Properties
- While is more complicated to define than f∗, 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 f∗. This implies that there are natural unit and counit morphisms and . These morphisms yield a natural adjunction correspondence:
- .
However, these morphisms are almost never isomorphisms. For example, if denotes the inclusion of a closed subset, the stalks 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 .
Read more about this topic: Inverse Image Functor
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)