# Theorems A and B on Stein manifolds

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

This lesson proves H. Cartan's Theorems A and B for Stein manifolds: on a complex manifold with a smooth strictly plurisubharmonic exhaustion, every coherent analytic sheaf has vanishing cohomology in positive degrees (Theorem B) and is generated at every point by its global sections (Theorem A). The proof follows A. Andreotti and H. Grauert. A sublevel set of the exhaustion is enlarged to a bigger one through finitely many small bumps, each inside a coordinate ball, where the local results of the previous lessons apply. The Mayer–Vietoris sequence shows that cohomology in positive degrees does not change along the way. Since the smallest sublevel sets are empty, all these groups vanish, and an abstract Mittag-Leffler argument passes to the whole manifold. Schwartz's finiteness theorem enters once, to make the relevant groups Hausdorff.

We use [Plurisubharmonic functions and Stein manifolds](plurisubharmonic-functions-and-stein-manifolds.md), [Stein domains in complex space](stein-domains-in-complex-space.md) and [Fréchet spaces of sections and Schwartz's theorem](frechet-spaces-of-sections-and-schwartzs-theorem.md). The Mayer–Vietoris sequence for two open sets is [Stacks, Tag 01EB](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cohomology.html#cohomology-lemma-mayer-vietoris).

Basic references are [Demailly] and [Andreotti–Grauert 1962].

Throughout, \(X\) is a complex manifold with a smooth strictly plurisubharmonic exhaustion \(\psi\), \(X_c=\{\psi<c\}\), and \(\mathcal S\) is a coherent analytic sheaf on \(X\). A **resolving ball** is an open subset of \(X\) identified by a chart with a ball \(B\subset\mathbf C^n\), such that \(\mathcal S\) has a finite free resolution on a neighbourhood of the closure of \(B\) in the chart. By the local syzygy theorem [Coherent sheaves and Oka's coherence theorem, Theorem 3.4](coherent-sheaves-and-okas-theorem.md#3-the-category-of-coherent-analytic-sheaves), every point has arbitrarily small resolving balls. Every Stein open subset of a resolving ball is a Stein domain in the chart on which \(\mathcal S\) has a finite free resolution, so by [Stein domains in complex space, Theorem 4.1](stein-domains-in-complex-space.md#4-coherent-sheaves-on-stein-domains) it carries no higher cohomology of \(\mathcal S\).

**Cohomological inputs.** [Cohomology of sheaves on ringed spaces, Theorem 5.1](course:AG-QC/cohomology-of-sheaves-on-ringed-spaces#5-two-open-sets-and-the-connecting-map) proves the two-open Mayer–Vietoris sequence from a flasque resolution. It applies to the coherent module sheaves used here. Its difference map is restriction from the first open minus restriction from the second; the map used in Section 3 below has the opposite sign. Multiplying the intersection term by minus one, and changing the connecting map by the same sign, identifies these conventions without changing exactness. [Čech cohomology, Theorem 3.2](course:AG-QC/cech-cohomology#3-comparing-overlaps-with-derived-cohomology) gives the refinement-compatible comparison for covers with acyclic finite intersections used in Sections 3 and 4. For Exercise 6.1, [Theorem 5.1 of that lesson](course:AG-QC/cech-cohomology#5-torsors-are-first-cohomology-classes) proves injectivity from fixed-cover Čech cohomology to sheaf cohomology in degree one; that injection does not require the cover to be acyclic. These are the precise programme inputs for the cohomological steps, in addition to the analytic and functional-analytic results cited below.

## 1. Local approximation

**Lemma 1.1.** Let \(W\) be a Stein open subset of a resolving ball and \(W'\subset W\) open. Suppose that for every compact \(L\subset W'\) there are a smooth strictly plurisubharmonic exhaustion \(\psi'\) of \(W\) and a number \(b\) with \(L\subset\{\psi'<b\}\) and \(\overline{\{\psi'<b\}}\subset W'\). Then the restriction \(\mathcal S(W)\to\mathcal S(W')\) has dense image.

**Proof.** Let \(W_b=\{\psi'<b\}\) for such a pair. Both \(W\) and \(W_b\) are Stein domains in the chart, so by Theorem 4.1 of [Stein domains in complex space](stein-domains-in-complex-space.md#4-coherent-sheaves-on-stein-domains) the maps \(\mathcal O(W)^{p_0}\to\mathcal S(W)\) and \(\mathcal O(W_b)^{p_0}\to\mathcal S(W_b)\) from the first term of the resolution are surjective, and the second is continuous and open by the definition of the Fréchet topology. By the approximation theorem [Stein domains in complex space, Theorem 3.1](stein-domains-in-complex-space.md#3-approximation-on-sublevel-sets), \(\mathcal O(W)^{p_0}\to\mathcal O(W_b)^{p_0}\) has dense image. Hence \(\mathcal S(W)\to\mathcal S(W_b)\) has dense image.

