Inverse image functor


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 edit

Suppose we are given a sheaf   on   and that we want to transport   to   using a continuous map  .

We will call the result the inverse image or pullback sheaf  . 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:


(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.

When dealing with morphisms   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 edit

  • While   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:

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 edit

  • Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190. See section II.4.