Exposé IX. Algebraic geometry and formal geometry

The goal of this Exposé is to generalize, to the case of a morphism that is not proper, Theorems 3.4.2 and 4.1.5 of EGA III.

1. The comparison theorem

Let $f:X\to X^{\prime }$ be a separated morphism of preschemes of finite type. Suppose that $X$ is locally noetherian. Let $Y^{\prime }$ be a closed subset of $X^{\prime }$ and let $Y=f^{-1}(Y^{\prime })$.

Let $\hat{X}$ and ${\hat{X}}^{\prime }$ be the formal completions of $X$ and $X^{\prime }$ along $Y$ and $Y^{\prime }$. Let $\hat{f}$ be the morphism deduced from $f$ by passing to the completions.

    X ◀───── Y               X̂ ──j──▶  X
    │                        │           │
    │ f                      │ f̂         │ f
    ▼                        ▼           ▼
    X′ ◀──── Y′      ,       X̂′ ──i──▶ X′.

We denote by $j$ (resp. $i$) the homomorphism from $\hat{X}$ into $X$ (resp. from ${\hat{X}}^{\prime }$ into $X^{\prime }$). It is known that $i$ and $j$ are flat.

Let $I^{\prime }$ be an ideal of definition of $Y^{\prime }$, and let $J=f\ast (I^{\prime })\cdot {\mathcal{O}}_X$; this is an ideal of definition of $Y$. One therefore has:

X̂′ = (Y′, lim_{k ∈ ℕ} 𝒪_{X′}/I′^{k+1}),    X̂ = (Y, lim_{k ∈ ℕ} 𝒪_X/J^{k+1}).

For every $k\in \mathbb{N}$, set:

Y′_k = (Y′, 𝒪_{X′}/I′^{k+1}),    Y_k = (Y, 𝒪_X/J^{k+1}).

Let $F$ be a coherent ${\mathcal{O}}_X$-Module. For every $k\in \mathbb{N}$, we set:

F_k = F/J^{k+1}F,    F̂ = j*(F) ≃ lim_{k} F_k.

If we set:

R^i f_*(F)^∧ = lim_{k ∈ ℕ} (R^i f_*(F) ⊗_{𝒪_{X′}} 𝒪_{Y′_k}),    i ∈ ℤ,

one has a natural homomorphism:

r_i: i*(R^i f_*(F)) → R^i f_*(F)^∧,

which is an isomorphism when $R^i{f}_{\ast }(F)$ is coherent.

As is explained in EGA III 4.1.1, one has a commutative diagram:

                       ρ_i
   i*(R^i f_*(F))  ──────▶  R^i f̂_*(F̂)
        │                       │
     r_i│                       │ψ_i
        ▼          ϕ_i          ▼
   R^i f_*(F)^∧  ──────▶  lim_{k ∈ ℕ} R^i f_*(F_k).

In loc. cit. one finds a commutative diagram, since one knows that $R^i{f}_{\ast }(F)$ is coherent, and one identifies the source and target of (1.6). In our case, $R^i{f}_{\ast }(F)$ will be coherent only for certain values of $i$, for which we shall study (1.7).

Consider the graded ring

S = ⨁_{k ∈ ℕ} I′^k,

and the graded $S$-Module:

H^i = ⨁_{k ∈ ℕ} R^i f_*(J^k F),    i ∈ ℤ,

whose $S$-Module structure is defined as follows.

The sheaf $R^i{f}_{\ast }(J^{k}F)$ is associated with the presheaf which, to every affine open $U^{\prime }$ of $X^{\prime }$, associates:

H^i(f⁻¹(U′), J^k F | f⁻¹(U′)).

