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
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