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''on an algebraic variety''

The theory of intersections of algebraic subvarieties and cycles. Let  $  X $
be a smooth [[Algebraic variety|algebraic variety]] of dimension  $  n $
over a field  $  k $, 
while  $  Y $
and  $  Z $
are subvarieties of  $  X $
of codimension  $  i $
and  $  j $, 
respectively. If  $  Y $
and  $  Z $
intersect transversally, then  $  Y \cap Z $
is a smooth subvariety of codimension  $  i+ j $, 
which is denoted by  $  Y \cdot Z $. 
In the general case, the pair  $  ( Y, Z) $
constitutes an [[Algebraic cycle|algebraic cycle]]  $  Y \cdot Z $
of codimension  $  i + j $. 
The idea behind this definition is that  $  Y $
and  $  Z $
are replaced by cycles  $  Y  ^  \prime  $
and  $  Z  ^  \prime  $
that are equivalent in some sense but that are in general position, and one then takes the intersection of  $  Y  ^  \prime  $
and  $  Z  ^  \prime  $; 
of course, the cycle  $  Y  ^  \prime  \cdot Z  ^  \prime  $
is also defined up to equivalence.

Let  $  A  ^ {i} ( X) $
be the group of classes of algebraic cycles of codimension  $  i $
on  $  X $
modulo rational equivalence; let  $  A( X) = \oplus _ {i \geq  0 }  A  ^ {i} ( X) $. 
Chow's intersection theory consists of the construction:

a) of a graded commutative ring structure on  $  A( X) $
for each smooth quasi-projective variety  $  X $;

b) of a homomorphism of graded rings  $  f ^ { \star } :  A( Y) \rightarrow A( X) $
for each morphism  $  f:  X \rightarrow Y $(
inverse image); and

c) of a homomorphism of groups  $  f _  \star  :  A( X) \rightarrow A( Y) $
of degree  $   \mathop{\rm dim}  Y -  \mathop{\rm dim}  X $
for each [[Proper morphism|proper morphism]]  $  f:  X \rightarrow Y $(
direct image).

Between the constructions a), b), c) there are a number of relations, of which the main ones are as follows:

the projection formula: For a proper morphism  $  f:  X \rightarrow Y $
and cycles  $  \alpha \in A( X) $
and  $  \beta \in A( Y) $,

$$ 
f _  \star  ( \alpha \cdot f ^ { \star } ( \beta ))  =  f _  \star  ( \alpha )
\cdot \beta ;
$$

reduction to the diagonal: If  $  \Delta :  X \rightarrow X \times X $
is the diagonal morphism, and  $  \alpha , \beta \in A( X) $, 
then  $  x \cdot y = \Delta  ^  \star  ( \alpha \times \beta ) $.

Also, there exists a natural homomorphism

$$ 
c _ {1} :   \mathop{\rm Pic} ( X)  \rightarrow  A  ^ {1} ( X),
$$

which enables one to construct the theory of Chern classes (cf. [[Chern class|Chern class]]) with values in Chow rings, and in particular the [[Chern character|Chern character]]

$$ 
 \mathop{\rm ch} :  K( X)  \rightarrow  A( X) \otimes \mathbf Q ,
$$

which is a ring homomorphism.

It is simplest to determine the direct image homomorphism  $  f _  \star  $. 
Let  $  Z \subset  X $
be an irreducible subvariety; if  $   \mathop{\rm dim}  f( Z) <  \mathop{\rm dim}  Z $, 
then  $  f _  \star  ( Z) = 0 $, 
while if  $   \mathop{\rm dim}  f( Z) =  \mathop{\rm dim}  Z $, 
then  $  f _  \star  ( Z) = d \cdot f ( Z) $, 
where  $  d $
is the degree of  $  Z $
over  $  f( Z) $. 
By linearity, the definition is extended to cycles and classes of cycles. The inverse image homomorphism  $  f ^ { \star } $
amounts to multiplication of cycles in accordance with

$$ 
f ^ { \star } ( \alpha )  =  p _  \star  ( \Gamma _ {f} \cdot ( X \times
\alpha )),
$$

