In mathematics, the inverse image functor is a covariant construction of sheaves. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
Suppose we are given a sheaf
G
on
Y
and that we want to transport
G
to
X
using a continuous map
f
:
X
→
Y
.
We will call the result the inverse image or pullback sheaf
f
−
1
G
. If we try to imitate the direct image by setting
f
−
1
G
(
U
)
=
G
(
f
(
U
)
)
for each open set
U
of
X
, we immediately run into a problem:
f
(
U
)
is not necessarily open. The best we could do is to approximate it by open sets, and even then we will get a presheaf and not a sheaf. Consequently, we define
f
−
1
G
to be the sheaf associated to the presheaf:
U
↦
lim
→
V
⊇
f
(
U
)
G
(
V
)
.
(Here
U
is an open subset of
X
and the colimit runs over all open subsets
V
of
Y
containing
f
(
U
)
.)
For example, if
f
is just the inclusion of a point
y
of
Y
, then
f
−
1
(
F
)
is just the stalk of
F
at this point.
The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limits.
When dealing with morphisms
f
:
X
→
Y
of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of
O
Y
-modules, where
O
Y
is the structure sheaf of
Y
. Then the functor
f
−
1
is inappropriate, because in general it does not even give sheaves of
O
X
-modules. In order to remedy this, one defines in this situation for a sheaf of
O
Y
-modules
G
its inverse image by
f
∗
G
:=
f
−
1
G
⊗
f
−
1
O
Y
O
X
.
While
f
−
1
is more complicated to define than
f
∗
, the stalks are easier to compute: given a point
x
∈
X
, one has
(
f
−
1
G
)
x
≅
G
f
(
x
)
.
f
−
1
is an exact functor, as can be seen by the above calculation of the stalks.
f
∗
is (in general) only right exact. If
f
∗
is exact, f is called flat.
f
−
1
is the left adjoint of the direct image functor
f
∗
. This implies that there are natural unit and counit morphisms
G
→
f
∗
f
−
1
G
and
f
−
1
f
∗
F
→
F
. These morphisms yield a natural adjunction correspondence:
H
o
m
S
h
(
X
)
(
f
−
1
G
,
F
)
=
H
o
m
S
h
(
Y
)
(
G
,
f
∗
F
)
.
However, these morphisms are almost never isomorphisms. For example, if
i
:
Z
→
Y
denotes the inclusion of a closed subset, the stalk of
i
∗
i
−
1
G
at a point
y
∈
Y
is canonically isomorphic to
G
y
if
y
is in
Z
and
0
otherwise. A similar adjunction holds for the case of sheaves of modules, replacing
i
−
1
by
i
∗
.
