Analytic prerequisites for algebraic–analytic comparison

Locally finite smooth partitions and cutoffs

smooth-partition-and-cutoffs

Our manifolds are finite dimensional, Hausdorff and second countable. A smooth manifold is a space with charts whose transition functions are smooth. Its tangent vectors are coordinate velocities, identified by the derivatives of chart transitions. A Lie group is a smooth manifold whose multiplication and inversion are smooth. These are definitions; no subgroup or quotient theorem is included in them.

The support of a function is the closure of the set on which it is nonzero. A compact exhaustion is a sequence of compact sets \(A_n\) with \(A_n\subset\operatorname{int}A_{n+1}\) and \(\bigcup_{n\geq0}A_n=M\). We write \(U(a,r)\) for an open Euclidean ball and \(B(a,r)\) for its closed ball.

Lemma 0.5 (a flat smooth cutoff factor). The function \(\eta(t)=e^{-1/t^2}\) for \(t>0\), extended by zero for \(t\leq0\), is smooth and all its derivatives vanish at zero.

Lemma 3.B (subordinate atlas). Suppose the topology of the manifold \(M\) has a countable basis, and let \((W_i)_{i\in I}\) be an open cover of \(M\). There is a countable family of compatible charts \(\alpha_j:U_j\to V_j\subset\mathbb R^d\), \(j\in J\), with radii \(0<\delta_j<\epsilon_j\) and \(B(0,\epsilon_j)\subset V_j\), such that

1. every \(U_j\) lies in some member \(W_{i(j)}\) of the cover;
2. the sets \(\alpha_j^{-1}(U(0,\delta_j))\), \(j\in J\), cover \(M\);
3. every point of \(M\) has a neighbourhood meeting only finitely many \(U_j\); in particular, every point lies in only finitely many \(U_j\).

Definition 3.C (partition of unity). On a topological space \(X\), a family \((h_j)_{j\in J}\) of functions \(h_j:X\to[0,1]\) is a partition of unity if every point of \(X\) has a neighbourhood on which all but finitely many \(h_j\) vanish identically, and \(\sum_{j\in J}h_j(x)=1\) for every \(x\in X\). The first condition makes each of these sums finite. The partition is called continuous, differentiable or smooth when every \(h_j\) is.

Definition 3.D (subordinate partition). For an open cover \((W_i)_{i\in I}\) of a topological space \(X\), a partition of unity \((h_j)_{j\in J}\) on \(X\) is subordinate to the cover if for every \(j\in J\) the support of \(h_j\) lies in some member \(W_{i(j)}\).

Theorem 3.E (partition of unity). Suppose the topology of the differentiable manifold \(M\) has a countable basis. Every open cover of \(M\) has a subordinate partition of unity consisting of continuously differentiable functions, whose supports are compact and lie in coordinate domains contained in members of the cover. If \(M\) is smooth, the partition can be chosen smooth.

Theorem 3.1 (smooth partition and cutoff). Every open cover of a smooth manifold admits a locally finite smooth partition of unity subordinate to it. Its supports can be chosen compact and contained in coordinate neighbourhoods lying in members of the cover. If \(K\) is closed and \(U\) is an open neighbourhood of \(K\), there is a smooth function \(\chi:M\to[0,1]\) equal to one on a neighbourhood of \(K\), with support contained in \(U\).

Statement text: GPT-6 Astra (OpenAI), Local tools for bundles and transport (original text CC0). The Section 3 construction follows Holger Brenner and Wikiversity contributors, Lecture 22 of Differentialgeometrie (Osnabrück 2023), revision 1052940.

Statements and proofs in Local tools for bundles and transport

Lemma 0.5: the flat smooth cutoff factor and its proof

Connections

• The Dolbeault complex

  Result used: Theorem 3.1

  Connection: Required input

  Hypotheses for this use: Hausdorff second-countable smooth manifold and any open cover of any open subset.

  Locally finite smooth subordinate partition with supports closed in the working open set; weighted local lifts extend by zero.

