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:
“And all the popular statesmen say
That purity built up the State
And after kept it from decay;
Admonish us to cling to that
And let all base ambition be,
For intellect would make us proud....”
—William Butler Yeats (1865–1939)
“When man has nothing but his will to assert—even his good-will—it is always bullying. Bolshevism is one sort of bullying, capitalism another: and liberty is a change of chains.”
—D.H. (David Herbert)