Theorem on formal functions

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

Let

f

:

X

→

S

{\displaystyle f:X\to S}

be a proper morphism of noetherian schemes with a coherent sheaf

F

{\displaystyle {\mathcal {F}}}

on X. Let

S

0

{\displaystyle S_{0}}

be a closed subscheme of S defined by

I

{\displaystyle {\mathcal {I}}}

and

X

^

,

S

^

{\displaystyle {\widehat {X}},{\widehat {S}}}

formal completions with respect to

X

0

=

f

−

1

(

S

0

)

{\displaystyle X_{0}=f^{-1}(S_{0})}

and

S

0

{\displaystyle S_{0}}

. Then for each

p

≥

0

{\displaystyle p\geq 0}

the canonical (continuous) map:

(

R

p

f

∗

F

)

∧

→

lim

←

k

⁡

R

p

f

∗

F

k

{\displaystyle (R^{p}f_{*}{\mathcal {F}})^{\wedge }\to \varprojlim _{k}R^{p}f_{*}{\mathcal {F}}_{k}}

is an isomorphism of (topological)

O

S

^

{\displaystyle {\mathcal {O}}_{\widehat {S}}}

-modules, where

The left term is

lim

←

⁡

R

p

f

∗

F

⊗

O

S

O

S

/

I

k

+

1

{\displaystyle \varprojlim R^{p}f_{*}{\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}{\mathcal {O}}_{S}/{{\mathcal {I}}^{k+1}}}

.

F

k

=

F

⊗

O

S

(

O

S

/

I

k

+

1

)

{\displaystyle {\mathcal {F}}_{k}={\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}({\mathcal {O}}_{S}/{\mathcal {I}}^{k+1})}

The canonical map is one obtained by passage to limit.

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

∈

S

{\displaystyle s\in S}

, topologically,

(

(

R

p

f

∗

F

)

s

)

∧

≃

lim

←

⁡

H

p

(

f

−

1

(

s

)

,

F

⊗

O

S

(

O

s

/

m

s

k

)

)

{\displaystyle ((R^{p}f_{*}{\mathcal {F}})_{s})^{\wedge }\simeq \varprojlim H^{p}(f^{-1}(s),{\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}({\mathcal {O}}_{s}/{\mathfrak {m}}_{s}^{k}))}

where the completion on the left is with respect to

m

s

{\displaystyle {\mathfrak {m}}_{s}}

.

Corollary: Let r be such that

dim

⁡

f

−

1

(

s

)

≤

r

{\displaystyle \operatorname {dim} f^{-1}(s)\leq r}

for all

s

∈

S

{\displaystyle s\in S}

. Then

R

i

f

∗

F

=

0

,

i

>

r

.

{\displaystyle R^{i}f_{*}{\mathcal {F}}=0,\quad i>r.}

Corollay: For each

s

∈

S

{\displaystyle s\in S}

, there exists an open neighborhood U of s such that

R

i

f

∗

F

|

U

=

0

,

i

>

dim

⁡

f

−

1

(

s

)

.

{\displaystyle R^{i}f_{*}{\mathcal {F}}|_{U}=0,\quad i>\operatorname {dim} f^{-1}(s).}

Corollary: If

f

∗

O

X

=

O

S

{\displaystyle f_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S}}

, then

f

−

1

(

s

)

{\displaystyle f^{-1}(s)}

is connected for all

s

∈

S

{\displaystyle s\in S}

.

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.