• Stein domains in complex space

  Result used: Lemma 3.A and Theorem 3.1

  Connection: Required input

  Hypotheses for this use: A compact closed sublevel inside a larger relatively compact sublevel, and a compact exhaustion of the approximation domain.

  Compactly supported smooth cutoff equal to one near the inner closure; cofinal nested compact sets for one diagonal sequence.

• Hörmander's L² estimates on pseudoconvex domains

  Location: Exact cutoff and partition inputs (Section 3)

  Result used: Lemmas 3.B, 0.5; Theorems 3.E and 3.1

  Connection: Required input

  Hypotheses for this use: Smooth second-countable manifold and an open cover; closed K inside open U for the cutoff assertion.

  Locally finite smooth compactly supported partitions; cutoff one near K. With U=int K_(n+1), compact support and derivative support away from K_n; local finiteness supplies the smooth majorant.

All cohomology connections

Compact exhaustion of a manifold

compact-exhaustion

Our manifolds are finite dimensional, Hausdorff and second countable. A smooth manifold is a space with charts whose transition functions are smooth. Its tangent vectors are coordinate velocities, identified by the derivatives of chart transitions. A Lie group is a smooth manifold whose multiplication and inversion are smooth. These are definitions; no subgroup or quotient theorem is included in them.

The support of a function is the closure of the set on which it is nonzero. A compact exhaustion is a sequence of compact sets \(A_n\) with \(A_n\subset\operatorname{int}A_{n+1}\) and \(\bigcup_{n\geq0}A_n=M\). We write \(U(a,r)\) for an open Euclidean ball and \(B(a,r)\) for its closed ball.

Lemma 3.A (compact exhaustion). Every manifold \(M\) whose topology has a countable basis admits a compact exhaustion.

Statement text: GPT-6 Astra (OpenAI), Local tools for bundles and transport (original text CC0). The Section 3 construction follows Holger Brenner and Wikiversity contributors, Lecture 22 of Differentialgeometrie (Osnabrück 2023), revision 1052940.

Statements and proofs in Local tools for bundles and transport

Connections

• Stein domains in complex space

  Result used: Lemma 3.A and Theorem 3.1

  Connection: Required input

  Hypotheses for this use: A compact closed sublevel inside a larger relatively compact sublevel, and a compact exhaustion of the approximation domain.

  Compactly supported smooth cutoff equal to one near the inner closure; cofinal nested compact sets for one diagonal sequence.

• Hörmander's L² estimates on pseudoconvex domains

  Location: Exact cutoff and partition inputs (Section 3)

  Result used: Lemma 3.A

  Connection: Required input

  Hypotheses for this use: A finite-dimensional Hausdorff second-countable manifold; here an open subset of real Euclidean space.

  Compact sets K_n with K_n inside the interior of K_(n+1) and union the whole domain.

All cohomology connections

Two open sets and the connecting map

AG-QC-01.theorem-5-1

Theorem 5.1 (Mayer–Vietoris). If \(X=U\cup V\) is an open cover, there is a natural exact sequence \[ \begin{aligned} 0&\to H^0(X,\mathcal F) \to H^0(U,\mathcal F)\oplus H^0(V,\mathcal F) \to H^0(U\cap V,\mathcal F)\\ &\to H^1(X,\mathcal F) \to H^1(U,\mathcal F)\oplus H^1(V,\mathcal F) \to H^1(U\cap V,\mathcal F)\to\cdots. \end{aligned} \] The map from the two opens to their intersection is restriction from \(U\) minus restriction from \(V\).

Full statement and proof in AG-QC-01

Used by

• Theorems A and B on Stein manifolds

  Hypotheses for this use: An arbitrary two-open cover and a module sheaf.

  Exact sequence and connecting class from a flasque resolution; the u-g convention here gives minus the displayed Cech class.

All cohomology connections

A smaller complex without repeated indices

AG-QC-02.theorem-2-1

Theorem 2.1. The ordered and alternating complexes are isomorphic, and their inclusion into the full complex is a homotopy equivalence. No division by a factorial is involved.

