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YCor
YCor
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Geometric intuition for $pf$pf in $fppf$fppf-topology

I am trying to learn a bit about algebraic spaces and the various topologies on categories of schemes in general. And, as it seems to always be the case, I am struggling with understanding how exactly some algebraic condition, in this case being locally of finite presentation, translates into geometry.

One geometric aspect that certainly plays a role is Chevalley's theorem on constructible sets. But other than that, how should one think geometrically or topologically about $fppf$fppf-coverings? An answer to 'What is the purpose of the flat/fppf/fpqc topologies?' says

The $fppf$fppf topology is similar to the etale topology, but now allowing maps $Y\to X$ which are open.

I can see why this is true to a certain extent, but surely being flat and locally of finite presentation is more than being open?

So I guess what I am trying to find out is how should one think of $fppf$fppf-coverings more accurately than flat and open coverings? Or, approaching the question from the 'opposite' direction, what would break if we tried to define a topology by defining coverings as collections of morphisms $\{f_i:U_i\to X\}_{i\in I}$ of schemes such that each $f_i$ is flat, (universally) open and such that $X=\bigcup f_i(T_i)$?

Geometric intuition for $pf$ in $fppf$-topology

I am trying to learn a bit about algebraic spaces and the various topologies on categories of schemes in general. And, as it seems to always be the case, I am struggling with understanding how exactly some algebraic condition, in this case being locally of finite presentation, translates into geometry.

One geometric aspect that certainly plays a role is Chevalley's theorem on constructible sets. But other than that, how should one think geometrically or topologically about $fppf$-coverings? An answer to 'What is the purpose of the flat/fppf/fpqc topologies?' says

The $fppf$ topology is similar to the etale topology, but now allowing maps $Y\to X$ which are open.

I can see why this is true to a certain extent, but surely being flat and locally of finite presentation is more than being open?

So I guess what I am trying to find out is how should one think of $fppf$-coverings more accurately than flat and open coverings? Or, approaching the question from the 'opposite' direction, what would break if we tried to define a topology by defining coverings as collections of morphisms $\{f_i:U_i\to X\}_{i\in I}$ of schemes such that each $f_i$ is flat, (universally) open and such that $X=\bigcup f_i(T_i)$?

Geometric intuition for pf in fppf-topology

I am trying to learn a bit about algebraic spaces and the various topologies on categories of schemes in general. And, as it seems to always be the case, I am struggling with understanding how exactly some algebraic condition, in this case being locally of finite presentation, translates into geometry.

One geometric aspect that certainly plays a role is Chevalley's theorem on constructible sets. But other than that, how should one think geometrically or topologically about fppf-coverings? An answer to 'What is the purpose of the flat/fppf/fpqc topologies?' says

The fppf topology is similar to the etale topology, but now allowing maps $Y\to X$ which are open.

I can see why this is true to a certain extent, but surely being flat and locally of finite presentation is more than being open?

So I guess what I am trying to find out is how should one think of fppf-coverings more accurately than flat and open coverings? Or, approaching the question from the 'opposite' direction, what would break if we tried to define a topology by defining coverings as collections of morphisms $\{f_i:U_i\to X\}_{i\in I}$ of schemes such that each $f_i$ is flat, (universally) open and such that $X=\bigcup f_i(T_i)$?

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Sergey Guminov
Sergey Guminov
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Geometric intuition for $pf$ in $fppf$-topology

I am trying to learn a bit about algebraic spaces and the various topologies on categories of schemes in general. And, as it seems to always be the case, I am struggling with understanding how exactly some algebraic condition, in this case being locally of finite presentation, translates into geometry.

One geometric aspect that certainly plays a role is Chevalley's theorem on constructible sets. But other than that, how should one think geometrically or topologically about $fppf$-coverings? An answer to 'What is the purpose of the flat/fppf/fpqc topologies?' says

The $fppf$ topology is similar to the etale topology, but now allowing maps $Y\to X$ which are open.

I can see why this is true to a certain extent, but surely being flat and locally of finite presentation is more than being open?

So I guess what I am trying to find out is how should one think of $fppf$-coverings more accurately than flat and open coverings? Or, approaching the question from the 'opposite' direction, what would break if we tried to define a topology by defining coverings as collections of morphisms $\{f_i:U_i\to X\}_{i\in I}$ of schemes such that each $f_i$ is flat, (universally) open and such that $X=\bigcup f_i(T_i)$?