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Theorem on formal functions
Theorem in algebraic geometry

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. 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.

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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 ′ : X ^ → X , i : S ^ → S {\displaystyle i':{\widehat {X}}\to X,i:{\widehat {S}}\to S} be the canonical maps. Then we have the base change map of O S ^ {\displaystyle {\mathcal {O}}_{\widehat {S}}} -modules

i ∗ R q f ∗ F → R p f ^ ∗ ( i ′ ∗ F ) {\displaystyle i^{*}R^{q}f_{*}{\mathcal {F}}\to R^{p}{\widehat {f}}_{*}(i'^{*}{\mathcal {F}})} .

where f ^ : X ^ → S ^ {\displaystyle {\widehat {f}}:{\widehat {X}}\to {\widehat {S}}} is induced by f : X → S {\displaystyle f:X\to S} . Since F {\displaystyle {\mathcal {F}}} is coherent, we can identify i ′ ∗ F {\displaystyle i'^{*}{\mathcal {F}}} with F ^ {\displaystyle {\widehat {\mathcal {F}}}} . Since R q f ∗ F {\displaystyle R^{q}f_{*}{\mathcal {F}}} is also coherent (as f is proper), doing the same identification, the above reads:

( R q f ∗ F ) ∧ → R p f ^ ∗ F ^ {\displaystyle (R^{q}f_{*}{\mathcal {F}})^{\wedge }\to R^{p}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}} .

Using f : X n → S n {\displaystyle f:X_{n}\to S_{n}} where X n = ( X 0 , O X / J n + 1 ) {\displaystyle X_{n}=(X_{0},{\mathcal {O}}_{X}/{\mathcal {J}}^{n+1})} and S n = ( S 0 , O S / I n + 1 ) {\displaystyle S_{n}=(S_{0},{\mathcal {O}}_{S}/{\mathcal {I}}^{n+1})} , one also obtains (after passing to limit):

R q f ^ ∗ F ^ → lim ← ⁡ R p f ∗ F n {\displaystyle R^{q}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}\to \varprojlim R^{p}f_{*}{\mathcal {F}}_{n}}

where F n {\displaystyle {\mathcal {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)

Notes

Further reading

References

  1. Grothendieck & Dieudonné 1961, 4.1.5 - Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085. http://www.numdam.org/item/PMIHES_1961__11__5_0

  2. Grothendieck & Dieudonné 1961, 4.2.1 - Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085. http://www.numdam.org/item/PMIHES_1961__11__5_0

  3. Hartshorne 1977, Ch. III. Corollary 11.2 - Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 https://mathscinet.ams.org/mathscinet-getitem?mr=0463157

  4. The same argument as in the preceding corollary

  5. Hartshorne 1977, Ch. III. Corollary 11.3 - Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 https://mathscinet.ams.org/mathscinet-getitem?mr=0463157