Full statement and proof in AG-QC-02

Used by

• Theorems A and B on Stein manifolds

  Hypotheses for this use: Any totally ordered open-cover index set; arbitrary coefficient groups.

  Ordered/full homotopy comparison, alternating extension and zero on repeated indices, without factorial division.

• Finiteness on compact complex spaces

  Hypotheses for this use: Finite totally ordered cover with m members.

  Ordered complex computes the same Cech cohomology and vanishes in degrees at least m.

• Fréchet spaces of sections and Schwartz's theorem

  Location: Section 5: The comparison used here; Cohomology on manifolds.

  Hypotheses for this use: A totally ordered open cover and a sheaf of modules; this application uses a countable Leray cover and a coherent analytic sheaf.

  The ordered and full Cech complexes are homotopy equivalent by signed extension, with zero on repeated indices and without division by factorials. This comparison does not itself establish a topology on cohomology or Hausdorffness.

All cohomology connections

Comparing overlaps with derived cohomology

AG-QC-02.theorem-3-2

Theorem 3.2 (acyclic covers). If \(H^q(U_{i_0\ldots i_p},\mathcal F)=0\) for all \(q>0\) and every nonempty finite intersection, then the natural comparison is an isomorphism \[ \check H^p(\mathcal U,\mathcal F)\cong H^p(X,\mathcal F) \] for every \(p\).

Full statement and proof in AG-QC-02

Used by

• Theorems A and B on Stein manifolds

  Hypotheses for this use: Every nonempty finite intersection acyclic for the given module sheaf.

  Cech comparison on finite or countable covers; refinement compatible with the global-section augmentation.

• Finiteness on compact complex spaces

  Hypotheses for this use: All nonempty finite intersections acyclic for the module sheaf.

  Nested-cover map is the identity under the natural sheaf-cohomology identifications.

• Fréchet spaces of sections and Schwartz's theorem

  Location: Section 5: The comparison used here; Cohomology on manifolds.

  Hypotheses for this use: A coherent analytic sheaf on a complex manifold and a countable cover whose every nonempty finite intersection is acyclic for that sheaf.

  The Cech complex computes sheaf cohomology, and refinement corresponds to the identity on sheaf cohomology. Continuity of restrictions and the separate open-mapping argument supply independence of the quotient topology; these cohomology groups need not be Hausdorff.

All cohomology connections

The obstruction that cohomology records

AG-QC-01.long-exact-cohomology-sequence

Statement and derived-functor route in AG-QC-01

Used by

• Theorems A and B on Stein manifolds

  Hypotheses for this use: Short exact sequence of module sheaves.

  Vanishing H1 of the kernel gives surjectivity on global sections.

• Stein domains in complex space

  Hypotheses for this use: Short exact module-sheaf sequences restricted to the same open domain.

  Dimension shifts and initial connecting map give positive-degree vanishing and surjectivity.

All cohomology connections

Torsors are first cohomology classes

AG-QC-02.theorem-5-1

Theorem 5.1. Isomorphism classes of \(\mathcal F\)-torsors are naturally classified by \(H^1(X,\mathcal F)\). The comparison \(\check H^1(\mathcal U,\mathcal F)\to H^1(X,\mathcal F)\) is injective, and its image consists exactly of classes of torsors trivial on every \(U_i\).

Full statement and proof in AG-QC-02

Used by

• Theorems A and B on Stein manifolds

  Location: Exercise 6.1

  Hypotheses for this use: A sheaf of abelian groups and any open cover; no acyclicity assumption on the cover. Here the sheaf is the underlying abelian sheaf of holomorphic functions on (C*)², and H¹ vanishes by the preceding Stein theorem.

  The fixed-cover Čech H¹ injects into sheaf H¹, so the two-open Čech group in Exercise 6.1 is zero. This degree-one injection does not invoke the acyclic-cover theorem used elsewhere in the lesson.

