z be the part of X·V supported on Z (§ 6.1). If N _X Y is generated by its sections, there is an open set T (Z) of sections such that for each, s ^! [C] is a (k - d)-cycle \( \begin{array}{*{20}{c}} {\lim } \\ {t \to 0} \end{array}\left( {{X_t} \cdot {V_t}} \right) \) and the part of s ^! [C] which is supported on Z is precisely (X·V) ^Z Thus (X·V) ^Z is represented by the part of the limit cycle supported on Z, for generic deformations, i.e., deformations whose characteristic section is in T (Z). Knowing (X·V) ^Z for all Z is the same as knowing the equivalences of the distinguished varieties.

William Fulton

Chapter 12. Positivity

Abstract

We have constructed intersection classes by intersecting a cone C in a normal bundle N with the zero-section. If Σm_i [C_i] is the cycle of C, the intersection class has a corresponding decomposition into Σm_iα_i, \( {\alpha _i} \in {A_*}\left( {{Z_i}} \right) \), \( {Z_i} = Supp({C_i}) \). If the bundle N is suitably positive, one can deduce corresponding positivity of the intersection classes, even if the intersections are not proper.

Assume for simplicity that the restriction N _i of N to Z _i; is generated by its sections. Then α_i is represented by a non-negative cycle. If N _i; is also ample, a_i is represented by a positive cycle. If is generated by its sections, for an ample line bundle L, then using L to compute degrees, the degree of α_i is bounded below by the degree of Z _i .

For intersections on a non-singular variety X, the positivity of its tangent bundle will imply corresponding positivity for all intersection classes on X. For \( X = I{P^N} \), V, subvarieties, a refined Bézout’s theorem follows:

$$ {V_1}{._{...}}.{V_r} = \sum\nolimits_{{m_i}{\alpha _i}} {} $$

,

$$ \mathop \prod \limits_{j = 1}^r \deg \left( {{V_J}} \right) = \sum\limits_{} {{m_i}\deg } \left( {{\alpha _i}} \right) \geqslant \sum\limits_{} {{m_i}\deg } \left( {{Z_i}} \right) $$

where the Z_i are the distinguished varieties; all irreducible components of ∩_j V_j are included among the Z_j.

There are also applications to intersection multiplicities. For example, if…, V_r meet properly at a non-singular point P of an n-dimensional variety X, and \( \tilde V \subset \tilde X \) are the blow-ups at P, then

$$ i\left( {p,{V_1}{._{...}}.{V_r};} \right) = \prod\limits_{j = 1}^r {{e_p}} \left( {{V_j}} \right) + \deg \left( {{{\tilde V}_1}{._{...}}.{{\tilde V}_1}} \right) $$

Here the intersection class is in A _o (E), \( E \cong I{P^{n - 1}} \) the exceptional divisor. The degree of is always non-negative, and one has lower bounds as in the refined Bézout’s theorem, e.g.

$$ \deg \left( {{{\tilde V}_1}{._{...}}.\tilde V} \right) \geqslant \sum\limits_{i = 1}^s {\deg \left( {{W_i}} \right)} $$

, where are the irreducible components of the intersection ∩_j P(C_pV_j) of the projective tangent cones. Such positivity is noteworthy since the \( {\widetilde V_i}\) may have excess intersections, and general intersections on \( \widetilde X\) can be negative. There are similar inequalities for proper intersections of divisors on a possibly singular variety.

Notation. Acycle Σn_i [V_i] on a scheme X is non-negative if each n_i is non-negative, and positive if, in addition, at least one n_i is positive. Let A_k ^≥X (resp.A) _k ⁺ X) denote the set of classes in A_k X which can be represented by non-negative (resp.positive) cycles. Thus

$$ A_k^ \geqslant X = A_k^ + X \cup \{ 0\}.$$

, Both sets are clesed under addition.

Let L be a line bundle on a complete scheme X. For a k-cycle or cycle class αon X,he L-degree of α,denoted deg_L(α), is defined by

$$ de{s_L}(\alpha ) = \int\limits_X {{c_1}} {(L)^k} \cap \alpha . $$

.

If V a subvariety of X,the L-degree of V, deg_L(L),is defined by

$$ {\deg _L}(V) = {\deg _L}([V]) = \int\limits_x {{c_1}} {(L)^{\dim V}} \cap [V].$$

.