where  $  p:  X \times Y \rightarrow X $
is the projection and  $  \Gamma _ {f} \subset  X \times Y $
is the graph of  $  f $. 
The definition of multiplication of cycles is given in two stages. Let, initially,  $  Y $
and  $  Z $
be irreducible subvarieties in  $  X $
that intersect properly (i.e. the codimension of  $  Y \cap Z $
is equal to the sum of the codimensions of  $  Y $
and  $  Z $). 
Each component  $  W $
of the intersection  $  Y \cap Z $
is ascribed some positive integer  $  i( Y, Z;  W) $, 
which is the local multiplicity of the intersection. There are several definitions of  $  i( Y, Z;  W) $, 
for example, Serre's Tor-formula:

$$ 
i( Y, Z;  W)  =  \sum _ { k\geq  } 0 (- 1)  ^ {k} l(  \mathop{\rm Tor} _ {k}  ^ {A} ( A /
\mathfrak a , A / \mathfrak b )),
$$

where  $  A $
is the local ring  $  {\mathcal O} _ {X,W} $, 
$  \mathfrak a $
and  $  \mathfrak b $
are ideals of  $  Y $
and  $  Z $, 
and  $  l $
is the length of the  $  A $-
module. After this, one puts

$$ 
Y \cdot Z  =  \sum _ { W } i( Y, Z;  W) \cdot W,
$$

where  $  W $
runs through the irreducible components of  $  Y \cap Z $.

The second stage is Chow's moving lemma: For any  $  Y $
and  $  Z $
on a quasi-projective variety  $  X $
there exists a cycle  $  Z  ^  \prime  $
that is rationally equivalent to  $  Z $
and that intersects properly with  $  Y $; 
moreover, the rational equivalence class of  $  Y \cdot Z  ^  \prime  $
is independent of  $  Z  ^  \prime  $.

The most interesting case is that of a projective variety  $  X $; 
applying the direct image functor to the structure morphism  $  X \rightarrow  \mathop{\rm Spec}  k $
one obtains the mapping  $   \mathop{\rm deg} :  A( X) \rightarrow \mathbf Z $. 
In essence, the degree of a cycle is the number of points in its zero-dimensional component. The composition of multiplication with the degree enables one to measure an intersection numerically. For example, if  $  Y $
and  $  Z $
have complementary dimensions, then one obtains the [[Intersection index (in algebraic geometry)|intersection index (in algebraic geometry)]] (the intersection number) of  $  Y $
and  $  Z $. 
Similarly, one obtains an intersection index for  $  n $
divisors  $  D _ {1} \dots D _ {n} $:

$$ 
( D _ {1} \dots D _ {n} )  =   \mathop{\rm deg}  ( D _ {1} \dots D _ {n} ).
$$

For example, the [[Chow ring|Chow ring]] for the projective space  $  P  ^ {n} $
is generated by the class of a hyperplane  $  H $, 
where  $  ( H  ^ {n} ) = ( H \dots H) = 1 $. 
Therefore, if  $  D _ {1} \dots D _ {n} $
are hypersurfaces of degrees  $  d _ {1} \dots d _ {n} $, 
then  $  ( D _ {1} \dots D _ {n} ) = d _ {1} \dots d _ {n} $(
Bezout's theorem). The degree of a projective variety  $  Y \subset  P  ^ {n} $
of dimension  $  k $
is defined as the intersection index of  $  Y $
with a linear subspace  $  P  ^ {n-} k $
of complementary dimension; if the varieties  $  Y $
and  $  Z $
intersect transversally, then the degree of  $  Y \cap Z $
is the product of the degrees of  $  Y $
and  $  Z $.

For properly-intersecting effective divisors  $  ( D _ {1} \dots D _ {n} ) \geq  0 $, 
but in the general case this is not true. For example, for an exceptional curve (cf. [[Exceptional subvariety|Exceptional subvariety]])  $  E $
on a surface,  $  ( E, E) = - 1 $.

