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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
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