William Fulton

Chapter 13. Rationality

Abstract

Refined intersection products can be used to prove the existence of rational solutions of algebraic equations, either in the given ground field K, or in extensions of restricted degrees.

Suppose V _I ,…, V _r , are subvarieties of a complete nonsingular variety X, with codim (V_i, X) = dim (X). By our construction, the intersection cycle V _I ,…, V _r is represented by a 0-cycle on Therefore there are points P₁,…, P_t in and integers n ₁ ,…, n _r such that

$$ \sum\limits_{i = 1} {{n_i}} [K({P_i}):K] = \deg ([{V_1}]{ \cdot _{ \cdot \cdot \cdot }} \cdot [{V_r}]) $$

For example, if and the right side is odd, must contain real points. If some part of the intersection class is known, similar conclusions are valid for the rest of it. Each isolated point P of appears in (*), with coefficient the intersection multiplicity of the V _j at P. With suitable positivity assumptions on the tangent bundle of X, the coefficients n _i , can all be taken to be non-negative, even when the intersections are improper.

Notation. If W is a complete scheme, i.e. W is proper over the ground field K, and α = Σ n _p [P] is a 0-cycle on W, the degree of a is the sum

$$ \deg (\alpha ) = \sum {{n_P}} [K(P):K] $$

where K(P) is the residue field of at P, and [K(P): K] is the degree of the field extension. Rationally equivalent 0-cycles have the same degree (§ 1.4).

For example, if is the field of real numbers (or an arbitrary real closed field), then if P is a real point of W, K(P) = if P is complex, and

$$ \deg (\alpha ) = \sum\limits_{Preal} {{n_p}} + 2\sum\limits_{Pcomplex} {{n_p}} $$

In particular, if deg (a) is odd, W must contain real points.

Let i: X → Y be a regular imbedding of codimension d, V a pure d-dimensional subscheme of Y, W= X∩V. Let N be the restriction of N _X Y to W, C = C _W V the normal cone to Win V,[C] = Σ m _i [C _i ] its cycle on N. Recall is the largest divisible subgroup of. (Use Examples 13.12 and 1.6.6.)

(d) If any two points of can be joined by a chain of rational curves (e.g. if X_c is a unirational variety), then

$${{\tilde{A}}_{0}}\left( X \right) = {{\left( {\mathbb{Z}/2\mathbb{Z}} \right)}^{{s - 1}}} $$

Two points of X(R) are rationally equivalent if and only if they belong to the same connected component of X(R). (Use Example 10.1.6.)

Using results of M. Knebusch and H. Delfs, Colliot-Thélène and Ischebeck (1) prove analogous results for arbitrary real closed fields.

William Fulton

Chapter 14. Degeneracy Loci and Grassmannians

Abstract

Let σ: E → F be a homomorphism of vector bundles of ranks e and f on a variety X, and let k min (e,f). The degeneracy locus

$$ {D_k}\left(\sigma \right) = \left\{{x \in X\left| {rank} \right.\left({\sigma \left(x \right)} \right) \leqq k} \right\} $$

has codimension at most (e - k) (f - k) in X, if it is not empty. We construct a class

$$ {\mathbb{D}_k}\left(\sigma \right) \in {A_m}\left({{D_k}\left(\sigma \right)} \right) $$

m = dim (X) - (e - k) (f - k), whose image in A _m (X) is given by the Thom-Porteous formula:

$$ {\mathbb{D}_k}\left(\sigma \right) = \Delta _{f - k}^{\left({e - k} \right)}\left({c\left({F - K} \right)} \right) \cap \left[X \right] $$

Here Δ _q ^(p) (c) denotes the determinant of the p by p matrix \( {\left({{c_{q + j - i}}} \right)_{1 \leqq i,j \leqq p}} \) If dim (D _k (σ) = m, and X is non-singular, or, more generally, if a suitable depth condition is satisfied, then \( {\mathbb{D}_k}\left(\sigma \right) \) is the m-cycle determined by the natural scheme structure on \( {\mathbb{D}_k}\left(\sigma \right) \)In general the formation of \( {\mathbb{D}_k}\left(\sigma \right) \)commutes with other intersection operations. These properties determine \( {\mathbb{D}_k}\left(\sigma \right) \)in case dim \( \left({{D_k}\left(\sigma \right)} \right) > m \).