Let $U^{\prime }$ then be an affine open of $X^{\prime }$, set

$$U=f^{-1}(U^{\prime }),$$

and let $x^{\prime }\in I^{\prime m}(U^{\prime })$. Let $x$ be the image of $x^{\prime }$ in $J^m(U)$. The homothety of ratio $x$ on $F|U$ maps $J^{k}F|U$ into $J^{k+m}F|U$, whence, by functoriality, a morphism:

μ^i_{x′, k}(U′): H^i(U, J^k F | U) → H^i(U, J^{k+m} F | U),

defined for every $i\in \mathbb{Z}$ and every $k\in \mathbb{N}$, which gives, by passing to the associated sheaf, the graded $S$-Module structure on $H^i$.

Theorem.

Let $n$ be an integer. Suppose that the graded $S$-Module $H^i$ is of finite type for $i=n-1$ and $i=n$. Then:

(0) $r_n$ and $r_{n-1}$ are isomorphisms, and $R^i{\hat{f}}_{\ast }(\hat{F})$ is coherent for $i=n-1$;

(1) for $i=n-1$, ${\rho }_i$, ${\phi }_i$, and ${\psi }_i$ are topological isomorphisms (in particular, the filtration defined on $R^{n-1}{f}_{\ast }(F)$ by the kernels of the homomorphisms

R^{n−1} f_*(F) → R^{n−1} f_*(F_k)

is $I^{\prime }$-good);

(2) for $i=n$, ${\rho }_i$, ${\phi }_i$, and ${\psi }_i$ are monomorphisms; furthermore, the filtration on $R^n{f}_{\ast }(F)$ is $I^{\prime }$-good and ${\psi }_n$ is an isomorphism;

(3) the projective system of the $R^i{f}_{\ast }(F_k)$ satisfies, for $i=n-2,n-1$, the uniform Mittag-Leffler condition, i.e. there exists a fixed integer $k⩾0$ such that, for every $p⩾0$ and every $p^{\prime }⩾p+k$, one has:

Im[R^i f_*(F_{p′}) → R^i f_*(F_p)] = Im[R^i f_*(F_{p+k}) → R^i f_*(F_p)].

Proceeding as in EGA III 4.1.8, it is easy to reduce to the case where $X^{\prime }$ is the spectrum of a noetherian ring $A$. In this case, one knows that

R^i f_*(F) = H^i(X, F)^~     (cf. 1.10).

Let $I$ be the ideal of $A$ such that $\tilde{I}=I^{\prime }$, and let

S = ⨁_{k ∈ ℕ} I^k,

H^i = ⨁_{k ∈ ℕ} H^i(X, J^k F),    i ∈ ℤ,

where $H^i$ is equipped with the graded $S$-module structure defined by 1.11, where one has taken $U^{\prime }=X^{\prime }$.

The proof is modelled on that of EGA III 4.1.5; let us give a summary.

We work on ${\phi }_i$ and ${\psi }_i$, which correspond to homomorphisms of modules:

                              H^i(X̂, F̂)
                                  │
                                  │ψ_i
                  ϕ_i             ▼
   H^i(X, F)^∧  ──────▶  lim_{k} H^i(X, F_k).

(a) We assume only that $H^i$ is a graded $S$-module of finite type. We deduce that the filtration defined on $H^i(X,F)$ by the modules:

R^i_k = ker(H^i(X, F) → H^i(X, F_k))

is $I$-good. For this, we use the cohomology exact sequence:

H^i(X, J^{k+1} F) → H^i(X, F) → H^i(X, F_k),

which shows that the graded $S$-module ${⨁}_{k\in \mathbb{N}}{R}_k^i$ is a quotient of the graded $S$-submodule

⨁_{k ∈ ℕ} H^i(X, J^{k+1} F)

of $H^i$, hence is of finite type, since $S$ is noetherian. Whence this first point.

Set:

M^i = H^i(X, F),    H^i_k = H^i(X, F_k).

