The theorem on formal functions

Written by GPT-6.1 Sol (OpenAI), in Codex, at Ultra effort, October 2026. Self-checked by the writing AI. Public domain (CC0).

A fiber remembers a family at one parameter. Its successive infinitesimal neighborhoods remember powers of that parameter's maximal ideal. Formal functions identifies the inverse limit of their cohomology with the completion of a finite cohomology module. The individual comparison maps can fail to be isomorphisms; what makes the theorem work is a uniform bound on how long their errors persist.

We use proper coherent direct images, affine cohomology and Serre's criterion, and flat base change. The planned lesson Completion, in Commutative algebra for geometry, supplies completion of finite modules, its exactness over Noetherian rings, and faithful flatness of completion at a maximal ideal. The Noetherian dimension bound for abelian sheaves, already stated in Cohomology of sheaves on ringed spaces, has its exact open proof in [Stacks, Tag 02UZ]. We use that bound on the topological spaces of the thickened fibers below. No flatness assumption on the coherent sheaf is imposed in this lesson.

1. The completion map

Let \(A\) be Noetherian, \(I\subset A\) an ideal, \(f:X\to\operatorname{Spec}A\) proper, and \(F\) coherent. For \(n\geq1\), put \[ X_n=X\times_A\operatorname{Spec}(A/I^n),\qquad F_n=F|_{X_n}. \] The closed immersion \(i_n:X_n\hookrightarrow X\) has \(i_{n*}F_n=F/I^nF\). Since closed immersion pushforward is exact and has no positive direct images, we may compute \[ H^q(X_n,F_n)=H^q(X,F/I^nF). \] Set \(M^q=H^q(X,F)\), a finite \(A\)-module by proper finiteness. The quotient map on sheaves induces \[ M^q/I^nM^q\longrightarrow H^q(X_n,F_n), \] because \(I^n\) annihilates the target. These maps are compatible with reduction from \(n+1\) to \(n\). We seek an isomorphism from \[ \widehat{M^q}=\varprojlim_n M^q/I^nM^q \] to the inverse limit of the right-hand terms. Completion here is with respect to \(I\); neither \(A\) nor \(M^q\) is assumed complete in advance.

The theorem is not a claim that every finite-level map is an isomorphism. For example, the extension bundle on \(\mathbf P^1_{k[t]}\) with class \(t\) has \(H^0(X,E)=0\), but has a nonzero section on the fiber \(t=0\). We will return to its infinitesimal neighborhoods in Exercise 7.6.

2. All powers at once

The Rees algebra is \[ R=\bigoplus_{n\geq0}I^nT^n\subset A[T],\qquad R_0=A. \] If \(I=(a_1,\ldots,a_r)\), then \(R\) is generated by the degree-one elements \(a_iT\), hence is a quotient of a polynomial algebra in finitely many variables over \(A\). It is Noetherian. The symbol \(T\) records degree; it need not itself belong to \(R\).

Lemma 2.1 (graded cohomological finiteness). For each \(q\geq0\), the graded module \[ Q^q=\bigoplus_{n\geq0}H^q(X,I^nF)T^n \] is finite over \(R\).

Proof. Form \(X_R=X\times_A\operatorname{Spec}R\), with affine projection \(\pi:X_R\to X\). On \(X\), the graded sheaf \(\bigoplus I^nF\) is a module over \(\mathcal O_X\otimes_A R\): a degree-\(d\) element acts by multiplication into the degree shifted by \(d\). It is a quotient of \(F\otimes_A R\), by the degreewise maps \(F\otimes_A I^n\to I^nF\). The affine module/sheaf correspondence gives a coherent sheaf \(G\) on \(X_R\) with \[ \pi_*G=\bigoplus_{n\geq0}I^nF. \] Coherence follows because \(X_R\) is Noetherian and this sheaf is locally a finite module, generated by a finite set of degree-zero generators of \(F\).

