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.
- 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.
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