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:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)