Properties
As with group extensions, if we fix F and H, then all (equivalence classes of) possible extensions of H by F form an abelian group. This group is isomorphic to the Ext group, where the identity element in corresponds to the trivial extension.
In the case where H is the structure sheaf, we have, so the group of extensions of by F is also isomorphic to the first sheaf cohomology group with coefficients in F.
Read more about this topic: Sheaf Extension
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)