• The Dolbeault complex

  Location: Exercise 4.3

  Hypotheses for this use: A sheaf of abelian groups and any open cover; no acyclicity assumption on the cover. Here X = C² minus the origin, the sheaf is the underlying abelian sheaf of holomorphic functions, and the two opens are z₁ ≠ 0 and z₂ ≠ 0.

  The Laurent coefficient argument in Exercise 4.3 shows that 1/(z₁z₂) represents a nonzero fixed-cover Čech H¹ class. The comparison injection makes its sheaf H¹ class nonzero. The theorem supplies a programme proof of the comparison cited in the exercise.

All cohomology connections

Coefficients and higher direct images

AG-QC-01.theorem-3-1

Theorem 3.1 (independence of the coefficient category). The underlying groups of \(H^q(U,\mathcal F)\), computed in modules over \(\mathcal O_X\), are naturally the cohomology groups of the underlying abelian sheaf.

Full statement and proof in AG-QC-01

Used by

• The Dolbeault complex

  Hypotheses for this use: An arbitrary sheaf of modules on a ringed space and any open subset.

  Module-category cohomology has the same underlying groups as abelian-sheaf cohomology; forgetting injectivity is not assumed.

• Cartan's coherence theorem and complex spaces

  Hypotheses for this use: An arbitrary ringed space and a sheaf of modules.

  Abelian-sheaf comparison gives precisely the module-category cohomology in the original theorem.

All cohomology connections

Extending sections removes the obstruction

AG-QC-01.theorem-2-3+corollary-2-4

Theorem 2.3 (flasque acyclicity). For a flasque sheaf of modules \(\mathcal F\), \[ H^q(U,\mathcal F)=0\qquad(q>0) \] on every open \(U\subset X\).

Corollary 2.4 (flasque resolutions). An exact resolution \(0\to\mathcal F\to\mathcal L^0\to\mathcal L^1\to\cdots\) with all \(\mathcal L^r\) flasque computes cohomology on every open by taking sections.

Full statements and proofs in AG-QC-01

Used by

• Cartan's coherence theorem and complex spaces

  Hypotheses for this use: Exact resolution by flasque sheaves; all open subsets.

  Exact closed pushforward produces a flasque resolution with the identical global section complex.

All cohomology connections

Local differential rules for distributions

distribution-local-calculus

Support-relative pairing

Theorem 3.1 (support-relative test extension). Let \(u\in\mathcal D'(X)\) and let \(F\subset X\) be relatively closed, with \(\operatorname{supp}u\subset F\). On the vector space

there is a unique linear extension \(\widetilde u\) of \(u\) that vanishes whenever \(\operatorname{supp}f\cap F=\varnothing\). If \(\chi\in C_c^\infty(X)\) equals one near \(F\cap\operatorname{supp}f\), its formula is

When \(u\) has order at most \(k\), the same construction extends its canonical \(C_c^k\) pairing to the space defined by (3.1) with \(C^k(X)\) in place of \(C^\infty(X)\).

Local operations

Define the singular support \(\operatorname{sing\,supp}u\) as the complement of the union of open sets where \(u\) is represented by a smooth function.

For \(a\in C^\infty(X)\), define

Proposition 4.1 (local differential rules). The operations (4.2) give distributions, commute with restriction, and are continuous in both the weak and strong dual topologies for fixed \(a\). They satisfy

Partial derivatives commute. Both formulas in (4.2) also hold for every \(f\in\mathcal E_{\operatorname{supp}u}(X)\), using the extended pairings on their left sides.

If \(a\in C^k(X)\) and \(u\) has order at most \(k\), the second formula still defines \(au\), by the \(C_c^k\) pairing. It has order at most \(k\), with support contained in the same support intersection. The singular-support assertions require a smooth multiplier.

Statement and argument in Local data and compatible products

Theorem 3.1: support-relative pairing and proof

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Measure and distribution inputs, at the weights used here (Sections 2–3)

  Result used: Section 4, Proposition 4.1

  Connection: Required input

  Hypotheses for this use: Distributions on an open Euclidean set and smooth multipliers; ordinary distribution derivatives.