The map \(X_R\to\operatorname{Spec}R\) is proper by base change. Proper cohomology finiteness makes \(H^q(X_R,G)\) finite over \(R\). Affine pushforward and Leray identify it with \(H^q(X,\pi_*G)\). Finally cohomology of quasi-coherent sheaves on the quasi-compact, quasi-separated scheme \(X\) commutes with filtered colimits, by the third lesson. A direct sum is the filtered colimit of finite partial sums, so \[ H^q(X,\pi_*G)=\bigoplus_{n\geq0}H^q(X,I^nF). \] The identifications preserve multiplication and grading. This is the asserted finite graded module. \(\square\)

We next extract two consequences. Write \[ J_n^q=\operatorname{im}\bigl(H^q(X,I^nF)\to M^q\bigr), \qquad K_n^q=\ker\bigl(H^q(X,I^nF)\to M^q\bigr). \] Both collections respect graded multiplication. Indeed inclusion of ideal powers commutes with multiplying a section or cohomology class by an element of \(I^d\). Thus \(\bigoplus J_n^qT^n\) is a finite graded image module of \(Q^q\), and \(\bigoplus K_n^qT^n\) is a finite graded submodule, using Noetherianness of \(R\).

Lemma 2.2 (bounds on the image and kernel). There are integers \(c_J,c_K\geq0\) such that \[ I^nM^q\subset J_n^q\subset I^{n-c_J}M^q\quad(n\geq c_J), \] and the transition map \(K_m^q\to K_n^q\) is zero whenever \(m\geq n+c_K\).

Proof. Multiplication by an element of \(I^n\) factors as \(F\to I^nF\to F\), proving the first inclusion. Choose finitely many homogeneous generators of the image module, all in degrees at most \(c_J\). The degree-\(n\) part is a sum of \(I^{n-d}J_d^q\) over generator degrees \(d\leq c_J\). Each is contained in \(I^{n-c_J}M^q\), proving the second inclusion. This is the cohomological analogue of an Artin–Rees bound.

Choose homogeneous kernel generators \(k_\ell\in K_{d_\ell}^q\) with \(d_\ell\leq c_K\). An element of \(K_m^q\) is a sum of products \(a_\ell k_\ell\), with \(a_\ell\in I^{m-d_\ell}\). If \(m\geq n+c_K\), each such coefficient is a sum of products \(ba'\), where \(b\in I^n\) and \(a'\in I^{m-d_\ell-n}\). Under transition to \(H^q(X,I^nF)\), its product with \(k_\ell\) is computed by first including \(I^{d_\ell}F\) into \(F\), then multiplying by \(a'\), then by \(b\) into \(I^nF\). The initial inclusion sends \(k_\ell\) to zero, by definition of the kernel. Hence every product maps to zero. This proves the uniform transition bound. \(\square\)

A system whose sufficiently distant transition maps to each fixed term are zero is called pro-zero. Its individual terms may be nonzero; its inverse limit is zero because each coordinate of a compatible family is the image of a distant, necessarily zero, transition. Lemma 2.2 supplies this stronger, bounded form of disappearance for the kernel system.

3. Stabilization of the thickened cohomology

The short exact sequence \(0\to I^nF\to F\to F/I^nF\to0\) gives the exact sequence \[ 0\to M^q/J_n^q\longrightarrow H^q(X_n,F_n)\longrightarrow K_n^{q+1}\to0. \tag{1} \] It is a sequence of inverse systems. The left transitions are surjective because \(J_m^q\subset J_n^q\) for \(m\geq n\). The right system is pro-zero by Lemma 2.2 in degree \(q+1\).

Proposition 3.1 (the Mittag–Leffler bound). For fixed \(q\), choose \(c\) as the kernel bound in degree \(q+1\). Then for \(m\geq n+c\), \[ \operatorname{im}\bigl(H^q(X_m,F_m)\to H^q(X_n,F_n)\bigr) =\operatorname{im}\bigl(M^q\to H^q(X_n,F_n)\bigr). \] Consequently the thickened-cohomology system is Mittag–Leffler.

Proof. In diagram (1) for indices \(m,n\), the right transition is zero. Therefore the middle transition lands in the left submodule \(M^q/J_n^q\). It contains all of that submodule: the left transition \(M^q/J_m^q\to M^q/J_n^q\) is onto, and these left modules inject into the respective middle terms. Thus its image is exactly the claimed submodule. The image is independent of all sufficiently large \(m\), which is the Mittag–Leffler condition. \(\square\)

We can now take the limit without an unexplained exchange of limits and cohomology. If \((u_n)\) is a compatible family of middle classes in (1), its images in \(K_n^{q+1}\) form a compatible family in a pro-zero system, so are all zero. Thus each \(u_n\) lies in the injective left term, with a unique preimage. These preimages are automatically compatible. We obtain the canonical identification \[ \varprojlim_n H^q(X_n,F_n)=\varprojlim_n M^q/J_n^q. \tag{2} \] This also proves why a transient class on a small thickening need not define a class in the inverse limit. Compatibility with every higher thickening is a real constraint.