Now let \(s\in\mathcal S(W')\). Choose relatively compact open sets \(W'_1\subset W'_2\subset\cdots\) exhausting \(W'\). The map \(\mathcal S(W')\to\prod_k\mathcal S(W'_k)\) is a topological embedding, by the open mapping theorem, since its image is closed; so a neighbourhood of \(s\) contains a set \(\{t:\ t|_{W'_k}\in N\}\) for some \(k\) and some neighbourhood \(N\) of \(s|_{W'_k}\). Apply the hypothesis to \(L=\overline{W'_k}\): approximate \(s|_{W_b}\) by restrictions of elements of \(\mathcal S(W)\), and restrict further to \(W'_k\subset W_b\). \(\square\)

## 2. Enlarging a sublevel set by bumps

**Lemma 2.1.** Let \(c<d\). There are open sets \(X_c=G_0\subset G_1\subset\cdots\subset G_s=X_d\) and open sets \(U_0,\ldots,U_{s-1}\) such that, for each \(j\):

1. \(G_{j+1}=G_j\cup U_j\), and \(G_j=\{\psi_j<c_j\}\) for a smooth strictly plurisubharmonic exhaustion \(\psi_j\) of \(X\);
2. \(U_j\) is a Stein open subset of a resolving ball;
3. \(G_j\cap U_j\) is Stein, and for every compact \(L\subset G_j\cap U_j\) there are a smooth strictly plurisubharmonic exhaustion \(\psi'\) of \(U_j\) and \(b\) with \(L\subset\{\psi'<b\}\) and \(\overline{\{\psi'<b\}}\subset G_j\cap U_j\).

**Proof.** First fix \(c<c'\leq d\) with \(c'-c\leq\varepsilon_0\), where \(\varepsilon_0\) is chosen below. Cover the compact set \(\overline{X_d}\setminus X_c\) by finitely many resolving balls \(A_0,\ldots,A_{s-1}\), with centres \(a_j\) and radii \(r_j\) in their charts, and choose smooth \(\theta_j\geq0\) with compact support in \(A_j\), \(\sum_j\theta_j\leq1\), and \(\sum_j\theta_j=1\) near \(\overline{X_d}\setminus X_c\). By the perturbation lemma [Plurisubharmonic functions and Stein manifolds, Lemma 2.4](plurisubharmonic-functions-and-stein-manifolds.md#2-stein-manifolds) there is \(\varepsilon_0>0\) such that \(\psi-\sum_j\varepsilon_j\theta_j\) is strictly plurisubharmonic whenever \(0\leq\varepsilon_j\leq\varepsilon_0\). Put \(\varepsilon=c'-c\) and

\[
\psi_j=\psi-\varepsilon\sum_{k<j}\theta_k,\qquad G_j=\{\psi_j<c\}\qquad(0\leq j\leq s).
\]

Each \(\psi_j\) is a strictly plurisubharmonic exhaustion (it differs from \(\psi\) by a bounded function). We have \(G_0=X_c\) and \(G_j\subset G_{j+1}\). Moreover \(G_s=X_{c'}\): where \(\psi<c\) we have \(\psi_s<c\); where \(c\leq\psi<c'\), the point lies in \(\overline{X_d}\setminus X_c\), so \(\psi_s=\psi-\varepsilon<c\); and where \(\psi\geq c'\), \(\psi_s\geq\psi-\varepsilon\geq c\). Since \(\psi_{j+1}-\psi_j=-\varepsilon\theta_j\) is supported in \(A_j\), we get \(G_{j+1}=G_j\cup U_j\) with \(U_j=G_{j+1}\cap A_j\).

\(U_j\) is the intersection of the Stein sets \(G_{j+1}\) (a sublevel set of \(\psi_{j+1}\)) and \(A_j\), so it is Stein, with the exhaustion \(\phi_j=1/(c-\psi_{j+1})+1/(r_j^2-|z-a_j|^2)\); similarly \(G_j\cap U_j=G_j\cap A_j\) is Stein. For (3), let \(L\subset G_j\cap U_j\) be compact, \(a=\max_L\psi_j<c\), and \(a<b<c\). For \(\eta>0\) the function \(\psi'=\psi_j+\eta\phi_j\) is a smooth strictly plurisubharmonic exhaustion of \(U_j\), as \(\psi_j\) is bounded on the relatively compact set \(U_j\). Since \(\phi_j\) is bounded on \(L\), \(L\subset\{\psi'<b\}\) for small \(\eta\). Since \(\phi_j>0\), every point of \(U_j\) with \(\psi'\leq b\) has \(\psi_j\leq b<c\); and \(\{\psi'\leq b\}\) is compact in \(U_j\). So \(\overline{\{\psi'<b\}}\subset G_j\cap U_j\).

