Proper base change theorem
- There is also a proper base change theorem in topology. For that, see base change map.
In algebraic geometry, there are at least two versions of proper base change theorems: one for ordinary cohomology and the other for étale cohomology.
In ordinary cohomology
The proper base change theorem states the following: let be a proper morphism between noetherian schemes, and S-flat coherent sheaf on . If , then there is a finite complex of finitely generated projective A-modules and a natural isomorphism of functors
on the category of -algebras.
There are several corollaries to the theorem, some of which are also referred to as proper base change theorems: (the higher direct image is coherent since f is proper.)
Corollary 1 (semicontinuity theorem): Let f and as in the theorem (but S may not be affine). Then we have:
- (i) For each , the function is upper semicontinuous.
- (ii) The function is locally constant, where denotes the Euler characteristic.
Corollary 2: Assume S is reduced and connected. Then for each the following are equivalent
- (i) is constant.
- (ii) is locally free and the natural map
- is an isomorphism for all .
- Furthermore, if these conditions hold, then the natural map
- is an isomorphism for all .
Corollary 3: Assume that for some p for all . Then the natural map
- is an isomorphism for all .
In étale cohomology
In nutshell, the proper base change theorem states that the higher direct image of a torsion sheaf along a proper morphism f commutes with base change. A closely related, the finiteness theorem states that the étale cohomology groups of a constructible sheaf on a complete variety are finite. Two theorems are usually proved simultaneously.
Theorem (finiteness): Let X be a variety over a separably closed field and a constructible sheaf on . Then are finite in each of the following cases: (i) X is complete, or (ii) has no p-torsion, where p is the characteristic of k.
References
- Brian Conrad's handout
- Robin Hartshorne, Algebraic Geometry. Theorem 12.11
- Grothendieck, A. Éléments de géométrie algébrique. III. Etude cohomologique des faisceaux cohérents. II, Section 7.7
- David Mumford, Abelian Varieties.
- Vakil's notes
- SGA 4
- Milne, Étale cohomology
- Gabber, "Finiteness theorems for étale cohomology of excellent schemes"