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:ABf: A \to B be a ring homomorphism. We say ff is flat if the functor

AB:A-ModB-Mod -\otimes _A B: A\text{-Mod} \to B\text{-Mod}

is exact.

Definition

A morphism p:YXp: Y \to X is flat if for all yYy \in Y and x=f(y)x = f(y), the map OX,xOY,y\mathcal{O}_{X, x} \to \mathcal{O}_{Y, y} is flat. This in particular implies the pullback functor

p:QCoh(Y)QCoh(X) p^*: \operatorname{QCoh}(Y) \to \operatorname{QCoh}(X)

is exact.

If XX is quasi-compact and quasi-separated, then pp^* being exact implies pp 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:ABf: A \to B is a flat map of rings and g:ACg:A \to C is any map of rings, then CBACC \to B \otimes _A C is flat. But if MM_\bullet is a exact sequence of chain complexes, then

BACCM=BAM, B \otimes _A C \otimes _C M_\bullet = B \otimes _A M_\bullet ,

where we think of MM_\bullet as an AA-module via gg. So this is exact.

Proof

Theorem

Let p:YXp: Y \to X be flat. Then the following are equivalent:

  1. pp^* is faithful, i.e. if h:FFh: \mathcal{F} \to \mathcal{F}' is a morphism of quasi-coherent sheaves over YY, and ph=0p^*h = 0, then h=0h = 0.

  2. If pF=0p^* \mathcal{F} = 0, then F=0\mathcal{F} = 0.

  3. pp^* reflects exactness, i.e. if a sequence F\mathcal{F}_\bullet is such that pFp^* \mathcal{F}_\bullet is exact, then so is F\mathcal{F}_\bullet .

  4. pp is surjective.

When these hold, we say pp is faithfully flat.

Proof

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

\begin{useimager} 
  \[
    \begin{tikzcd}
      Y \times_X X' \ar[d, "p'"] \ar[r, "f'"] & Y \ar[d, "p"]\\
      X' \ar[r, "f"] & X
    \end{tikzcd}
  \]
\end{useimager}

with ff faithfully flat, then pp has property P iff pp' does.

Theorem

Flat morphisms satisfy fpqc descent.

Proof
Suppose pp' is flat. If we have a sequence F\mathcal{F}_\bullet of quasi-coherent sheaves on XX, then pfFp'^* f^* \mathcal{F}_\bullet is exact since ff and pp' are flat, and since pf=fpp'^* f^* = f'^* p^*, we know pFp^* \mathcal{F}_\bullet 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:RSf: R \to S is faithfully flat and MM is an RR-module. We want to show that MRSM \otimes _R S being finitely-generated implies MM is finitely generated.

Suppose y1,ymy_1, \ldots y_m generate MRSM \otimes _R S, and yj=xi,jfi,jy_j = \sum x_{i, j} \otimes f_{i, j}. Then the xi,jx_{i, j} generate MM, since they generate MRSM \otimes _R S as an SS-module and RSR \to S is faithfully flat, hence reflects surjectivity.

Proof