For general \(c<d\), apply this to a subdivision \(c=c_0<c_1<\cdots<c_N=d\) with steps at most \(\varepsilon_0\) (the number \(\varepsilon_0\) depends only on the chosen balls and functions for the pair \(c,d\), which may be used for every step), and concatenate. \(\square\)

## 3. Comparison of sublevel sets

**Proposition 3.1.** Let \(c<d\). Then:

1. \(H^k(X_c,\mathcal S)\) is finite-dimensional and Hausdorff for \(k\geq1\);
2. the restriction \(H^k(X_d,\mathcal S)\to H^k(X_c,\mathcal S)\) is bijective for \(k\geq1\);
3. the restriction \(\mathcal S(X_d)\to\mathcal S(X_c)\) has dense image.

**Proof.** *Surjectivity.* Take the sets of Lemma 2.1. Since \(U_j\) and \(G_j\cap U_j\) are Stein open subsets of a resolving ball, \(H^k(U_j,\mathcal S)=H^k(G_j\cap U_j,\mathcal S)=0\) for \(k\geq1\). The Mayer–Vietoris sequence for \(G_{j+1}=G_j\cup U_j\),

\[
\mathcal S(G_j)\oplus\mathcal S(U_j)\to\mathcal S(G_j\cap U_j)\xrightarrow{\ \partial\ }H^1(G_{j+1},\mathcal S)\to H^1(G_j,\mathcal S)\to0\to\cdots\to H^k(G_{j+1},\mathcal S)\to H^k(G_j,\mathcal S)\to0,
\tag{3.1}
\]

shows that \(H^k(G_{j+1},\mathcal S)\to H^k(G_j,\mathcal S)\) is bijective for \(k\geq2\) and surjective for \(k=1\). Composing, \(H^k(X_d,\mathcal S)\to H^k(X_c,\mathcal S)\) is surjective for \(k\geq1\).

*Finiteness.* Choose a Leray covering \(\mathcal V=(V_\alpha)\) of \(X_d\) by Stein open subsets of resolving balls, and finitely many of its members \(V_{\alpha_1},\ldots,V_{\alpha_p}\) together with open \(V'_{\alpha_i}\) with compact closure in \(V_{\alpha_i}\), such that the \(V'_{\alpha_i}\) cover the compact set \(\overline{X_c}\). Let \(\mathcal W\) be a Leray covering of \(X_c\) refining the covering \((V'_{\alpha_i}\cap X_c)\). The refinement map \(C^\bullet(\mathcal V,\mathcal S)\to C^\bullet(\mathcal W,\mathcal S)\) factors through restrictions \(\mathcal S(V_{\alpha_{i_0}\ldots\alpha_{i_k}})\to\mathcal S(V'_{\alpha_{i_0}}\cap\cdots\cap V'_{\alpha_{i_k}})\) between finitely many factors, which are compact by [Fréchet spaces of sections, Proposition 5.2(3)](frechet-spaces-of-sections-and-schwartzs-theorem.md#5-the-topology-on-sections-of-coherent-sheaves); so it is compact in each degree, and it induces the surjection \(H^k(X_d,\mathcal S)\to H^k(X_c,\mathcal S)\). By the finiteness criterion [Fréchet spaces of sections, Theorem 2.2](frechet-spaces-of-sections-and-schwartzs-theorem.md#2-schwartz-s-theorem), \(H^k(X_c,\mathcal S)\) is finite-dimensional and Hausdorff for \(k\geq1\). The same argument applied to the exhaustion \(\psi_j\) shows that each \(H^k(G_j,\mathcal S)\), \(k\geq1\), is Hausdorff.

*Injectivity in degree one and density.* All maps in (3.1) are continuous: computed on Leray coverings refining \(\{G_j,U_j\}\), they are induced by continuous maps of Čech complexes. By Lemma 1.1 and Lemma 2.1(3), \(\mathcal S(U_j)\to\mathcal S(G_j\cap U_j)\) has dense image. The composite of this map with \(\partial\) is zero, and \(H^1(G_{j+1},\mathcal S)\) is Hausdorff, so \(\partial=0\). Hence \(H^1(G_{j+1},\mathcal S)\to H^1(G_j,\mathcal S)\) is injective, and

\[
\mathcal S(G_{j+1})\to\mathcal S(G_j)\oplus\mathcal S(U_j)\xrightarrow{\ (g,u)\mapsto u-g\ }\mathcal S(G_j\cap U_j)\to0
\]

is exact; the last map is open, being a continuous surjection of Fréchet spaces. Given \(g\in\mathcal S(G_j)\), choose \(u_\nu\in\mathcal S(U_j)\) with \(u_\nu|_{G_j\cap U_j}\to g|_{G_j\cap U_j}\). By openness there are \((g'_\nu,u'_\nu)\to0\) with \(u'_\nu-g'_\nu=u_\nu-g\) on \(G_j\cap U_j\). Then \(g-g'_\nu\) and \(u_\nu-u'_\nu\) agree on \(G_j\cap U_j\) and glue to \(f_\nu\in\mathcal S(G_{j+1})\) with \(f_\nu|_{G_j}=g-g'_\nu\to g\). So \(\mathcal S(G_{j+1})\to\mathcal S(G_j)\) has dense image, and composing gives (3). Composing the bijections and injections gives (2). \(\square\)