If \( {A_1} \subset \ldots \subset {A_d} \subset E \) is a flag of sub-bundles of E, the determinantal locus is

$$ \Omega \left({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} ;\sigma} \right) = \left\{{x \in X\left| {\dim \left({Ker\left({\sigma \left(x \right)} \right) \cap {A_i}\left(x \right)} \right) \geqq ifor1 \leqq i \leqq d} \right.} \right\} $$

Similarly, there are classes \( \Omega \left({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} ;\sigma} \right) \) in \( {A_ *}\left({\Omega \left({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} ;\sigma } \right)} \right) \), whose images in A _* (X) are given by certain determinants in Chern classes. If \( c\left({E/{A_i}} \right) = 1 \), the formula is

$$ \Omega \left({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{A} ;\sigma } \right) = {\Delta _\lambda }\left({c\left({F - E} \right)} \right) \cap \left[X \right]$$

where \( \lambda = \left({{\lambda _i}, \ldots,{\lambda _d}} \right)\), \( {\lambda _i} = f - rank\left({{A_i}} \right) + i\), and Δ_λis the Schur polynomial

$$ {\Delta _{\lambda \left(c \right)}} = \det \left({\begin{array}{*{20}{c}} {{c_{{\lambda _1}}}}{{c_{{\lambda _1} + 1}} \ldots {c_{{\lambda _1} + d - 1}}} \\ {{c_{{\lambda _2} - 1}}}{{c_{{\lambda _2}}} \ldots {c_{{\lambda _2} + d - 2}}} \\ {{c_{{\lambda _d} - d + 1}}}{{c_{{\lambda _d}}}} \end{array}} \right)$$

A special case of degeneracy locus is the zero set of a section of a vector bundle. In this case the degeneracy class localizes the top Chern class of the bundle (§ 14.1). The construction of general degeneracy loci is reduced to the case of sections of bundles on Grassmannians; proving the formulas requires some Gysin computations (§ 14.2).

Formal identities among Schur polynomials determine formulas for inter­secting determinantal classes. When applied to Grassmann bundles, these formulas yield generalizations of classical formulas of Schubert calculus: the basis theorem, duality, Pieri's formula, and Giambelli's formula.

Notation. The fibre of a vector bundle E over a scheme X at a point x ∈ X is denoted E (x); it is a vector space over \(\kappa (x) \). If σ: E→ F is a vector bundle homomorphism, ∧^k σ denotes the induced homomorphism on k ^th exterior powers.

If σ: E → F is a homomorphism of vector bundles on a scheme X, the zero scheme of σ will be denoted Z (σ). On an affine open set U where E and F are trivial, σ is defined by a matrix of elements in the coordinate ring of U, which generate the ideal of Z (σ) on U. In particular, if s is a section of a bundle E, i.e., a homomorphism from the trivial line bundle to E, its zero scheme is denoted Z(s) (cf. Appendix B.3.2).

More generally, for a non-negative integer k, we have the k ^th degeneracy locus

$$ {D_k}(\sigma ) = \{ x \in X|rank(\sigma (x)) \leqslant k\} = Z({ \wedge ^{k + 1}}(\sigma ))$$

The second description determines the scheme structure on D _k (σ): locally its ideal is generated by (k +1)-minors of a matrix for σ.

Let A be a flag of subbundles of E:

$$ 0 \subsetneqq {A_1} \subsetneqq ... \subsetneqq {{\rm A}_d} \subset E$$

Given a: E → F, set

$$ \Omega (\underline A ;\sigma ) = \{ x \in X|\dim (Ker(\sigma (x)) \cap {A_i}(x)) \geqq i,1 \leqq i \leqq d\} = \mathop \cap \limits_{i = 1}^d Z({ \wedge ^{{a_i} - i + 1}}({\sigma _i})).$$

Here σ_i:A _i → Fis the restriction of a to _i and a _i is the rank of A _i The second description determines the scheme structure.

For vector bundles E, F on X, set

$$ c(F - E) = c(F)/c(E) = 1 + ({c_1}(F) - {c_1}(E) + ...)$$

and let c _i (F— E) be the term of degree i in this expansion.

