Geometric mixed motives
We will fix a field k of characteristic 0 and let
A
=
Q
,
Z
{\displaystyle A=\mathbb {Q} ,\mathbb {Z} }
be our coefficient ring.
Given a smooth variety X and a variety Y call an integral closed subscheme
W
⊂
X
×
Y
{\displaystyle W\subset X\times Y}
which is finite over X and surjective over a component of Y a prime correspondence from X to Y. Then, we can take the set of prime correspondences from X to Y and construct a free A-module
C
A
(
X
,
Y
)
{\displaystyle C_{A}(X,Y)}
. Its elements are called finite correspondences. Then, we can form an additive category
S
m
C
o
r
{\displaystyle {\mathcal {SmCor}}}
whose objects are smooth varieties and morphisms are given by smooth correspondences. The only non-trivial part of this "definition" is the fact that we need to describe compositions. These are given by a push-pull formula from the theory of Chow rings.
Typical examples of prime correspondences come from the graph
Γ
f
⊂
X
×
Y
{\displaystyle \Gamma _{f}\subset X\times Y}
of a morphism of varieties
f
:
X
→
Y
{\displaystyle f:X\to Y}
.
From here we can form the homotopy category
K
b
(
S
m
C
o
r
)
{\displaystyle K^{b}({\mathcal {SmCor}})}
of bounded complexes of smooth correspondences. Here smooth varieties will be denoted
[
X
]
{\displaystyle [X]}
. If we localize this category with respect to the smallest thick subcategory (meaning it is closed under extensions) containing morphisms
[
X
×
A
1
]
→
[
X
]
{\displaystyle [X\times \mathbb {A} ^{1}]\to [X]}
and
[
U
∩
V
]
→
j
U
′
+
j
V
′
[
U
]
⊕
[
V
]
→
j
U
−
j
V
[
X
]
{\displaystyle [U\cap V]{\xrightarrow {j_{U}'+j_{V}'}}[U]\oplus [V]{\xrightarrow {j_{U}-j_{V}}}[X]}
then we can form the triangulated category of effective geometric motives
D
M
gm
eff
(
k
,
A
)
.
{\displaystyle {\mathcal {DM}}_{\text{gm}}^{\text{eff}}(k,A).}
Note that the first class of morphisms are localizing
A
1
{\displaystyle \mathbb {A} ^{1}}
-homotopies of varieties while the second will give the category of geometric mixed motives the Mayer–Vietoris sequence.
Also, note that this category has a tensor structure given by the product of varieties, so
[
X
]
⊗
[
Y
]
=
[
X
×
Y
]
{\displaystyle [X]\otimes [Y]=[X\times Y]}
.
Using the triangulated structure we can construct a triangle
L
→
[
P
1
]
→
[
Spec
(
k
)
]
→
[
+
1
]
{\displaystyle \mathbb {L} \to [\mathbb {P} ^{1}]\to [\operatorname {Spec} (k)]{\xrightarrow {[+1]}}}
from the canonical map
P
1
→
Spec
(
k
)
{\displaystyle \mathbb {P} ^{1}\to \operatorname {Spec} (k)}
. We will set
A
(
1
)
=
L
[
−
2
]
{\displaystyle A(1)=\mathbb {L} [-2]}
and call it the Tate motive. Taking the iterative tensor product lets us construct
A
(
k
)
{\displaystyle A(k)}
. If we have an effective geometric motive M we let
M
(
k
)
{\displaystyle M(k)}
denote
M
⊗
A
(
k
)
.
{\displaystyle M\otimes A(k).}
Moreover, this behaves functorially and forms a triangulated functor. Finally, we can define the category of geometric mixed motives
D
M
g
m
{\displaystyle {\mathcal {DM}}_{gm}}
as the category of pairs
(
M
,
n
)
{\displaystyle (M,n)}
for M an effective geometric mixed motive and n an integer representing the twist by the Tate motive. The hom-groups are then the colimit
Hom
D
M
(
(
A
,
n
)
,
(
B
,
m
)
)
=
lim
k
≥
−
n
,
−
m
Hom
D
M
g
m
eff
(
A
(
k
+
n
)
,
B
(
k
+
m
)
)
{\displaystyle \operatorname {Hom} _{\mathcal {DM}}((A,n),(B,m))=\lim _{k\geq -n,-m}\operatorname {Hom} _{{\mathcal {DM}}_{gm}^{\operatorname {eff} }}(A(k+n),B(k+m))}