One has a commutative diagram:

                     s_i
   H^i(X, F)^∧  ─────────▶  lim_{k} (M^i / R^i_k)
        \                         │
         \                        │
          \ ϕ_i                   │ t_i
           \                      ▼
            ──────────────▶  lim_{k} H^i_k,

in which $s_i$ is an isomorphism; indeed, the filtration of $H^i(X,F)$ is $I$-good. Moreover, $t_i$ is a monomorphism; indeed, the functor lim is left exact, and, for every $k⩾0$, the natural morphism $M^i/R_k^i\to H_k^i$ is a monomorphism, by definition of $R_k^i$.

To study the surjectivity of $t_i$, we introduce:

Q^i_k = coker(H^i(X, F) → H^i(X, F_k)),

whence a projective system of exact sequences:

0 → R^i_k → M^i → H^i_k → Q^i_k → 0.

Using the cohomology exact sequence:

H^i(X, F) → H^i(X, F_k) → H^{i+1}(X, J^{k+1} F),

one sees that the graded $S$-module

Q^i = ⨁_{k ∈ ℕ} Q^i_k

is a graded $S$-submodule of $H^{i+1}$. Moreover, for every $k⩾0$, one has:

$$I^{k+1}{Q}_k^i=0,$$

since $Q_k^i$ is the image of $H_k^i$.

(b) We assume only that $H^{i+1}$ is of finite type, and we focus on $t_i$ (forgetting $s_i$). Since $S$ is noetherian, $Q^i$ is of finite type; since $I^{k+1}{Q}_k^i$ vanishes, we find that there exist an integer $r⩾0$ and an integer $k_0⩾0$ such that

I^r Q^i_k = 0    for k ⩾ k₀.

It follows that the projective system $(Q_k^i{)}_{k\in \mathbb{N}}$ is essentially zero, and hence the projective system $(H_k^i{)}_{k\in \mathbb{N}}$ satisfies the uniform Mittag-Leffler condition. From the exact sequence (1.22) one deduces the exact sequence

0 → M^i / R^i_k → H^i_k → Q^i_k → 0,

whence the exact sequence:

0 → lim_{k} M^i / R^i_k  ──t_i──▶  lim_{k} H^i_k → lim_{k} Q^i_k.

Now the projective system $(Q_k^i{)}_{k\in \mathbb{N}}$ is essentially zero, hence $t_i$ is an isomorphism.

(c) Let us prove that, if $H^i$ is of finite type, then ${\psi }_i$ is an isomorphism. It suffices to apply EGA 0_III 13.3.1, taking as a basis of opens of $X$ the affine opens. This is legitimate;

indeed, by (b), the projective system $(H_k^{i-1}{)}_{k\in \mathbb{N}}$ satisfies the Mittag-Leffler condition.

The theorem follows formally from (a), (b), and (c). One will note that, in fact, the proof uses, at each step, the finiteness of $H^i$ only for a single value of $i$.

Let us give some examples in which the hypothesis of Theorem 1.1 is satisfied.

Corollary.

Suppose that $I^{\prime }$ is generated by a section $t^{\prime }$ of ${\mathcal{O}}_{X^{\prime }}$, and denote by $t$ the corresponding section of ${\mathcal{O}}_X$. Let $F$ be a coherent ${\mathcal{O}}_X$-module and let $n$ be an integer. Suppose that:

(i) $t$ is $F$-regular (i.e. the homothety of ratio $t$ on $F$ is a monomorphism).

(ii) $R^i{f}_{\ast }(F)$ is coherent for $i=n-1$ and $i=n$.

Then the hypothesis of Theorem 1.1 is satisfied.

Indeed, one observes that multiplication by $t^k$ defines an isomorphism $F\stackrel{\sim }{\to }{J}^{k}F$, and one deduces that

H^i ≃ R^i f_*(F) ⊗_{𝒪_{X′}} 𝒪_{X′}[T],

where $T$ is an indeterminate. Whence the conclusion.

Corollary.