If λ₁,…,λ_d are integers, and c ⁽¹⁾ …, c ^(d) are formal sums:

$$ {c^{(i)}} = \sum\limits_j {c_j^{(i)}}$$

with c _j ⁽ⁱ⁾ of degree j, \( j \in \mathbb{Z}\), Set

$$ {\Delta _{{\lambda _{1,}}...,{\lambda _d}}}({c^{(1)}},...,{c^{(d)}}) = \det \left\{ \begin{gathered} c_{{\lambda _1}}^{(1)}c_{{\lambda _{1 + 1}}}^{(1)}...c_{{\lambda _{1 + d - 1}}}^{(1)} \hfill \\ c_{{\lambda _2}}^{(2)} \hfill \\ c_{{\lambda _{d - d + 1}}}^{(d)}c_{{\lambda _d}}^{(d)} \hfill \\ \end{gathered} \right\}$$

If c ⁽¹⁾ =… = c ^(d) = c, we denote this Δ _λ1,…,λd (c), or Δ _λ (c), i.e. Δ _λ (c)=|c _λi+j-i |.

If, in addition, λ₁ =… =λ_d = e, this is Δ _e ^(d) (c).

William Fulton

Chapter 15. Riemann-Roch for Non-singular Varieties

Abstract

The Grothendieck-Riemann-Roch theorem (GRR) states that for a proper morphism f: X → Y of non-singular varieties,

$$ ch\left( {{f_*}\alpha } \right) \times td\left( {{T_Y}} \right) = {f_*}\left( {ch\left( \alpha \right) \times td\left( {{T_X}} \right)} \right) $$

for all α in the Grothendieck group of vector bundles, or of coherent sheaves, on X. When Y is a point, one recovers Hirzebruch’s formula (HRR) for the Euler characteristic of a vector bundle E on X:

$$ {\sum {\left( { - 1} \right)} ^i}\dim {H^i}\left( {X,E} \right) = \int\limits_X {ch} \left( E \right) \cdot td\left( {{T_X}} \right) $$

The aim of this chapter is to show how the geometry of the deformation to the normal cone leads to a simple proof of GRR when f is a closed imbedding. The same proof gives the corresponding theorem without denominators, which in turn yields a simple proof of the formula for blowing up Chern classes.

The reader of this chapter is assumed to have some familiarity with the cohomology of coherent sheaves, although the necessary facts are reviewed in the first section. In addition, the proof of GRR when f is a projection is only sketched briefly. The first nine sections of the article of Borel and Serre (1) are recommended for a detailed discussion of these points.

Although the theorem is stated here for arbitrary non-singular varieties, the proof in this chapter makes an additional assumption of projectivity. The general case will be considered, together with singular varieties, in Chap. 18.

William Fulton

Chapter 16. Correspondences

Abstract

A correspondence from X to Y, denoted \( \alpha :X \vdash Y \) is a subvariety, cycle, or equivalence class of cycles on X x Y. The graph of a morphism, or the closure of the graph of a rational map, are basic examples, but more general correspondences have played an important role in the development of algebraic geometry. On complete non-singular varieties correspondences have a product ß ° α, and a correspondence \( \alpha :X \vdash Y \) determines homomorphisms α_*from A(X) to A(Y), and α^* from A(Y) to A(X), these notions generalizing composition, push-forward, and pull-back for morphisms. The basic algebra of correspondences is deduced easily from the general theory of Chap. 8.

If X = Y has dimension n, and T is an n-dimensional correspondence, then the degree of the intersection class T·Δ of T with the diagonal is the virtual number of fixed points of T. In case there are non-isolated fixed points, the excess intersection formulas can be applied. (If T = V x W, with V, W sub-varieties of X, T·Δ= V·W is the intersection class studied in Chap. 8.) When one has explicit formulas for the equivalence class of [T] or of [Δ] on X x X, fixed point formulas for T·Δcan be deduced.

Notation. Unless otherwise stated, all ambient varieties X,Y,Z,… in this chapter are assumed to be complete and non-singular, i.e., proper and smooth over the given ground field.

William Fulton

Chapter 17. Bivariant Intersection Theory

Abstract

Our basic intersection construction has assigned to a regular imbedding (or 1.c.i. morphism) f: X → Y of codimension d a collection of homomorphisms

$$ {{f}^{!}}:{{A}_{k}}Y' \to {{A}_{{k - d}}}X' $$

for all Y’ → Y, X’ = X x _y Y’, all k. In this chapter we formalize the study of such operations. For any morphism f: X → Y, a bivariant class c in \( {A^p}\left( {X\xrightarrow{f}Y} \right) \) is a collection of homomorphisms from A _k Y’ to A _k-P X’, for all Y’ → Y, all k, compatible with push-forward, pull-back, and intersection products.