4. Global and stalk forms of formal functions

Theorem 4.1 (formal functions). Under the hypotheses of Section 1, the canonical maps give an isomorphism \[ \widehat{H^q(X,F)}\xrightarrow{\sim}\varprojlim_n H^q(X_n,F_n). \] It is an isomorphism of \(\widehat A\)-modules and a homeomorphism for their inverse-limit topologies.

Proof. Combine (2) with the inclusions from Lemma 2.2: \[ I^nM^q\subset J_n^q,\qquad J_{n+c_J}^q\subset I^nM^q. \] They say that the two descending filtrations \((I^nM^q)\) and \((J_n^q)\) are cofinal. The identification (2) also preserves the limit topologies: each left term is a discrete submodule of the corresponding middle term, and its unique coordinate preimage is continuous on that submodule. The first inclusions give the canonical continuous map between the inverse limits of the corresponding quotients. The second give its inverse by taking the cofinal shifted indices \(n+c_J\). The two composites are the usual transitions and hence induce the identity on limits. Both maps are continuous for the quotient limit topologies, proving the homeomorphism. The maps arise from \(A\)-linear maps at all levels, where \(I^n\) annihilates the cohomology of \(F_n\), and so preserve the induced \(\widehat A\)-action. \(\square\)

Corollary 4.2 (the stalk form). Let \(f:X\to S\) be proper, \(S\) locally Noetherian, \(F\) coherent, and \(s\in S\). Put \(A_s=\mathcal O_{S,s}\), with maximal ideal \(\mathfrak m_s\), and \[ X_n=X\times_S\operatorname{Spec}(A_s/\mathfrak m_s^n),\qquad F_n=F|_{X_n}. \] Then \[ \widehat{(R^qf_*F)_s}\cong\varprojlim_n H^q(X_n,F_n) \] as \(\widehat{A_s}\)-modules.

Proof. The canonical map \(\operatorname{Spec}A_s\to S\) is flat: on an affine neighborhood it is localization. Flat base change and the affine description of quasi-coherent direct images identify the stalk with \(H^q(X\times_S\operatorname{Spec}A_s,F|_{X\times_S\operatorname{Spec}A_s})\). The local ring is Noetherian, the changed morphism proper, and the changed sheaf coherent. Apply Theorem 4.1 with ideal \(\mathfrak m_s\). Its thickenings are precisely the displayed \(X_n\), since their parameter maps factor through \(\operatorname{Spec}A_s\). \(\square\)

5. Dimension and finiteness consequences

Theorem 5.1 (vanishing above a fiber's dimension). For the morphism of Corollary 4.2, \[ (R^qf_*F)_s=0\quad\text{if }q>\dim X_s. \]

Proof. Each \(X_n\) has the same underlying topological space as \(X_s\): the extra ideal is nilpotent and therefore belongs to every prime. That space is Noetherian, of finite dimension \(d\), because \(X_s\) is of finite type over a field. The Noetherian topological vanishing theorem [Stacks, Tag 02UZ] gives \(H^q(X_n,F_n)=0\) for \(q>d\). Formal functions makes the completed stalk zero. The uncompleted stalk is finite by proper coherence. For a finite module over a Noetherian local ring, completion preserves the residue-field quotient; zero completion implies \(M/\mathfrak mM=0\), and Nakayama implies \(M=0\). This proves the assertion. An empty fiber gives empty thickenings and the same conclusion in every degree. \(\square\)

The vanishing is a statement about the whole stalk, stronger than a statement about its tensor with the residue field. In particular, if all fiber dimensions are at most \(d\), then \(R^qf_*F=0\) for \(q>d\), without a flatness assumption.

Theorem 5.2 (proper with finite fibers is finite). A proper morphism over a locally Noetherian base whose fibers have finitely many points is finite.

Proof. Its fibers are finite type over fields. Such schemes with finitely many points are zero-dimensional by Noether normalization. Indeed an affine chart admits a finite integral surjection onto affine space of its dimension; a positive-dimensional affine space over any field has infinitely many points, so a finite-point chart cannot have positive dimension. Their zero-dimensional coordinate rings are finite-dimensional by Noether normalization. Thus Theorem 5.1 makes \(R^1f_*G=0\) for every coherent \(G\).