  Commuting partial derivatives, smooth cutoff Leibniz rule and local restriction; commuting complex derivatives imply dbar squared equals zero by wedge antisymmetry.

All cohomology connections

Zero distributional derivative on an interval

distributional-constants-on-an-interval

Theorem 1.1 (distributional constants). If \(I\subset\mathbb R\) is a nonempty open interval and \(u\in\mathcal D'(I)\) satisfies \(u'=0\), there is a unique scalar \(c\) such that

The interval may be unbounded. On a disconnected open set the constant can differ between its interval components.

Statements and proofs in Weak equations and classical functions

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Endpoint traces and the full adjoint argument for Exercise 6.1

  Result used: Theorem 1.1

  Connection: Required input

  Hypotheses for this use: A scalar distribution on a nonempty open interval whose distributional derivative vanishes.

  The distribution is constant. Applied to u minus the integral of its L2 weak derivative; the separate continuous-right-side Corollary 1.2 is not used.

All cohomology connections

The open mapping theorem for Fréchet spaces

complex-analytic-spaces-and-coherent-sheaves.frechet-spaces-of-sections-and-schwartzs-theorem.theorem-1-1

Theorem 1.1 (open mapping theorem). A continuous linear surjection \(A:E\to F\) between Fréchet spaces is open. A continuous linear bijection between Fréchet spaces is a topological isomorphism.

Statement and proof in Fréchet spaces of sections and Schwartz's theorem

Used by

• Theorems A and B on Stein manifolds

  Result used: Theorem 1.1

  Hypotheses for this use: Continuous linear surjection between Frechet spaces.

  Canonical section identifications are homeomorphisms and errors tending to zero have lifts tending to zero.

All cohomology connections

Compact cohomology and quotient topologies

complex-analytic-spaces-and-coherent-sheaves.frechet-spaces-of-sections-and-schwartzs-theorem.theorem-2-2+remark-2-3

Theorem 2.2 (finiteness criterion). Let \(\rho:(E^\bullet,d)\to(F^\bullet,\delta)\) be a morphism of complexes of Fréchet spaces with continuous differentials. If \(\rho^q\) is compact and \(H^q(\rho):H^q(E^\bullet)\to H^q(F^\bullet)\) is surjective, then \(H^q(F^\bullet)\) is finite-dimensional and \(\delta(F^{q-1})\) is closed in \(F^q\).

Proof. Consider \(g,h:Z^q(E^\bullet)\oplus F^{q-1}\to Z^q(F^\bullet)\), \(g(x,y)=\rho^q(x)+\delta y\), \(h(x,y)=-\rho^q(x)\), between Fréchet spaces (cocycle spaces are closed). Surjectivity of \(H^q(\rho)\) means that \(g\) is surjective, and \(h\) is compact. By Theorem 2.1, \(g+h=(x,y)\mapsto\delta y\) is a quasi-epimorphism, so \(\delta(F^{q-1})\) is closed and of finite codimension in \(Z^q(F^\bullet)\). \(\square\)

Remark 2.3. In the situation of Theorem 2.2 without compactness, if \(H^q(\rho)\) is surjective then the map \(g\) is open by Theorem 1.1, so \(H^q(\rho)\) is open for the quotient topologies; if \(H^q(\rho)\) is bijective it is a homeomorphism.

Full statements and arguments in Fréchet spaces of sections and Schwartz's theorem

Used by

• Theorems A and B on Stein manifolds

  Result used: Theorem 2.2 and Remark 2.3

  Hypotheses for this use: Continuous morphism of Frechet complexes, compact map in degree q and surjection in cohomology.

  Finite-dimensional Hausdorff target cohomology, with no prior closed-range assumption.

• Finiteness on compact complex spaces

  Result used: Theorem 2.2

  Hypotheses for this use: Morphism of Frechet complexes compact in degree q and onto in cohomology.

