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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.

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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. * 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). * The finest such that all representable functors are sheaves.
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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.
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  • List of topologies on the category of schemes
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