In algebraic geometry , a branch of mathematics , Chow groups are an important invariant of varieties.
definition
Let be a smooth, irreducible, projective variety over an algebraically closed field .
X
{\ displaystyle X}
The group of algebraic cycles of codimension i
Z
i
(
X
)
{\ displaystyle {\ mathcal {Z}} ^ {i} (X)}
is defined as the free Abelian group generated by the irreducible (not necessarily smooth) sub-varieties of the codimension . So an element is a finite sum
W.
⊂
X
{\ displaystyle W \ subset X}
i
{\ displaystyle i}
Z
∈
Z
i
(
X
)
{\ displaystyle Z \ in {\ mathcal {Z}} ^ {i} (X)}
Z
=
∑
α
n
α
W.
α
{\ displaystyle Z = \ sum _ {\ alpha} n _ {\ alpha} W _ {\ alpha}}
with and irreducible sub-variety of the codimension .
n
α
∈
Z
{\ displaystyle n _ {\ alpha} \ in \ mathbb {Z}}
W.
α
⊂
X
{\ displaystyle W _ {\ alpha} \ subset X}
i
{\ displaystyle i}
Two sub-varieties
Y
,
Z
⊂
X
{\ displaystyle Y, Z \ subset X}
are called rational equivalent if there is a sub-variety
V
⊂
X
×
P
1
{\ displaystyle V \ subset X \ times P ^ {1}}
which is flat over ,
P
1
{\ displaystyle P ^ {1}}
as well as with
a
,
b
∈
P
1
{\ displaystyle a, b \ in P ^ {1}}
p
r
X
(
V
∩
X
×
{
a
}
)
=
Y
,
p
r
X
(
V
∩
X
×
{
b
}
)
=
Z
{\ displaystyle pr_ {X} (V \ cap X \ times \ left \ {a \ right \}) = Y, pr_ {X} (V \ cap X \ times \ left \ {b \ right \}) = Z }
gives. Rational equivalence defines an equivalence relation on the cycle group .
Z
i
(
X
)
{\ displaystyle {\ mathcal {Z}} ^ {i} (X)}
The Chow group is defined as the quotient of the Zykel group modulo rational equivalence:
C.
H
i
(
X
)
{\ displaystyle CH ^ {i} (X)}
C.
H
i
(
X
)
=
Z
i
(
X
)
/
∼
{\ displaystyle CH ^ {i} (X) = {\ mathcal {Z}} ^ {i} (X) / \ sim}
.
Chow ring
The intersection product of sub-varieties (clearly: modulo rational equivalence, one puts sub-varieties in a general position and then takes their average) defines a mapping
×
{\ displaystyle \ times}
C.
H
i
(
X
)
⊗
C.
H
j
(
X
)
→
C.
H
i
+
j
(
X
)
{\ displaystyle CH ^ {i} (X) \ otimes CH ^ {j} (X) \ rightarrow CH ^ {i + j} (X)}
for everyone . The chow ring is the direct sum of the chow groups
i
,
j
{\ displaystyle i, j}
C.
H
(
X
)
=
⨁
i
=
0
∞
C.
H
i
(
X
)
{\ displaystyle CH (X) = \ bigoplus _ {i = 0} ^ {\ infty} CH ^ {i} (X)}
with the multiplication defined by the cut product.
By means of the cleavage product to define the global average product by
×
:
C.
H
k
(
X
)
⊗
C.
H
l
(
X
)
→
C.
H
k
+
l
(
X
)
{\ displaystyle \ times \ colon CH ^ {k} (X) \ otimes CH ^ {l} (X) \ to CH ^ {k + l} (X)}
⋅
:
C.
H
k
(
X
)
⊗
C.
H
l
(
X
)
→
C.
H
k
+
l
-
d
i
m
(
X
)
(
X
)
{\ displaystyle \ cdot \ colon CH ^ {k} (X) \ otimes CH ^ {l} (X) \ to CH ^ {k + l-dim (X)} (X)}
x
⋅
y
: =
Δ
∗
(
x
×
y
)
{\ displaystyle x \ cdot y: = \ Delta ^ {*} (x \ times y)}
for diagonal embedding .
Δ
:
X
→
X
×
X
{\ displaystyle \ Delta \ colon X \ to X \ times X}
Examples
For every smooth, irreducible variety is
C.
H
0
(
X
)
=
Z
{\ displaystyle CH ^ {0} (X) = \ mathbb {Z}}
.
C.
H
1
(
X
)
{\ displaystyle CH ^ {1} (X)}
is the Picard group
C.
H
1
(
X
)
=
P
i
c
(
X
)
{\ displaystyle CH ^ {1} (X) = Pic (X)}
.
For -dimensional affine space holds
n
{\ displaystyle n}
A.
n
{\ displaystyle A ^ {n}}
C.
H
k
(
A.
n
)
=
0
{\ displaystyle CH ^ {k} (A ^ {n}) = 0}
for ,
k
≠
0
,
n
{\ displaystyle k \ not = 0, n}
C.
H
n
(
A.
n
)
=
Z
{\ displaystyle CH ^ {n} (A ^ {n}) = \ mathbb {Z}}
.
For the -dimensional projective space applies
n
{\ displaystyle n}
P
n
{\ displaystyle P ^ {n}}
C.
H
k
(
P
n
)
=
Z
{\ displaystyle CH ^ {k} (P ^ {n}) = \ mathbb {Z}}
For
0
≤
k
≤
n
+
1
{\ displaystyle 0 \ leq k \ leq n + 1}
C.
H
k
(
P
n
)
=
0
{\ displaystyle CH ^ {k} (P ^ {n}) = 0}
For
k
>
n
+
1
{\ displaystyle k> n + 1}
Relationship to the algebraic K-theory
Let be the function field of the variety and the Milnor's K-theory of this field. Then
K
(
X
)
{\ displaystyle K (X)}
X
{\ displaystyle X}
K
∗
M.
(
K
(
X
)
)
{\ displaystyle K _ {*} ^ {M} (K (X))}
C.
H
k
(
X
)
=
K
O
k
e
r
n
(
⋃
x
∈
X
(
p
+
1
)
K
1
M.
(
K
(
X
)
)
→
⋃
x
∈
X
(
p
)
K
0
M.
(
K
(
X
)
)
)
,
{\ displaystyle CH ^ {k} (X) = Kokern (\ bigcup _ {x \ in X _ {(p + 1)}} K_ {1} ^ {M} (K (X)) \ to \ bigcup _ { x \ in X _ {(p)}} K_ {0} ^ {M} (K (X))),}
where is the set of all points of the dimension .
X
(
p
)
{\ displaystyle X _ {(p)}}
X
{\ displaystyle X}
p
{\ displaystyle p}
literature
Wei-Liang Chow : On Equivalence Classes of Cycles in an Algebraic Variety , Annals of Mathematics, Volume 64, 1956, pp. 450-479, ISSN 0003-486X
William Fulton : Intersection theory , results of mathematics and their border areas. 3rd episode. A Series of Modern Surveys in Mathematics 2, Berlin, New York: Springer-Verlag 1998, ISBN 978-0-387-98549-7 , MR 1644323
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