List of topologies on the category of schemes
The most fundamental item of study in modern algebraic geometry is the category of schemes. This category admits many different Grothendieck topologies, each of which is well-suited for a different purpose. This is a list of some of the topologies on the category of schemes.
- Zariski topology Essentially equivalent to the "ordinary" Zariski topology.
- Nisnevich topology Uses etale morphisms, but has an extra condition about isomorphisms between residue fields.
- Étale topology Uses etale morphisms.
- fppf topology Faithfully flat of finite presentation
- fpqc topology Faithfully flat quasicompact
- rh topology A variation of the h topology used to have a good theory of cohomology with compact support
- cdh topology rh + Nisnevich
- ldh topology used to apply Gabber's theorem on alterations
- h topology Coverings are universal topological epimorphisms. Also, h = rh + fppf.
- v-topology (also called universally subtrusive topology): coverings are maps which admit liftings for extensions of valuation rings
- qfh topology Similar to the h topology with a quasifiniteness condition. Used to encode finite correspondences topologically.
- Smooth topology Uses smooth morphisms, but is usually equivalent to the etale topology (at least for schemes).
- Canonical topology The finest such that all representable functors are sheaves.
See also
[edit]- Lists of mathematics topics
- List of topologies – List of concrete topologies and topological spaces
References
[edit]- Belmans, Pieter. Grothendieck topologies and étale cohomology
- Gabber, Ofer; Kelly, Shane (2015), "Points in algebraic geometry", J. Pure Appl. Algebra, 219 (10): 4667–4680, arXiv:1407.5782