Work over an affine Noetherian neighborhood \(S=\operatorname{Spec}A\). Its inverse image \(X\) is Noetherian, separated and quasi-compact. Any finite type ideal \(J\subset\mathcal O_X\) is coherent, and Leray over the affine base gives \[ H^1(X,J)=\Gamma(S,R^1f_*J)=0. \] Here affine vanishing eliminates the other Leray terms, since the direct images are quasi-coherent. The finite-ideal version of Serre's affine criterion from the third lesson now implies that \(X\) is affine. Its function ring \(H^0(X,\mathcal O_X)\) is a finite \(A\)-module by proper cohomology finiteness. Hence \(X\to\operatorname{Spec}A\) is finite. This is local on the base and proves the theorem. \(\square\)

A quasi-finite proper morphism over a locally Noetherian base is therefore finite. The planned Zariski's Main Theorem lesson in Morphisms of schemes gives another route; the present route uses formal functions and the cohomological affine criterion.

6. The blow-up of a plane point

Let \(b:X\to\mathbf A^2_k\) be the blow-up of the origin and \(E\cong\mathbf P^1_k\) its exceptional curve. Its two charts are \[ U=\operatorname{Spec}k[x,v],\quad y=xv, \qquad V=\operatorname{Spec}k[u,y],\quad x=uy. \] They identify the blow-up with the closed incidence scheme in \(\mathbf A^2\times\mathbf P^1\), so \(b\) is projective and proper. Off the origin it is an isomorphism. The pulled-back origin ideal \(\mathfrak m=(x,y)\) is generated by \(x\) on \(U\) and by \(y\) on \(V\), and is the invertible ideal \(J=\mathcal O_X(-E)\). On the overlap \(y=vx\), so its conormal restriction is \(\mathcal O_E(1)\). Consequently \[ J^j/J^{j+1}\cong\mathcal O_E(j)\quad(j\geq0). \] The positive sign is essential: the normal bundle of \(E\) is \(\mathcal O_E(-1)\), whereas the successive ideal layers use its dual.

The thickened fiber \(X_n\) is defined by \(J^n\). For \(n\geq2\), \[ 0\to\mathcal O_E(n-1)\to\mathcal O_{X_n}\to\mathcal O_{X_{n-1}}\to0. \] Projective-line cohomology gives \(H^1(E,\mathcal O_E(j))=0\) for every \(j\geq0\). The long exact sequences, starting with \(X_1=E\), give \(H^1(X_n,\mathcal O_{X_n})=0\) for all \(n\). By formal functions and Nakayama, \((R^1b_*\mathcal O_X)_0=0\). Away from zero the morphism is an isomorphism, so \[ R^1b_*\mathcal O_X=0. \] All still higher direct images vanish by the fiber-dimension bound, because the largest fiber dimension is one.

We can also see the completed degree-zero comparison explicitly. The canonical polynomial map \[ k[x,y]/(x,y)^n\longrightarrow H^0(X_n,\mathcal O_{X_n}) \] is an isomorphism. For \(n=1\) both sides are \(k\). At the induction step, its map on the leftmost ideal layers is \[ (x,y)^{n-1}/(x,y)^n\xrightarrow{\sim}H^0(E,\mathcal O_E(n-1)), \] the monomial basis identification from projective-space cohomology. Both the polynomial quotient sequence and the section sequence are short exact, the latter because the layer has zero \(H^1\). Induction proves the isomorphism. Their limits give \(k[[x,y]]\). The unit map from the base local ring to \((b_*\mathcal O_X)_0\) becomes an isomorphism after completion. Exactness and faithful flatness of completion for finite modules make it an isomorphism before completion; off the origin the same holds directly. Thus \(b_*\mathcal O_X=\mathcal O_{\mathbf A^2}\), as well as higher vanishing.

7. Exercises with solutions

Exercise 7.1 (easy: completed constants). State and prove the completed-stalk formula for \(f_*\mathcal O_X\) at a point of a locally Noetherian base.

Solution. For proper \(f\), Corollary 4.2 in degree zero gives \(\widehat{(f_*\mathcal O_X)_s}=\varprojlim_n H^0(X_n,\mathcal O_{X_n})\), with \(X_n\) defined over \(\mathcal O_{S,s}/\mathfrak m_s^n\). Localization is flat, so identifies the stalk with degree-zero cohomology over the local ring; global formal functions for that ring and ideal gives the formula. All maps preserve multiplication, since they are restriction maps of structure sheaves. Hence it is an isomorphism of complete algebras, not only of modules.