The group A ^-k (X → pt.) is canonically isomorphic to A _k (X). The other extreme \( {A^k}\left( {X\xrightarrow{{id}}X} \right) \) is defined to be the cohomology group A _K X. The bivariants groups have products

$$ {A^p}\left( {X\xrightarrow{f}Y} \right) \otimes {A^q}\left( {Y\xrightarrow{g}Z} \right)\xrightarrow{ \cdot }{A^{p + q}}\left( {X\xrightarrow{{gf}}Z} \right) $$

which specialize to give a ring structure on A ^* X, and a cap product action of A ^* X on A ^* X.If X is non-singular \( {A^ * }X \cong {A_ * }X \), There are also a proper push-forward and a pull-back operation for bivariant groups, generalizing the push-forward on A _* and defining a pull-back on A _* There are compatibilities among these three operations which allow one to manipulate bivariant classes symbolically with a freedom one is accustomed to with homology and cohomology in topology.

Many constructions of previous chapters actually produce classes in appropriate bivariant groups. For example, Chern classes of vector bundles on X live in A _* XFlat and l.c.i. morphisms f:X→Y determine canonical elements in \( {A^ * }\left( {X\xrightarrow{f}Y} \right) \), which are denoted [f]. An element c of A ^P (X→Y) determines generalized Gysin homomorphisms \( {A_k}Y\xrightarrow{{{c^ * }}}{A_{k - p}} \) and \( {A^k}X\xrightarrow{{{c_ * }}}{A^{k + p}}Y \) (for the latter f is assumed to be proper). Intersection formulas such as the excess and residual intersection formulas achieve their sharpest formulation in the bivariant language.

There is a useful criterion which implies that an operation which produces rational equivalence classes on X’ from subvarieties of Y’ (for all Y’→Y), passes to rational equivalence and defines a bivariant class (Theorem 17.1). This will be used in the next chapter to deduce the important properties of local Chem classes.

William Fulton

Chapter 18. Riemann-Roch for Singular Varieties

Abstract

The basic tool for a general Riemann-Roch theorem is MacPherson’s graph construction, applied to a complex E. of vector bundles on a scheme Y, exact off a closed subset X. This produces a localized Chern character¹ ch _x ^y (E.) which lives in the bivariant group \( A{\left( {X \to Y} \right)_\mathbb{Q}} \) For each class α∈A _* Y, this gives a class

$$ ch_X^Y\left( {E.} \right) \cap \alpha \in {A_ * }{X_\mathbb{Q}} $$

whose image in \( {A_ * }{Y_\mathbb{Q}} \) is \( {\sum {\left( { - 1} \right)} ^i}ch\left( {{E_i}} \right) \cap \alpha \) The properties needed for Riemann-Roch, in particular the invariance under rational deformation, follow from the bivariant nature of ch _x ^y E.

The general Riemann-Roch theorem constructs homomorphisms

$$ {{\tau }_{x}}:{{K}_{ \circ }}X \to {{A}_{*}}{{X}_{\mathbb{Q}}} $$

covariant for proper morphisms, such that \( {\tau _x}\left( {\beta \otimes \alpha } \right) = ch\left( \beta \right) \cap {\tau _x}\left( \alpha \right) \) for \( \beta \in {{K}^{ \circ }}X, \alpha \in {{K}_{ \circ }}X \) imbedded in a non-singular variety M, and a coherent sheaf ℐis resolved by a complex of vector bundles E. on M, then

$$ {\tau _x}\left( \mathcal{F} \right) = ch_X^M\left( {E.} \right) \cap Td\left( M \right) $$

where \( Td\left( M \right) = td\left( {{T_M}} \right) \cap \left[ M \right] \) Such t_xis constructed for quasi-projective schemes in the second section. The extension to arbitrary algebraic schemes, using Chow’s lemma, is carried out in the last section. As a corollary one has the GRR formula

$$ {f_ * }\left( {ch\left( \alpha \right) \cdot td\left( {{T_x}} \right)} \right) = ch\left( {{f_ * }\alpha } \right) \cdot td\left( {{T_Y}} \right) $$

for f: X→Y proper. X, Y arbitrary non-singular varieties, \( \alpha \in {{K}^{ \circ }}X \) In the singular case, there are refinements for f: X→Y a l.c.i. morphism.