<span id="sublevel-comparison-functional-inputs"></span>

**The functional and cohomological maps in Sections 1–3.** The topology used throughout is the canonical topology of [Fréchet spaces of sections, Propositions 5.1–5.2](frechet-spaces-of-sections-and-schwartzs-theorem.md#5-the-topology-on-sections-of-coherent-sheaves). In Lemma 1.1, the compatible families in the countable product \(\prod_k\mathcal S(W'_k)\) form a closed subspace: they satisfy the closed equations \(t_{k+1}|_{W'_k}=t_k\), since restriction is continuous and the section spaces are Hausdorff. The sheaf axiom identifies this subspace with \(\mathcal S(W')\). Restriction is a continuous linear bijection onto it, hence a topological isomorphism by [the open mapping theorem](frechet-spaces-of-sections-and-schwartzs-theorem.md#1-fréchet-spaces). A basic product neighbourhood uses finitely many coordinates. Taking their largest index and intersecting the inverse images of those finitely many neighbourhoods gives the single neighbourhood \(N\) used in the proof. This justifies the passage from approximation on a sublevel to approximation in the topology on all of \(W'\); no continuous linear section of the quotient presentation is needed.

For the bumps, the uniform perturbation constant is supplied by [the PSH perturbation lemma, Lemma 2.4](plurisubharmonic-functions-and-stein-manifolds.md#2-stein-manifolds). The same finite collection of compactly supported functions works at every step of the subdivision: each intermediate collar lies inside the original compact collar, where their sum is one. If that collar is empty, the two sublevels coincide and no bump is necessary. The barrier \(\phi_j\) in Lemma 2.1 is positive and strictly plurisubharmonic. A bounded sublevel keeps both denominators bounded away from zero; its closure is therefore a compact subset of \(G_{j+1}\cap A_j\). The original exhaustion \(\psi_j\) is bounded on the closure of the resolving ball, so \(\psi_j+\eta\phi_j\) is again an exhaustion. For a nonempty compact \(L\), choose
\[
0<\eta<\frac{b-\max_L\psi_j}{\max_L\phi_j}.
\]
For \(L=\varnothing\), choose any \(b<c\) and any \(\eta>0\); the containment of \(L\) is vacuous, and positivity of \(\phi_j\) gives the same closed-sublevel containment. The Stein and finite-resolution vanishing inputs are [PSH Proposition 2.2](plurisubharmonic-functions-and-stein-manifolds.md#2-stein-manifolds) and [Stein domains, Theorem 4.1](stein-domains-in-complex-space.md#4-coherent-sheaves-on-stein-domains), not the global Theorem B proved later in this lesson.

In the finiteness step, the closure of a finite intersection of the \(V'_i\) is contained in \(\bigcap_i\overline{V'_i}\), a compact subset of the corresponding \(\bigcap_i V_i\). Thus [Proposition 5.2(3)](frechet-spaces-of-sections-and-schwartzs-theorem.md#5-the-topology-on-sections-of-coherent-sheaves) applies to every selected restriction. There are only finitely many such factors in each degree. Their product is compact, and the subsequent map to the countable product of sections on the refining cover is continuous because each component is restriction. Its composite is therefore compact. Repeated assigned indices are interpreted by alternating extension, with zero on repetitions, as proved in [Čech cohomology, Theorem 2.1](course:AG-QC/cech-cohomology#2-a-smaller-complex-without-repeated-indices). [Theorem 3.2](course:AG-QC/cech-cohomology#3-comparing-overlaps-with-derived-cohomology) identifies the resulting map with restriction in sheaf cohomology. This is the compact map to which [the Fréchet finiteness criterion, Theorem 2.2](frechet-spaces-of-sections-and-schwartzs-theorem.md#2-schwartz-s-theorem), is applied. The same surjectivity-and-compactness proof applies to each perturbed exhaustion \(\psi_j\), before using degree-one injectivity or density, so it supplies Hausdorffness without circular use of Theorem B.