  Finite-dimensional Hausdorff target cohomology without a prior closed-range assumption.

All cohomology connections

The topology and compact restriction of coherent sections

complex-analytic-spaces-and-coherent-sheaves.frechet-spaces-of-sections-and-schwartzs-theorem.proposition-5-1+proposition-5-2

Let \(X\) be a complex space and \(\mathcal S\) a coherent \(\mathcal O_X\)-module. Call an open set \(U\subset X\) standard if it is of the form \(U=A\cap V\), where \(A\subset\Omega\subset\mathbf C^N\) is a distinguished patch for \(\mathcal S\) (Definition 4.2 of the previous lesson) with resolution beginning \(\mathcal O_\Omega^{p_0}\to i_*\mathcal S\to0\), and \(V\subset\Omega\) is a Stein domain. By Proposition 4.3 of the previous lesson, the sequence

is exact, and \(\mathcal Z_0(V)\) is closed by Section 4. Give \(\mathcal S(U)\) the quotient Fréchet topology.

Proposition 5.1. This topology does not depend on the choices. For an arbitrary open set \(W\subset X\), the topology on \(\mathcal S(W)\) induced by the injection \(\mathcal S(W)\to\prod_\alpha\mathcal S(U_\alpha)\), for any countable covering of \(W\) by standard open sets, is a Fréchet topology independent of the covering, and it agrees with the previous one when \(W\) is standard.

Proposition 5.2. Let \(\mathcal S\) be coherent on \(X\).

1. For every \(x\in W\), the map \(\mathcal S(W)\to\mathcal S_x\) is continuous for the Krull topology of the finitely generated \(\mathcal O_{X,x}\)-module \(\mathcal S_x\).
2. If \(\mathcal S'\subset\mathcal S\) is a coherent subsheaf, \(\mathcal S'(W)\) is closed in \(\mathcal S(W)\). A morphism of coherent sheaves induces continuous maps on sections, and restriction maps are continuous.
3. If \(W'\subset W\) is open with compact closure \(\overline{W'}\subset W\), the restriction \(\mathcal S(W)\to\mathcal S(W')\) is compact.

Full statements and arguments in Fréchet spaces of sections and Schwartz's theorem

Used by

• Theorems A and B on Stein manifolds

  Result used: Propositions 5.1-5.2

  Hypotheses for this use: Coherent sheaves on the countable-atlas complex spaces of this programme.

  Frechet topology, continuity of restrictions, closed compatible families and compact restriction under compact containment.

• Finiteness on compact complex spaces

  Result used: Propositions 5.1 and 5.2(3)

  Hypotheses for this use: Coherent analytic sheaf; smaller open set has compact closure inside larger open set.

  Frechet section spaces and compact restriction, including singular/nonreduced spaces.

All cohomology connections

Countable products of Fréchet spaces

analytic-Frechet-countable-products

Registered statement. Every countable product of real or complex Frechet spaces is Frechet for the product topology; finite and empty families included.

Statement and proof in Fréchet spaces of sections and Schwartz's theorem

Used by

• Theorems A and B on Stein manifolds

  Result used: Countable-product proof

  Hypotheses for this use: Countably many Frechet spaces, not necessarily separable or normable.

  Complete product topology, including zero and empty factors, for the levelwise cochain spaces.

All cohomology connections

The abstract Mittag-Leffler theorem

complex-analytic-spaces-and-coherent-sheaves.frechet-spaces-of-sections-and-schwartzs-theorem.proposition-3-1

Proposition 3.1. Let \((E^\bullet_\nu)_{\nu\in\mathbf N}\) be complexes of Fréchet spaces with continuous differentials and continuous morphisms \(E^\bullet_{\nu+1}\to E^\bullet_\nu\) with dense image in each degree, and let \(E^\bullet=\varprojlim E^\bullet_\nu\).