William Fulton

Chapter 19. Algebraic, Homological, and Numerical Equivalence

Abstract

Each k-dimensional complex variety V has a cycle class cl(V) in H _2k V, where H _* denotes homology with locally finite supports (Borel-Moore homology). If V is a subvariety of an n-dimensional complex manifold X, then \( {H_{2k}}\left( V \right) \cong {H^{2n - 2k}}\left( {X,X - V} \right) \) The resulting homomorphism from cycles to homology passes to algebraic equivalence. There results in particular a cycle map

$$ cl:{A_ * }X \to {H_ * }X $$

for complex schemes X, which is covariant for proper morphisms, and compatible with Chern classes of vector bundles.

If V and W are subvarieties of dimensions k and l of a non-singular n-dimensional variety X, a refined topological intersection product cl(V)·cl(W) is constructed in \( {H_{2m}}\left( {V \cap W} \right) \), \( m = k + l - n \) If cl ^x V is the class in \( {H^{2n - 2k}}\left( {X,X - V} \right) \) dual to cl(V),and similarly for cl ^x (W), then cl(V)·cl(W) is defined to be the class dual to

$$ c{l^x}\left( V \right) \cup c{l^x}\left( W \right) \in {H^{2n - 2k}}\left( {X,X - V \cap W} \right) $$

We show that the cycle map takes the refined intersection \( V \cdot W \in {A_m}\left( {V \cap W} \right) \) of Chap. 8 to the class cl(V)·cl(W). In particular, cl is a ring homomorphism from A ^* X toH ^* X More generally, if i: X→Y is a regular imbedding of codimension d, the cycle classes of the refined products i’ (α) of Chap. 6 are given by cap product with an orientation class uxy in H ^2d (y,y-x)

In the final section we discuss what is known about algebraic, homological, and numerical equivalence on non-singular projective varieties. Only a few salient facts are mentioned which relate most directly to other chapters, and few proofs are included. Together with the examples, this may serve as an introduction to the literature on the transcendental theory of algebraic cycles.

Notation. Unless otherwise stated, all schemes in this chapter are assumed to be complex algebraic schemes which admit a closed imbedding into some non-singular complex variety. All topological spaces will be locally compact Hausdorff spaces which admit a closed imbedding into some Euclidean space. As in preceding chapters, a k-cycle on X is a formal sum of algebraic subvarieties of X.

William Fulton

Chapter 20. Generalizations

Abstract

Much of the intersection theory developed in this text is valid for more general schemes than algebraic schemes over a field. A convenient category, sufficient for applications envisaged at present, is the category of schemes X of finite type over a regular base scheme S. Using an appropriate definition of relative dimension, one has a notion of k-cycle on X, and a graded group A _* (X) of rational equivalence classes, satisfying the main functorial properties of Chaps. 1-6. The Riemann-Roch theorem also holds; in particular

$$ A\begin{array}{*{20}{c}} {}\\ * \end{array}\left( X \right) \otimes \mathbb{Q} \cong K\begin{array}{*{20}{c}} {} \\ ^\circ \end{array}\left( X \right) \otimes \mathbb{Q}$$

The main missing ingredient in such generality is an exterior product If S is one-dimensional, however, there is such a product. In particular,

$$ A\begin{array}{*{20}{c}} {} \\ k \end{array}\left( X \right) \otimes A\begin{array}{*{20}{c}} {} \\ 1 \end{array}\left( Y \right) \to A\begin{array}{*{20}{c}} {} \\ {k + 1} \end{array}\left( {X \times \begin{array}{*{20}{c}} {} \\ s \end{array}Y} \right)$$

if X is smooth over S, then A _* (X) has a natural ring structure.

When S = Spec (R),R a discrete valuation ring, and X is a scheme over S, with general fibre X° and special fibre, there are specialization maps

$$ \sigma :{A_k}({X^0}) \to {A_k}(\overline X ),$$

which are compatible with all our intersection operations. If X is smooth over S, σ is a homomorphism of rings.

For proper intersections on a regular scheme, Serre has defined intersection numbers using Tor. For smooth schemes over a field, these numbers agree with those in § 8.2. Indeed, even for improper intersections, a Riemann-Roch construction shows how to recover intersection classes from Tor, at least with rational coefficients.

Although higher K-theory is outside the scope of this work, the chapter concludes with a brief discussion of Bloch’s formula

$$ {A^p}X = {H^p}(X,{\mathcal{Y}_p})$$

William Fulton

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