Suppose that $X^{\prime }=\mathrm{Spec}(A)$, where $A$ is a noetherian ring separated and complete for the $I$-adic topology. Suppose that the $S$-module $H^i$ is of finite type for $i=n-1$ and $i=n$ (cf. 1.14 and 1.15). Then the hypotheses of Theorem 1.1 are satisfied, and one finds a commutative diagram of isomorphisms:

                    ρ′_i
   H^i(X, F)  ─────────────▶  H^i(X̂, F̂)
         \                      /
          \                    /
           \ ϕ′_i         ψ_i /
            \                /
             ▼              ▼
              lim_{k} H^i(X, F_k)              for i = n − 1.

¹

One simply notes that $H^i(X,F)$ is of finite type, hence isomorphic to its completion. One obtains (1.1) by transcribing the diagram of Modules (1.7) into the category of $A$-modules, and replacing the left vertical by $H^i(X,F)$.

Proposition.

Let $A$ be a noetherian ring. Let $t\in A$ and suppose that $A$ is separated and complete for the (tA)-adic topology. Set:

X′ = Spec(A),    Y′ = V(t),    I = (tA).

Let $T$ be a closed subset of $X^{\prime }$; set

X = X′ − T,    Y = Y′ ∩ X = Y′ − (Y′ ∩ T).

Let $F$ be a coherent ${\mathcal{O}}_X$-Module. Finally, let

T′ = {x ∈ X′ | codim({x} ∩ T, {x}) = 1}.

Suppose that:

a) $t$ is $F$-regular,

b) $pro{f}_{T^{\prime }}(F)⩾n+1$,

c) $A$ is a quotient of a regular noetherian ring.

Then, in diagram (1.1), the morphisms ${\rho }_i^{\prime }$, ${\phi }_i^{\prime }$, and ${\psi }_i$ are isomorphisms for $i<n$ and monomorphisms for $i=n$. Moreover ${\psi }_n$ is an isomorphism.

By virtue of 1.3 and 1.2, it suffices to prove that $R^i{f}_{\ast }(F)$ is coherent for $i⩽n$, which follows from the finiteness theorem 2.1.²

In particular:

Example.

One will apply 1.4 when $A$ is a local ring and $t$ belongs to the radical $r(A)$ of $A$. One will then take $T=r(A)$. In this case, for $n=1$, one obtains the following statement:

If $A$ is noetherian, separated and complete for the $t$-adic topology, and a quotient of a regular ring (for example, if $A$ is complete), if moreover $t$ is $F$-regular and if $prof{F}_x⩾2$ for every $x\in \mathrm{Spec}(A)$ such that $\dim A/x=1$, then the natural homomorphism

Γ(X, F) → Γ(X̂, F̂)

is an isomorphism.

Indeed, keeping the notation of 1.4, one has $T=r(A)$, and formula (1.33) says that

T′ = {x ∈ Spec(A) | dim A/x = 1}.

2. The existence theorem

Let us first state EGA III 3.4.2 in a slightly more general form.

Let $f:X\to X^{\prime }$ be an adic morphism³ of formal preschemes, with $X^{\prime }$ noetherian. Let $I^{\prime }$ be an ideal of definition of $X^{\prime }$; since $f$ is adic, $f\ast I^{\prime }=J$ is⁴ an ideal of definition of $X$.

For every $n\in \mathbb{N}$, set

$$X_n=(X,{\mathcal{O}}_X/J^{n+1});$$

this is an ordinary prescheme having the same underlying topological space as $X$.

Let $F$ be a coherent ${\mathcal{O}}_X$-Module. For every $k\in \mathbb{N}$, the ${\mathcal{O}}_{X_k}$-Modules

$$F_k=F/J^{k+1}F$$

are coherent. For every $i$, one has a homomorphism

ψ_i: R^i f_*(F) → lim_{k} R^i f_*(F_k),

deduced by functoriality from the natural homomorphism:

$$F\to F_k.$$

Set

S = gr_{I′} 𝒪_{X′} = ⨁_{k ∈ ℕ} I′^k / I′^{k+1},

gr_J(F) = ⨁_{k ∈ ℕ} J^k F / J^{k+1} F,

K^i = R^i f_*(gr_J(F)) = ⨁_{k ∈ ℕ} R^i f_*(J^k F / J^{k+1} F).