Here is also an explicit continuity check for the connecting map. Write \(G=G_j\), \(U=U_j\), and choose a countable Leray refinement \(\mathcal W=(W_\alpha)\) of \(\{G,U\}\), labelling each member by \(\epsilon_\alpha=0\) or \(1\) according as its assigned containing open is \(G\) or \(U\). For \(s\in\mathcal S(G\cap U)\), put
\[
c(s)_{\alpha\beta}
=(\epsilon_\beta-\epsilon_\alpha)s|_{W_\alpha\cap W_\beta}
\quad(\alpha<\beta),
\]
where the expression is defined to be zero for equal labels. For unequal labels the overlap lies in \(G\cap U\), so every restriction is defined. On a triple overlap the three coefficients cancel, hence \(\delta c(s)=0\). Each component is continuous, and the product topology makes \(s\mapsto c(s)\) a continuous map to the closed cocycle space.

For the difference convention \(u-g\) in this lesson, the connecting class is \(-[c(s)]\). To check the sign, in an injective resolution choose degree-zero lifts \(a\) on \(G\) and \(b\) on \(U\) with \(b-a=s\); these exist by flasqueness. The sections \(da,db\) glue to the connecting cocycle \(h\), by the construction in [Cohomology of sheaves, Theorem 5.1](course:AG-QC/cohomology-of-sheaves-on-ringed-spaces#5-two-open-sets-and-the-connecting-map). The refinement cochain \(v_\alpha\) given by the chosen lift has \(\delta v=c(s)\) and \(dv=h\). With total differential \(D=\delta+(-1)^p d\), the identity \(Dv=c(s)+h\) gives precisely that minus sign. Passing to the quotient cocycle topology is continuous. Thus the connecting map used in Section 3 is continuous even before its target is known to be Hausdorff. Once Hausdorffness has been established, vanishing on the dense image indeed forces it to be zero.

Finally, openness of the surjective difference map provides the small corrections \((g'_\nu,u'_\nu)\) used for density, without asserting a continuous choice of lifts. Choose a decreasing zero-neighbourhood basis \(N_m\) in its Fréchet domain. Since the errors tend to zero and every image \(Q(N_m)\) is a zero-neighbourhood, sufficiently late errors lie in each \(Q(N_m)\). Choose their lifts in \(N_{m(\nu)}\) with \(m(\nu)\to\infty\). Those lifts tend to zero, exactly as the gluing argument requires.

*Exact functional/cohomological input comparison: GPT-6 Astra (OpenAI), Ultra, 13 October 2026; author self-check, CC0. The original bump and approximation arguments, statements and source attribution are retained.*

## 4. Theorem B

**Theorem 4.1 (Cartan's Theorem B).** If \(X\) is a complex manifold with a smooth strictly plurisubharmonic exhaustion, then \(H^k(X,\mathcal S)=0\) for every coherent analytic sheaf \(\mathcal S\) on \(X\) and every \(k\geq1\).

**Proof.** The exhaustion \(\psi\) is bounded below; choose \(c<\inf\psi\), so that \(X_c=\emptyset\). By Proposition 3.1(2), \(H^k(X_d,\mathcal S)\cong H^k(\emptyset,\mathcal S)=0\) for every \(d\) and \(k\geq1\).

Let \(\mathcal W=(W_\alpha)\) be a countable basis of the topology of \(X\) consisting of relatively compact Stein open subsets of resolving balls, and let \(\mathcal W_d\) be the family of those \(W_\alpha\) contained in \(X_d\). Then \(\mathcal W_d\) is a Leray covering of \(X_d\) and \(\mathcal W\) a Leray covering of \(X\) (Section 5 of the lesson on Fréchet spaces of sections). Put \(E^\bullet_\nu=C^\bullet(\mathcal W_{c+\nu},\mathcal S)\), \(\nu\in\mathbf N\). The restriction maps \(E^\bullet_{\nu+1}\to E^\bullet_\nu\) forget factors, so they are surjective; every finite family of members of \(\mathcal W\) has relatively compact union, hence all its members lie in some \(X_{c+\nu}\), so \(\varprojlim_\nu E^\bullet_\nu=C^\bullet(\mathcal W,\mathcal S)\), whose cohomology is \(H^\bullet(X,\mathcal S)\). For \(k\geq1\), \(H^k(E^\bullet_\nu)=H^k(X_{c+\nu},\mathcal S)=0\), so the maps \(H^k(E_{\nu+1})\to H^k(E_\nu)\) are injective; and the maps \(H^{k-1}(E_{\nu+1})\to H^{k-1}(E_\nu)\) have dense image, trivially for \(k\geq2\) and by Proposition 3.1(3) for \(k=1\). By the abstract Mittag-Leffler theorem [Fréchet spaces of sections, Proposition 3.1(3)](frechet-spaces-of-sections-and-schwartzs-theorem.md#3-an-abstract-mittag-leffler-theorem), \(H^k(X,\mathcal S)\to H^k(E^\bullet_0)=0\) is injective. \(\square\)

*Reference:* the bump argument is due to [Andreotti–Grauert 1962]; the arrangement follows [Demailly]. This theorem is the analytic vanishing theorem used in [Serre's comparison theorems and Chow's theorem, Section 1](course:AG-QC/serres-comparison-theorems-and-chows-theorem), in exactly that generality.