1. If all \(H^q(E^\bullet_{\nu+1})\to H^q(E^\bullet_\nu)\) are surjective, then \(H^q(E^\bullet)\to H^q(E^\bullet_0)\) is surjective.
2. If all \(H^q(E^\bullet_{\nu+1})\to H^q(E^\bullet_\nu)\) have dense image, then \(H^q(E^\bullet)\to H^q(E^\bullet_0)\) has dense image.
3. If all \(H^{q-1}(E^\bullet_{\nu+1})\to H^{q-1}(E^\bullet_\nu)\) have dense image and all \(H^q(E^\bullet_{\nu+1})\to H^q(E^\bullet_\nu)\) are injective, then \(H^q(E^\bullet)\to H^q(E^\bullet_0)\) is injective.
4. If \(\phi:F^\bullet\to E^\bullet\) is a morphism of Fréchet complexes with dense image in each degree and every \(H^q(F^\bullet)\to H^q(E^\bullet_\nu)\) has dense image, then \(H^q(F^\bullet)\to H^q(E^\bullet)\) has dense image.

Here density in cohomology refers to the quotient topology on cocycles modulo coboundaries.

Statement and proof in Fréchet spaces of sections and Schwartz's theorem

Used by

• Theorems A and B on Stein manifolds

  Result used: Proposition 3.1(3)

  Hypotheses for this use: Frechet complexes with degreewise dense continuous maps; dense H^(q-1) and injective H^q for q>=1.

  Injection from limit cohomology into the initial level through summably corrected primitives; no Hausdorff-cohomology assumption.

All cohomology connections

Injective embeddings, flasqueness and restriction

derived-categories-and-sheaf-operations.injective-modules-and-bounded-below-derived-functors.theorem-2-3+lemma-1-1

Theorem 2.3 (functorial injective embeddings). Let \(\mathcal O\) be any sheaf of rings on \(X\). Put \[ E(F)=\prod_{x\in X}x_*\bigl(I_{\mathcal O_x}(F_x)\bigr), \] where \(I_{\mathcal O_x}\) is the functor (2.1) for the ring \(\mathcal O_x\), and let \(\iota_F:F\to E(F)\) have \(x\)-component the morphism that corresponds under (5.2) of the preceding lesson to \(\iota_{F_x}:F_x\to I_{\mathcal O_x}(F_x)\). Then \(E(F)\) is injective, \(\iota_F\) is a monomorphism, and \(E\) is a functor with \(\iota\) natural. So \(\operatorname{Mod}(\mathcal O)\) has enough injectives, with functorial injective embeddings.

Lemma 1.1 (injectives are flasque and restrict to injectives). An injective module sheaf is flasque. Its restriction to any open subset is injective in the module category of that open subset.

Theorem 2.3: statement and proof in Injective modules, flasque sheaves and bounded-below derived functors

Lemma 1.1: flasqueness and restriction

Used by

• The Dolbeault complex

  Result used: Theorem 2.3; restriction in Lemma 1.1

  Hypotheses for this use: Modules over an arbitrary sheaf of rings; here smooth complex functions on the working open subset.

  Enough injectives for the smooth-module resolution; exact sections yield zero positive module-category cohomology.

• Cartan's coherence theorem and complex spaces

  Result used: Theorem 2.3 and Lemma 1.1

  Hypotheses for this use: Sheaves of modules over any sheaf of rings, including constant Z.

  Take an injective abelian-sheaf resolution; its terms are flasque.

All cohomology connections

Right derived functors on bounded-below complexes

derived-categories-and-sheaf-operations.injective-modules-and-bounded-below-derived-functors.theorem-4-1

Theorem 4.1. Let \(\mathcal A\) and \(\mathcal B\) be abelian, and assume that every object of \(\mathcal A\) embeds in an injective object. Let \(\Phi:\mathcal A\to\mathcal B\) be additive.

