Definition
Let X be a scheme, and let F, H be sheaves (of modules) on X. An extension of H by F is a short exact sequence of sheaves
Note that an extension is not determined by the sheaf G alone: The morphisms are also important.
A simple example of an extension of H by F is the sequence
where the second arrow is the inclusion and the fourth arrow is the projection onto the second summand. This extension is sometimes called trivial.
Read more about this topic: Sheaf Extension
Famous quotes containing the word definition:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)