**Corollary 4.2.** \(H^k(V,\mathcal S)=0\) for \(k\geq1\) whenever \(V\) is a Stein open subset of a complex manifold and \(\mathcal S\) is coherent on \(V\). This applies to polydiscs, balls and their finite intersections, to the sets \((\mathbf C^*)^a\times\mathbf C^{n-a}\), to the sublevel sets of strictly plurisubharmonic exhaustions, and to closed submanifolds of Stein manifolds and their products.

**Proof.** Theorem 4.1 and the examples of [Plurisubharmonic functions and Stein manifolds, Section 2](plurisubharmonic-functions-and-stein-manifolds.md#2-stein-manifolds). \(\square\)

<span id="theorem-b-exhaustion-limit-inputs"></span>

**The exhaustion-limit comparison in Theorem 4.1.** Give the countable basis \(\mathcal W\) one fixed ordering and put
\[
I_\nu=\{\alpha:W_\alpha\subset X_{c+\nu}\},\qquad
E^q_\nu=\prod_{\substack{\alpha_0<\cdots<\alpha_q\\\alpha_i\in I_\nu}}
\mathcal S(W_{\alpha_0}\cap\cdots\cap W_{\alpha_q}),
\qquad E^q_\nu=0\ (q<0).
\]
These are the ordered Čech complexes. [Fréchet spaces of sections, Section 5](frechet-spaces-of-sections-and-schwartzs-theorem.md#5-the-topology-on-sections-of-coherent-sheaves) gives Fréchet section spaces and continuous restrictions; [the countable-product proof in Section 1](frechet-spaces-of-sections-and-schwartzs-theorem.md#frechet-countable-product-input) supplies the product topology and completeness. The differential is continuous because every output coordinate is a finite alternating sum of restrictions. Transition maps \(r_\nu:E^q_{\nu+1}\to E^q_\nu\) are product projections, hence continuous and onto: extend a cochain by zero on the new coordinates. That degreewise extension need not commute with the differential; no chain-map splitting is asserted.

The sets indexed by \(I_\nu\) cover \(X_{c+\nu}\), by the basis property. Finite intersections are Stein by [PSH Proposition 2.2(4)](plurisubharmonic-functions-and-stein-manifolds.md#2-stein-manifolds) and lie in a resolving chart of any constituent member. The finite resolution from [Oka, Corollary 3.3 and Theorem 3.4](coherent-sheaves-and-okas-theorem.md#3-the-category-of-coherent-analytic-sheaves), restricted to that intersection, and [Stein domains, Theorem 4.1](stein-domains-in-complex-space.md#4-coherent-sheaves-on-stein-domains) prove acyclicity. This does not use the global theorem under proof. The [acyclic-cover comparison](course:AG-QC/cech-cohomology#3-comparing-overlaps-with-derived-cohomology) therefore applies at every level and on \(X\), and its naturality matches transition maps with restriction.

For each fixed finite tuple, the union of the closures \(\bigcup_i\overline{W_{\alpha_i}}\) is compact. Continuity of \(\psi\) bounds it above there, so some common integer \(N\) puts every member of the tuple in \(I_N\). It is this finite-family property, not just boundedness of the intersection, that puts every Čech coordinate at a finite stage. Restriction consequently gives an isomorphism of complexes
\[
C^\bullet(\mathcal W,\mathcal S)
\xrightarrow{\ \sim\ }\varprojlim_\nu E^\bullet_\nu.
\]
It is also a topological isomorphism in every degree: its forward maps are coordinate projections, and each coordinate of its inverse is a projection from any sufficiently large stage, independent of that choice by compatibility. Both product topologies are initial for these coordinates. Differentials agree coordinatewise.

By Proposition 3.1, all positive-degree cohomology at finite levels vanishes. For degree zero, gluing identifies \(H^0(E^\bullet_\nu)=Z^0(E^\bullet_\nu)\) topologically with \(\mathcal S(X_{c+\nu})\): the compatible-family image is closed in the product, and the continuous bijection from the Fréchet section space has continuous inverse by the open mapping theorem. Thus the density of restrictions in Proposition 3.1(3) is precisely the degree-zero density required by [the abstract Mittag-Leffler theorem, Proposition 3.1(3)](frechet-spaces-of-sections-and-schwartzs-theorem.md#3-an-abstract-mittag-leffler-theorem). For a positive degree \(k\), injectivity in degree \(k\) and density in degree \(k-1\) follow: for \(k\geq2\) both relevant groups vanish; for \(k=1\) use the just-identified section topology.

