11. The " fibers " are by definition the subspaces of that are the inverse images of points of. 12. The flatness of ? ensures that the inverse image of " Z " continues to have codimension one. 13. Then Zariski's connectedness theorem says that the inverse image of any normal point of " Y " is connected. 14. The easy way to remember the definitions above is to notice that finding an inverse image is used in both. 15. A cartesian section is thus a ( strictly ) compatible system of inverse images over objects of " E ". 16. For example, it can be shown that an inverse image f ^ {-1 } [ 0 ] is a non-Borel set. 17. There are also domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. 18. However, it is a direct consequence of the definition that two such inverse images are isomorphic in " F T ". 19. The twisted inverse image functor f ^ ! is, in general, only defined as a functor between Grothendieck duality and Verdier duality. 20. To do this requires two steps : First compute an inverse image of each point to be visited; then sort the values.
