Theorem on formal functions

In algebraic geometry, the theorem on formal functions states the following:[1]

Let f:X->S be a proper morphism of noetherian schemes with a coherent sheaf ℱ on X. Let S0 be a closed subscheme of S defined by ℐ and X^,S^ formal completions with respect to X0=f−1(S0) and S0. Then for each p≥0 the canonical (continuous) map:

(Rpf*ℱ)∧->lim<-kRpf*ℱk
is an isomorphism of (topological) 𝒪S^-modules, where

* The left term is lim<-Rpf*ℱ⊗𝒪S𝒪S/ℐk+1.
* ℱk=ℱ⊗𝒪S(𝒪S/ℐk+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:[2] For any s∈S, topologically,

((Rpf*ℱ)s)∧≃lim<-Hp(f−1(s),ℱ⊗𝒪S(𝒪s/𝔪sk))

where the completion on the left is with respect to 𝔪s.

Corollary:[3] Let r be such that dim⁡f−1(s)≤r for all s∈S. Then

Rif*ℱ=0,i>r.

Corollay:[4] For each s∈S, there exists an open neighborhood U of s such that

Rif*ℱ|U=0,i>dim⁡f−1(s).

Corollary:[5] If f*𝒪X=𝒪S, then f−1(s) is connected for all s∈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.

## Contents

* 1 The construction of the canonical map
* 2 Notes
* 3 References
* 4 Further reading

## 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 be the canonical maps. Then we have the base change map of 𝒪S^-modules

i*Rqf*ℱ->Rpf^*(i'*ℱ).

where f^:X^->S^ is induced by f:X->S. Since ℱ is coherent, we can identify i'*ℱ with ℱ^. Since Rqf*ℱ is also coherent (as f is proper), doing the same identification, the above reads:

(Rqf*ℱ)∧->Rpf^*ℱ^.

Using f:Xn->Sn where Xn=(X0,𝒪X/𝒥n+1) and Sn=(S0,𝒪S/ℐn+1), one also obtains (after passing to limit):

Rqf^*ℱ^->lim<-Rpf*ℱn

where ℱn 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

1. ↑ Grothendieck & Dieudonné 1961, 4.1.5
2. ↑ Grothendieck & Dieudonné 1961, 4.2.1
3. ↑ Hartshorne 1977, Ch. III. Corollary 11.2
4. ↑ The same argument as in the preceding corollary
5. ↑ Hartshorne 1977, Ch. III. Corollary 11.3

## References

* 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. http://www.numdam.org/item/PMIHES_1961__11__5_0.
* Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9

## Further reading

* Illusie, Luc. "Topics in Algebraic Geometry". http://staff.ustc.edu.cn/~yiouyang/Illusie.pdf.

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