The provider proof corrects primitives by cocycles and coboundaries until their discrepancies are summable in compatible complete metrics. Continuity of the differential then preserves their prescribed coboundary on taking limits. It assumes density in the quotient topology, not Hausdorffness of every intermediate cohomology group. In degree \(k=1\), the degree-minus-one correction space is zero, consistent with the nonnegative complexes above. Since \(c<\inf_X\psi\), the zeroth level is the zero complex, and the resulting injection into its positive cohomology proves the asserted vanishing on \(X\). If \(X\) is empty, the conclusion holds directly with all section and cochain spaces zero.

*Exhaustion-limit reconciliation: GPT-6 Astra (OpenAI), Ultra, 13 October 2026; author self-check, CC0. This verifies the exact application of the retained programme proofs; it is not independent whole-course certification.*

## 5. Theorem A

**Theorem 5.1 (Cartan's Theorem A).** Let \(X\) be a Stein manifold and \(\mathcal S\) a coherent sheaf on \(X\). For every \(x\in X\), the germs at \(x\) of the global sections \(\mathcal S(X)\) generate the stalk \(\mathcal S_x\) as an \(\mathcal O_{X,x}\)-module.

**Proof.** Let \(\mathcal I_x\subset\mathcal O_X\) be the ideal sheaf of the point \(x\), coherent by Cartan's theorem (near \(x\) it is generated by the coordinate functions). The image \(\mathcal I_x\mathcal S\) of \(\mathcal I_x\otimes\mathcal S\to\mathcal S\) is coherent, and \(\mathcal S/\mathcal I_x\mathcal S\) is the skyscraper sheaf at \(x\) with stalk \(\mathcal S_x/\mathfrak m_x\mathcal S_x\). By Theorem 4.1, \(H^1(X,\mathcal I_x\mathcal S)=0\), so \(\mathcal S(X)\to\mathcal S_x/\mathfrak m_x\mathcal S_x\) is surjective. By Nakayama's lemma, elements of \(\mathcal S_x\) whose classes span \(\mathcal S_x/\mathfrak m_x\mathcal S_x\) generate \(\mathcal S_x\). \(\square\)

**Corollary 5.2.** On a Stein manifold \(X\): global holomorphic functions separate points, and at every point some \(n\) global holomorphic functions form local coordinates; for every closed analytic subset \(A\subset X\), every holomorphic function on \(A\) (a section of \(\mathcal O_A\)) extends to a holomorphic function on \(X\); and every coherent sheaf is, on every relatively compact open subset, a quotient of a free sheaf of finite rank.

**Proof.** For \(x\neq y\), apply Theorem 4.1 to the coherent ideal sheaf \(\mathcal I_{\{x,y\}}\): \(\mathcal O(X)\to\mathcal O_X/\mathcal I_{\{x,y\}}\cong\mathbf C_x\oplus\mathbf C_y\) is surjective, so some \(f\) has \(f(x)=0\neq f(y)\). Applying it to \(\mathcal I_x^2\), every element of \(\mathfrak m_x/\mathfrak m_x^2\), in particular a basis, is the differential at \(x\) of a global function. Extension from \(A\): \(H^1(X,\mathcal I_A)=0\) makes \(\mathcal O(X)\to(\mathcal O_X/\mathcal I_A)(X)=\mathcal O_A(A)\) surjective. For the last statement, by Theorem 5.1 and Lemma 1.3 of the lesson on Oka's theorem, each point has a neighbourhood on which finitely many global sections generate \(\mathcal S\); finitely many such neighbourhoods cover a compact closure. \(\square\)

<span id="theorem-a-point-and-jet-inputs"></span>

**The point, jet and generation inputs in Section 5.** The algebra used here is [Nakayama's generator criterion, Theorem 4.2](course:AG-CA/AG-CA-02#4-finite-generators-near-a-point), proved by the determinant trick in the preceding Lemma 4.1. Apply it to the finite module \(\mathcal S_x\) over the local ring \(\mathcal O_{X,x}\), with maximal ideal \(\mathfrak m_x\). Neither a freeness assumption on \(\mathcal S_x\) nor a choice of global generators in advance is required.

The coherence operations have an earlier proof in [Analytification, Proposition 1.1](course:AG-QC/complex-analytic-spaces-and-analytification#1-analytic-equations-and-their-sheaves): use its first two paragraphs, the abstract argument over a coherent ring sheaf. They prove coherence of finite-type submodules, kernels, images, cokernels, extensions and tensor products by local generators and their relations. This argument precedes its application to analytic quotient models; it uses neither Theorem A nor Theorem B. Together with [Oka's theorem](coherent-sheaves-and-okas-theorem.md#2-oka-s-coherence-theorem), it justifies the products and images used here. The point ideal is locally of finite type: at \(x\) it is generated by centered coordinates, as in [the local-ring description, Proposition 1.1](the-local-ring-of-holomorphic-germs.md#1-the-ring-of-germs), and off \(x\) it is the whole structure sheaf. The same argument covers a finite set of distinct points and powers of its ideal. For general analytic subsets retain the stronger Cartan ideal-coherence theorem already cited in the original proof.

