Inverse image functor explained
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map
, the inverse image
functor is a functor from the category of sheaves on
Y to the category of sheaves on
X. The
direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
Definition
Suppose we are given a sheaf
on
and that we want to transport
to
using a continuous map
.
. If we try to imitate the
direct image by setting
for each open set
of
, we immediately run into a problem:
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
to be the sheaf associated to the presheaf:
U\mapsto\varinjlimV\supseteql{G}(V).
(Here
is an open subset of
and the
colimit runs over all open subsets
of
containing
.)
For example, if
is just the inclusion of a point
of
, then
is just the
stalk of
at this point.
The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limits.
of
locally ringed spaces, for example
schemes in
algebraic geometry, one often works with
sheaves of
-modules, where
is the structure sheaf of
. Then the functor
is inappropriate, because in general it does not even give sheaves of
-modules. In order to remedy this, one defines in this situation for a sheaf of
-modules
its inverse image by
.
Properties
is more complicated to define than
, the
stalks are easier to compute: given a point
, one has
.
is an
exact functor, as can be seen by the above calculation of the stalks.
is (in general) only right exact. If
is exact,
f is called
flat.
is the
left adjoint of the
direct image functor
. This implies that there are natural unit and counit morphisms
and
. These morphisms yield a natural adjunction correspondence:
HomSh(X)(f-1lG,lF)=HomSh(Y)(lG,f*lF)
.However, the morphisms
and
are
almost never isomorphisms. For example, if
denotes the inclusion of a closed subset, the stalk of
at a point
is canonically isomorphic to
if
is in
and
otherwise. A similar adjunction holds for the case of sheaves of modules, replacing
by
.
References