It is clear that $K^i$ is equipped with a graded $S$-Module structure.

Theorem.

Suppose that $K^i$ is a graded $S$-Module of finite type for $i=n-1$, $i=n$, $i=n+1$. Then:

(i) $R^n{f}_{\ast }(F)$ is coherent.

(ii) The homomorphism ${\psi }_n$ (2.3) is a topological isomorphism. The natural filtration of the right-hand side of (2.3) is $I^{\prime }$-good.

(iii) The projective system of the $R^n{f}_{\ast }(F_k)$ satisfies the uniform Mittag-Leffler condition.

The proof is very easy from EGA 0_III 13.7.7 (cf. EGA III 3.4.2), provided one corrects the text on page 78 as indicated in (EGA III 2, Err_III 24).

Theorem.

⁵

Let $A$ be a noetherian adic ring and let $I$ be an ideal of definition of $A$. Let $T$ be a closed subset of $X^{\prime }=\mathrm{Spec}(A)$. Suppose that $I$ is generated by a $t\in A$. Take up the notation 1.31, 1.32, and 1.33. Let $F$ be a coherent ${\mathcal{O}}_{\hat{X}}$-Module. Set

$$F_0=F/JF,$$

where $J=t{\mathcal{O}}_{\hat{X}}$ is an ideal of definition of $\hat{X}$. Suppose that $A$ is a quotient of a regular noetherian ring and that:

(1) $t$ is $F$-regular,

(2) $pro{f}_{T^{\prime }}{F}_0⩾2$.

Then there exists a coherent ${\mathcal{O}}_X$-Module $F$ such that $F^{\wedge }\simeq F$.

It suffices to prove that ${\hat{f}}_{\ast }(F)$ is a coherent ${\mathcal{O}}_{{\hat{X}}^{\prime }}$-Module, where $\hat{f}:\hat{X}\to {\hat{X}}^{\prime }$ is the morphism of formal preschemes deduced from the injection of $X$ into $X^{\prime }$ by completion with respect to $t$. Indeed, $A$ is separated and complete for the $t$-adic topology, so there will exist an $A$-module $F^{\prime }$ whose completion will be isomorphic to ${\hat{f}}_{\ast }(F)$. Since $X$ is an open of $X^{\prime }$, one will be able to take $F=F^{\prime }|X$.

It remains to show that 2.1 is applicable to the morphism of formal preschemes $\hat{f}$ and to $F$. Now, by hypothesis (1), for every $k\in \mathbb{N}$ one has an isomorphism:

J^k F / J^{k+1} F → F/J F,

whence it follows that the hypothesis of 2.1 will be satisfied if one knows that

R^i f_*(F₀) is coherent for i ⩽ 1.

Now this follows from (2) and from the finiteness theorem 2.1.² Whence the conclusion.

It remains to specialize this statement by supposing that $A$ is a local ring.

Corollary.

Let $A$ be a noetherian local ring and let $t\in r(A)$ be an element of the radical of $A$. Suppose that $A$ is separated and complete for the $t$-adic topology and, moreover, a quotient of a regular ring (for example, suppose that $A$ is a complete noetherian local ring). Set

X′ = Spec A,    T = {r(A)},

and take up the notation (1.31), (1.32), and (1.33). Let $F$ be an ${\mathcal{O}}_{\hat{X}}$-Module. Suppose that:

(1) $t$ is $F$-regular,

(2) $pro{f}_{T^{\prime }}{F}_0⩾2$, with $F_0=F/JF$ and $J=t{\mathcal{O}}_{\hat{X}}$.

Then there exists a coherent ${\mathcal{O}}_X$-Module $F$ such that $F^{\wedge }\simeq F$.

Note that here $T^{\prime }$ is the set of prime ideals $p$ of $A$ such that $\dim A/p=1$.