For clarity, the quotient \(\mathcal Q=\mathcal S/\mathcal I_x\mathcal S\) has stalk zero away from \(x\), and stalk \(\mathcal S_x/\mathfrak m_x\mathcal S_x\) at \(x\). Sending a section on an open containing \(x\) to its germ, and a section on an open avoiding \(x\) to zero, gives a morphism to that skyscraper sheaf. It is an isomorphism on every stalk, hence an isomorphism of sheaves. Thus its global sections are exactly the indicated finite-dimensional residue module. The long exact sequence from [Cohomology of sheaves, Section 1](course:AG-QC/cohomology-of-sheaves-on-ringed-spaces#1-the-obstruction-that-cohomology-records) and Theorem 4.1 give the surjection from global sections. Lifting a finite residue basis and applying Nakayama proves Theorem 5.1.

For the local-coordinate assertion, the quotient by \(\mathcal I_x^2\) remembers the constant and linear Taylor coefficients. Prescribe zero constant term and a basis of \(\mathfrak m_x/\mathfrak m_x^2\); vanishing gives global lifts \(f_1,\ldots,f_n\). Their Jacobian at \(x\) is invertible, so [the holomorphic inverse-function proof, Lemma 2.2](course:AG-QC/complex-analytic-spaces-and-analytification#2-local-analytic-algebra) makes them local coordinates. In dimension zero the empty coordinate family describes the open singleton. For the final finite-generation assertion, [Oka, Lemma 1.3](coherent-sheaves-and-okas-theorem.md#1-coherent-modules-on-a-ringed-space) spreads each finite generating family to a neighbourhood. A finite subcover of the compact closure supplies one finite list of global sections generating on the relatively compact open set. On the empty open set the empty list suffices.

*Exact Section 5 input comparison: GPT-6 Astra (OpenAI), Ultra, 13 October 2026; author self-check, CC0. The original statements, proofs, source credit and alternative routes are retained.*

## 6. Exercises

**Exercise 6.1.** Let \(X=(\mathbf C^*)^2\) and \(U_\pm=X\cap\{z_1\neq\pm1\}\). Explain why the Čech group \(\check H^1(\{U_+,U_-\},\mathcal O)\) vanishes, and why Exercise 4.3 of the Dolbeault lesson does not contradict Theorem 4.1.

*Solution.* \(X\) is Stein (Corollary 4.2), so \(H^1(X,\mathcal O)=0\), and for any covering the Čech \(H^1\) injects into \(H^1(X,\mathcal O)\), so it vanishes as well. In the Dolbeault lesson the space was \(\mathbf C^2\setminus\{0\}\), which is not Stein (Exercise 3.2 of the lesson on Stein manifolds); its nonzero \(H^1\) is the obstruction detected by the Laurent coefficient of \(z_1^{-1}z_2^{-1}\).

**Exercise 6.2.** Let \(A=\{z_1z_2=1\}\subset\mathbf C^2\). Use Corollary 5.2 to show that every holomorphic function on \(A\) is the restriction of an entire function on \(\mathbf C^2\), and find such an extension of \(f(z_1,z_2)=z_1^{-1}\) on \(A\).

*Solution.* \(A\) is a closed submanifold of the Stein manifold \(\mathbf C^2\), so Corollary 5.2 applies. On \(A\), \(z_1^{-1}=z_2\), so the polynomial \(z_2\) is an extension.

**Exercise 6.3.** Show in two ways that a compact connected complex manifold of positive dimension is not Stein.

*Solution.* On a compact connected manifold, global holomorphic functions are constant by the maximum principle, so they separate no points, which contradicts Corollary 5.2 if the manifold were Stein. (Directly: a strictly plurisubharmonic function attains a maximum on a compact manifold, and at a maximum its complex Hessian is negative semidefinite in every direction, a contradiction.)

## References

- [Andreotti–Grauert 1962] A. Andreotti and H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, *Bulletin de la Société Mathématique de France* 90 (1962), 193–259. <https://www.numdam.org/item/BSMF_1962__90__193_0/>
- [Demailly] J.-P. Demailly, *Complex Analytic and Differential Geometry*, version of 21 June 2012, freely available from the author with permission to copy, modify and redistribute with credit. <https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf>
- [Stacks] The Stacks project, cited by tag; each tag links to the same result in the AI Integrated Stacks Project. <https://stacks.math.columbia.edu/>