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Daniel Miller
Daniel Miller
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The profinite fundamental group of X_{fppf}$X_{fppf}$ as you define it is again the etale fundamental group of X. More precisely, the functor (of points)

f.et./X --> Sh_{fppf}(X)$f : X_{et} \to \mathrm{Sh}_{fppf}(X)$

is fully faithful and has essential image the locally finite constant sheaves (image clearly contained there, as finite etale maps are even etale locally finite constant, let alone fppf locally so). Proof in 3 steps:

  1. It is fully faithful by Yoneda (note also well-defined by fppf descent for morphsisms).

  2. Both sides are fppf sheaves (stacks) in X$X$, by classical fppf descent.

  3. Combining 1 and 2, it suffices to show that a sheaf we want to hit is just fppf locally hit, which is obvious since locally it's finite constant.

Note that the same proof also works for X_{et}$X_{et}$ or anything in between -- once your topology splits finite etale maps it doesn't really matter what it is. So we usually just work with the minimal one, the small etale topology. As Mike Artin said to me apropos of something like this, "Why pack a suitcase when you're just going around the corner?"

The profinite fundamental group of X_{fppf} as you define it is again the etale fundamental group of X. More precisely, the functor (of points)

f.et./X --> Sh_{fppf}(X)

is fully faithful and has essential image the locally finite constant sheaves (image clearly contained there, as finite etale maps are even etale locally finite constant, let alone fppf locally so). Proof in 3 steps:

  1. It is fully faithful by Yoneda (note also well-defined by fppf descent for morphsisms).

  2. Both sides are fppf sheaves (stacks) in X, by classical fppf descent.

  3. Combining 1 and 2, it suffices to show that a sheaf we want to hit is just fppf locally hit, which is obvious since locally it's finite constant.

Note that the same proof also works for X_{et} or anything in between -- once your topology splits finite etale maps it doesn't really matter what it is. So we usually just work with the minimal one, the small etale topology. As Mike Artin said to me apropos of something like this, "Why pack a suitcase when you're just going around the corner?"

The profinite fundamental group of $X_{fppf}$ as you define it is again the etale fundamental group of X. More precisely, the functor (of points)

$f : X_{et} \to \mathrm{Sh}_{fppf}(X)$

is fully faithful and has essential image the locally finite constant sheaves (image clearly contained there, as finite etale maps are even etale locally finite constant, let alone fppf locally so). Proof in 3 steps:

  1. It is fully faithful by Yoneda (note also well-defined by fppf descent for morphsisms).

  2. Both sides are fppf sheaves (stacks) in $X$, by classical fppf descent.

  3. Combining 1 and 2, it suffices to show that a sheaf we want to hit is just fppf locally hit, which is obvious since locally it's finite constant.

Note that the same proof also works for $X_{et}$ or anything in between -- once your topology splits finite etale maps it doesn't really matter what it is. So we usually just work with the minimal one, the small etale topology. As Mike Artin said to me apropos of something like this, "Why pack a suitcase when you're just going around the corner?"

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Dustin Clausen
Dustin Clausen
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The profinite fundamental group of X_{fppf} as you define it is again the etale fundamental group of X. More precisely, the functor (of points)

f.et./X --> Sh_{fppf}(X)

is fully faithful and has essential image the locally finite constant sheaves (image clearly contained there, as finite etale maps are even etale locally finite constant, let alone fppf locally so). Proof in 3 steps:

  1. It is fully faithful by Yoneda (note also well-defined by fppf descent for morphsisms).

  2. Both sides are fppf sheaves (stacks) in X, by classical fppf descent.

  3. Combining 1 and 2, it suffices to show that a sheaf we want to hit is just fppf locally hit, which is obvious since locally it's finite constant.

Note that the same proof also works for X_{et} or anything in between -- once your topology splits finite etale maps it doesn't really matter what it is. So we usually just work with the minimal one, the small etale topology. As Mike Artin said to me apropos of something like this, "Why pack a suitcase when you're just going around the corner?"