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:
“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)
“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)