The Étale Fundamental GroupFaithfully flat morphisms
A Faithfully flat morphisms
In this appendix, we document some important facts about flat and faithfully flat morphisms.
Definition
Let f:A→B be a ring homomorphism. We say f is flat if the functor
−⊗AB:A-Mod→B-Mod
is exact.
Definition
A morphism p:Y→X is flat if for all y∈Y and x=f(y), the map OX,x→OY,y is flat. This in particular implies the pullback functor
p∗:QCoh(Y)→QCoh(X)
is exact.
If X is quasi-compact and quasi-separated, then p∗ being exact implies p being flat.
Lemma
Compositions and pullbacks of flat maps are flat.
Proof
□
We only have to show that the pullback of flat maps is flat. Since flatness is local, it suffices to show that if f:A→B is a flat map of rings and g:A→C is any map of rings, then C→B⊗AC is flat. But if M∙ is a exact sequence of chain complexes, then
B⊗AC⊗CM∙=B⊗AM∙,
where we think of M∙ as an A-module via g. So this is exact.
Proof
□
Theorem
Let p:Y→X be flat. Then the following are equivalent:
p∗ is faithful, i.e. if h:F→F′ is a morphism of quasi-coherent sheaves over Y, and p∗h=0, then h=0.
If p∗F=0, then F=0.
p∗ reflects exactness, i.e. if a sequence F∙ is such that p∗F∙ is exact, then so is F∙.
p is surjective.
When these hold, we say p is faithfully flat.
Proof
□
(1) ⇒ (2): Take h=id:F→F.
(2) ⇒ (3): Apply (2) to the homology groups of F∙.
(3) ⇒ (1): h=0 iff F→hF′→1F′ is exact.
(4) ⇒ (2): Take F=0. We may assume F is in fact coherent, for F contains a coherent subsheaf G and p∗ preserves subsheaves by flatness. So if p∗G=0, then p∗F=0.
Pick x∈X such that Fx=0. By surjectivity, there is a field k and a map x~:Speck→X that sends the unique point to x and has a lift to Y (e.g. by first picking a map to Y that hits a preimage of x). This means the pullback Y×SpeckX is non-empty. Moreover, x~∗F=0 by Nakayama, and is free since Speck is a field. So the pullback of F to Y×SpeckX is non-zero. Hence p∗F=0.
(2) ⇒ (4): Let p∈X, and SpecA⊆X an affine open containing p. Set F∣SpecA=Ap/pAp and extend by zero. Then p∗F=0 implies there is some affine open SpecB⊆Y such that B⊗ApApAp=Bp/pBp=0. Then a prime of Bp/pBp is a prime of B that gets mapped to p under p.
Proof
□
Using (4), it is clear that
Lemma
Compositions and pullbacks of faithfully flat maps are faithfully flat.
We say a property P of morphisms satisfies fpqc descent if whenever we have a pullback diagram
with f faithfully flat, then p has property P iff p′ does.
Theorem
Flat morphisms satisfy fpqc descent.
Proof
□
Suppose p′ is flat. If we have a sequence F∙ of quasi-coherent sheaves on X, then p′∗f∗F∙ is exact since f and p′ are flat, and since p′∗f∗=f′∗p∗, we know p∗F∙ is exact by faithfulness.
Proof
□
Theorem
Finite morphisms satisfy fpqc descent.
Proof
□
[Proof sketch] The pullback of a finite morphism is clearly finite. For the other direction, We will prove the affine version. The gluing step part (which is the hard step) is annoying and will be omitted.
Suppose that f:R→S is faithfully flat and M is an R-module. We want to show that M⊗RS being finitely-generated implies M is finitely generated.
Suppose y1,…ym generate M⊗RS, and yj=∑xi,j⊗fi,j. Then the xi,j generate M, since they generate M⊗RS as an S-module and R→S is faithfully flat, hence reflects surjectivity.