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.

• cdh topology A variation of the h topology
• Étale topology Uses etale morphisms.
• fppf topology Faithfully flat of finite presentation
• fpqc topology Faithfully flat quasicompact
• h topology Coverings are universal topological epimorphisms

• v-topology (also called universally subtrusive topology): coverings are maps which admit liftings for extensions of valuation rings
• l′ topology A variation of the Nisnevich topology
• Nisnevich topology Uses etale morphisms, but has an extra condition about isomorphisms between residue fields.
• qfh topology Similar to the h topology with a quasifiniteness condition.
• Zariski topology Essentially equivalent to the "ordinary" Zariski topology.
• 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

• Lists of mathematics topics
• List of topologies – List of concrete topologies and topological spaces

References

• 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