Theorem on formal functions explained

In algebraic geometry, the theorem on formal functions states the following:

Let

f:X\toS

be a proper morphism of noetherian schemes with a coherent sheaf

l{F}

on X. Let

S0

be a closed subscheme of S defined by

l{I}

and

\widehat{X},\widehat{S}

formal completions with respect to

X0=f-1(S0)

and

S0

. Then for each

p\ge0

the canonical (continuous) map:

(Rpf*l{F})\wedge\to\varprojlimkRpf*l{F}k

is an isomorphism of (topological)

l{O}\widehat{S

}-modules, where

\varprojlimRpf*l{F}l{OS}

k+1
l{O}
S/{l{I}
}.

l{F}k=l{F}l{OS}

k+1
(l{O}
S/{l{I}}

)

The theorem is used to deduce some other important theorems: Stein factorization and a version of Zariski's main theorem that says that a proper birational morphism into a normal variety is an isomorphism. Some other corollaries (with the notations as above) are:

Corollary: For any

s\inS

, topologically,

((Rpf*

\wedge
l{F})
s)

\simeq\varprojlimHp(f-1(s),l{F} ⊗ l{OS}(l{O}s/ak{m}

k))
s
where the completion on the left is with respect to

ak{m}s

.

Corollary: Let r be such that

\operatorname{dim}f-1(s)\ler

for all

s\inS

. Then

Rif*l{F}=0,i>r.

Corollay:[1] For each

s\inS

, there exists an open neighborhood U of s such that

Rif*l{F}|U=0,i>\operatorname{dim}f-1(s).

Corollary: If

f*l{O}X=l{O}S

, then

f-1(s)

is connected for all

s\inS

.

The theorem also leads to the Grothendieck existence theorem, which gives an equivalence between the category of coherent sheaves on a scheme and the category of coherent sheaves on its formal completion (in particular, it yields algebralizability.)

Finally, it is possible to weaken the hypothesis in the theorem; cf. Illusie. According to Illusie (pg. 204), the proof given in EGA III is due to Serre. The original proof (due to Grothendieck) was never published.

The construction of the canonical map

Let the setting be as in the lede. In the proof one uses the following alternative definition of the canonical map.

Let

i':\widehat{X}\toX,i:\widehat{S}\toS

be the canonical maps. Then we have the base change map of

l{O}\widehat{S

}-modules

i*Rqf*l{F}\toRp\widehat{f}*(i'*l{F})

.where

\widehat{f}:\widehat{X}\to\widehat{S}

is induced by

f:X\toS

. Since

l{F}

is coherent, we can identify

i'*l{F}

with

\widehat{l{F}}

. Since

Rqf*l{F}

is also coherent (as f is proper), doing the same identification, the above reads:

(Rqf*l{F})\wedge\toRp\widehat{f}*\widehat{l{F}}

.Using

f:Xn\toSn

where

Xn=(X0,

n+1
l{O}
X/l{J}

)

and

Sn=(S0,

n+1
l{O}
S/l{I}

)

, one also obtains (after passing to limit):

Rq\widehat{f}*\widehat{l{F}}\to\varprojlimRpf*l{F}n

where

l{F}n

are as before. One can verify that the composition of the two maps is the same map in the lede. (cf. EGA III-1, section 4)

Further reading

Notes and References

  1. The same argument as in the preceding corollary