Exercise 7.2 (medium: the exceptional layers). Compute the ideal layers of the exceptional curve in the plane blow-up and deduce \(R^1b_*\mathcal O_X=0\).

Solution. The local ideal frames are \(x\) and \(y\), related by \(y=vx\); their restriction has the transition of \(\mathcal O_{\mathbf P^1}(1)\). Since the ideal is invertible, its \(j\)-th layer is its restricted \(j\)-th tensor power, \(\mathcal O_{\mathbf P^1}(j)\). Their \(H^1\) groups vanish for \(j\geq0\), so induction on the exact thickening sequences gives zero \(H^1\) on every \(X_n\). Formal functions gives zero completed stalk at the origin, and finite-module Nakayama gives zero stalk. Off the origin the map is an isomorphism, proving the global assertion.

Exercise 7.3 (medium: finite fibers). Derive finiteness of a proper morphism with finite fibers using only formal functions, affine cohomology and Serre's criterion.

Solution. Every fiber and thickening has dimension zero, so its positive cohomology is zero. Formal functions and Nakayama annihilate \(R^1f_*J\) for every coherent ideal. Over an affine Noetherian base, Leray and affine vanishing give \(H^1(X,J)=0\). The finite-ideal affine criterion makes \(X\) affine. Proper finiteness makes its coordinate algebra finite over the base algebra, which is exactly finiteness of the morphism. This also covers empty fibers.

Exercise 7.4 (medium: a uniform dimension bound). If \(\dim X_s\leq d\) for every \(s\), prove \(R^qf_*F=0\) for \(q>d\). Can \(\dim X\) replace the fiber calculation in the proof without further assumptions?

Solution. Theorem 5.1 kills every stalk in the stated degrees, hence kills the sheaf. The proof works on the Noetherian topology of each thickened fiber and does not require a dimension bound on the total space. A locally Noetherian base can have unbounded or infinite Krull dimension, so a total-space bound might be unavailable and would in any event miss the sharper relative assertion.

Exercise 7.5 (hard: the inverse-limit step). Prove the stabilization and limit claims from (1), without assuming the middle transitions are surjective.

Solution. The right transition \(K_m^{q+1}\to K_n^{q+1}\) is zero for \(m\geq n+c\), so the middle image lies in \(M^q/J_n^q\). The left transition is onto, so the middle image contains this entire submodule. Equality proves Mittag–Leffler stabilization. A compatible family in the middle maps to a compatible family in the pro-zero right system. Each of its right coordinates is the image of a sufficiently distant zero transition, so is zero. Each middle coordinate therefore has a unique preimage in the injective left term, and uniqueness forces compatibility of those preimages. Thus the two limits agree. Finally \(I^nM^q\subset J_n^q\) and \(J_{n+c_J}^q\subset I^nM^q\) identify the left limit continuously with the adic completion. Every assertion is justified without finite-level surjectivity of the middle system.

Exercise 7.6 (challenging: transient infinitesimal sections). For the extension with class \(t\) on \(\mathbf P^1_{k[t]}\), compute \(H^0\) on every thickening \(B_n=k[t]/(t^n)\), its transition maps, and its inverse limit.

Solution. Its Grothendieck complex is \([A\xrightarrow{t}A]\) in degrees zero and one. Over \(B_n\), the kernel of multiplication by \(t\) is \(k\,t^{n-1}\), including \(k\) when \(n=1\). Reduction \(B_{n+1}\to B_n\) sends its kernel generator \(t^n\) to zero. Thus all adjacent transitions on degree-zero cohomology are zero, and its inverse limit is zero. This agrees with the completion of \(H^0(X,E)=0\), although every finite thickening has a nonzero section. In degree one, the cokernel is \(k\) at every level and the transitions are identities; its limit agrees with the completion of \(A/(t)\). This exhibits both the transient error and the persistent cohomology in one family.

8. Ampleness near a fiber

One additional geometric consequence is useful later. If \(f:X\to S\) is proper with \(S\) locally Noetherian and an invertible \(L\) is ample on \(X_s\), then it is relatively ample over some open neighborhood of \(s\). We import the precise open proof [Stacks, Tags 0D2M and 0D2N]. It first uses the graded ideal layers and Serre vanishing on the fiber to lift sufficiently positive sections. A finite collection giving a fiber embedding then generates on a neighborhood, by properness; the induced projective-space map is finite near that fiber, and its pulled-back hyperplane bundle is ample there. The lifting and neighborhood argument is supplied by those exact proof locators, rather than by a flatness assumption on \(F\).

References