Base Change

In mathematics, especially in algebraic geometry, base change refers to a number of similar theorems concerning the cohomology of sheaves on algebro-geometric objects such as varieties or schemes.

The situation of a base change theorem typically is as follows: given two maps of, say, schemes, let and be the projections from the fiber product to and, respectively. Moreover, let a sheaf on X' be given. Then, there is a natural map (obtained by means of adjunction)

Depending on the type of sheaf, and on the type of the morphisms g and f, this map is an isomorphism (of sheaves on Y) in some cases. Here denotes the higher direct image of under g. As the stalk of this sheaf at a point on Y is closely related to the cohomology of the fiber of the point under g, this statement is paraphrased by saying that "cohomology commutes with base extension".

Image functors for sheaves
direct image f
inverse image f
direct image with compact support f!
exceptional inverse image Rf!

Read more about Base Change:  Flat Base Change For Quasi-coherent Sheaves, Proper Base Change For Etale Sheaves, Smooth Base Change For Etale Sheaves

Famous quotes containing the words base and/or change:

    I am sure my bones would not rest in an English grave, or my clay mix with the earth of that country. I believe the thought would drive me mad on my death-bed could I suppose that any of my friends would be base enough to convey my carcass back to her soil. I would not even feed her worms if I could help it.
    George Gordon Noel Byron (1788–1824)

    A child... who has learned from fairy stories to believe that what at first seemed a repulsive, threatening figure can magically change into a most helpful friend is ready to believe that a strange child whom he meets and fears may also be changed from a menace into a desirable companion.
    Bruno Bettelheim (20th century)