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:
“Time, force, and death
Do to this body what extremes you can,
But the strong base and building of my love
Is as the very centre of the earth,
Drawing all things to it.”
—William Shakespeare (1564–1616)
“That man is a creature who needs order yet yearns for change is the creative contradiction at the heart of the laws which structure his conformity and define his deviancy.”
—Freda Adler (b. 1934)