Other theories have many of the formal properties of the theory of Chow rings: cycles modulo algebraic or numerical equivalence,  $  K $-
theory, singular cohomology theory  $  H  ^  \star  (  , \mathbf Z ) $(
in the case  $  k = \mathbf C $), 
and [[L-adic-cohomology| $  l $-
adic cohomology]] theory (see also [[Weil cohomology|Weil cohomology]]). This leads to the axiomatic construction of intersection theory by putting each variety  $  X $(
from some category) into correspondence with a ring  $  C( X) $
and homomorphisms  $  f ^ { \star } $
and  $  f _  \star  $
related by axioms of the type of projection or reduction-to-the-diagonal formulas (see [[#References|[1]]]). The comparison of different intersection theories leads to useful results. For example, in the complex case, the concept of a [[Fundamental cycle|fundamental cycle]] enables one to define an intersection-theory homomorphism  $  A( X) \rightarrow H  ^  \star  ( X, \mathbf Z ) $, 
which enables one to use transcendental methods. Comparison of [[K-theory| $  K $-
theory]] with Chow's theory leads to the Riemann–Roch–Grothendieck theorem (cf. [[Riemann–Roch theorem|Riemann–Roch theorem]]). An important part is played here by the intersection theory for monoidal transformations [[#References|[2]]], [[#References|[6]]]. Another application of intersection theory is related to the foundations of Schubert's geometric calculus [[#References|[3]]]. This branch of geometry can be regarded as the theory of Chow rings for various varieties that classify geometric objects: Grassmann manifolds, flag manifolds, etc.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> , ''Anneaux de Chow et applications'' , ''Sem. Chevalley'' (1958) {{MR|}} {{ZBL|0098.13101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on algebraic geometry" , '''1''' , Moscow (1970) (In Russian) {{MR|0284434}} {{ZBL|0204.21302}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> "Hilbert's problems" ''Bull. Amer. Math. Soc.'' , '''8''' (1902) pp. 437–479 (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M. Baldassarri, "Algebraic varieties" , Springer (1956) {{MR|0082172}} {{ZBL|0995.14003}} {{ZBL|0075.15902}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.-P. Serre, "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1965) {{MR|0201468}} {{ZBL|0142.28603}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , ''Théorie des intersections et théorème de Riemann–Roch (SGA 6)'' , ''Lect. notes in math.'' , '''225''' , Springer (1971)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1982)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> W. Fulton, "Intersection theory" , Springer (1984) {{MR|0735435}} {{MR|0732620}} {{ZBL|0541.14005}} </TD></TR></table>

====Comments====
W. Fulton has defined Chow groups for singular varieties [[#References|[a1]]]. A more refined intersection theory has been developed by Fulton and R. MacPherson: given cycles  $  Y $
and  $  Z $
on  $  X $, 
it gives a well-defined element  $  X \cdot Y $
of  $  A ( Y \cap Z ) $[[#References|[a2]]].

A new development is the intersection theory for arithmetic varieties, i.e. flat schemes over the ring of integers of number fields with suitable extra data for the infinite places, [[#References|[a3]]]–.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Fulton, "Rational equivalence on singular varieties" ''Publ. Math. IHES'' , '''45''' (1975) pp. 147–167 {{MR|0404257}} {{ZBL|0332.14002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Fulton, R.D. MacPherson, "Defining algebraic intersections" L.D. Olson (ed.) , ''Algebraic geometry'' , ''Lect. notes in math.'' , '''687''' , Springer (1978) pp. 1–30 {{MR|0527228}} {{ZBL|0405.14003}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G. Faltings, "Calculus on arithmetic surfaces" ''Ann. of Math.'' , '''119''' (1984) pp. 387–424 {{MR|0740897}} {{ZBL|0559.14005}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S. Arakelov, "Intersection theory of divisors on an arithmetical surface" ''Math. USSR Izv.'' , '''8''' (1974) pp. 1167–1180 ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''38''' : 6 (1974) pp. 1179–1192 {{MR|472815}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> H. Gillet, "An introduction to higher dimensional Arakelov theory" K.A. Ribet (ed.) , ''Current trends in arithmetical algebraic geometry'' , ''Contemp. Math.'' , '''67''' , Amer. Math. Soc. (1987) pp. 209–228 {{MR|0902594}} {{ZBL|0621.14004}} </TD></TR></table>