1. Every bounded-below complex \(K\) has a qis \(K\to I\) into a bounded-below complex of injectives. If \(K^n=0\) for \(n<a\), it can be chosen injective in each degree, with \(I^n=0\) for \(n<a\).
2. Such an \(I\) is K-injective (Lemma 3.2). Hence \(\operatorname{Hom}_{K(\mathcal A)}(L,I)=\operatorname{Hom}_{D(\mathcal A)}(L,I)\) for every \(L\). A morphism into \(I\) extends along a qis, uniquely up to homotopy.
3. The rule \(R\Phi(K)=\Phi(I)\) gives a triangulated functor \(D^+(\mathcal A)\to D^+(\mathcal B)\), and this functor is the right derived functor \(R\Phi\). If \(\Phi\) is left exact, \(R^0\Phi=\Phi\) and \((R^i\Phi)_i\) is a universal \(\delta\)-functor.
4. (Leray) If \(\Phi\) is left exact and \(J\) is a bounded-below complex with \(R^i\Phi(J^n)=0\) for all \(i>0\) and all \(n\), then \(\Phi(J)\to R\Phi(J)\) is an isomorphism. Left exactness cannot be dropped. For a general additive \(\Phi\), the terms must also satisfy \(\Phi(J^n)\cong R^0\Phi(J^n)\), which holds by (3) when \(\Phi\) is left exact. For \(\Phi=-\otimes_{\mathbb Z}\mathbb Z/2\) on \(\operatorname{Mod}(\mathbb Z)\) and \(J=\mathbb Z\) in degree \(0\), the injective resolution \(\mathbb Q\to\mathbb Q/\mathbb Z\) gives \(R\Phi(\mathbb Z)=0\), because \(\mathbb Q\) and \(\mathbb Q/\mathbb Z\) are divisible, while \(\Phi(\mathbb Z)=\mathbb Z/2\).
5. For \(\mathcal A=\operatorname{Mod}(\mathcal O)\), with \(\mathcal O\) any sheaf of rings, (1)–(4) hold by Theorem 2.3. This gives \(Rf_*\), \(R\Gamma(U,-)\), and \(R\operatorname{Hom}_{\mathcal O}(G,-)\) on \(D^+\). The support functor is treated in Sections with support and the localization triangle.

Full statement of Theorem 4.1 in Injective modules, flasque sheaves and bounded-below derived functors

Proof of part (1): constructing an injective resolution

Proof of parts (2)–(3): comparison and derived functors

Proof of parts (4)–(5): acyclic resolutions

Used by

• The Dolbeault complex

  Result used: Theorem 4.1(4)-(5)

  Hypotheses for this use: Left exact section functor and bounded-below complex whose individual terms are acyclic; the Dolbeault resolution is finite.

  Compute all degrees of the full bundle Dolbeault isomorphism, with the zero incoming term in degree zero.

• Stein domains in complex space

  Result used: Theorem 4.1(4)-(5)

  Hypotheses for this use: The finite exact locally L2 Dolbeault resolution viewed in abelian or holomorphic module sheaves.

  The section complex computes Hq(U,O); its global positive-degree exactness on a Stein domain gives vanishing.

• Stein domains in complex space

  Result used: Theorem 4.1

  Hypotheses for this use: Left exact section functor and bounded-below complexes with enough injectives.

  Existing derived-functor proof supplies the cohomological long exact sequence.

All cohomology connections

Fréchet metrics, closed subspaces and finite products

AN03-Frechet-metric-and-closed-subspaces

Registered conclusion. Metric and seminorm convergence and Cauchy conditions coincide; equivalent countable families preserve completeness; closed subspaces and finite products are Frechet.

Statement and argument in Banach estimates, quotient spaces and compact parameter arguments

Connections

• Fréchet spaces of sections and Schwartz's theorem

  Location: Section 1; Lemma 1.3; Theorem 2.2

  Result used: Section 14.1

  Connection: Required input

  Conclusion: Metric and seminorm convergence and Cauchy conditions coincide; equivalent countable families preserve completeness; closed subspaces and finite products are Frechet.

  Hypotheses for this use: Real or complex vector space with a countable separating family of seminorms; completeness for the associated translation-invariant metric.

  The consumer uses increasing separating seminorms, a special case of this general family. The kernels and cocycle spaces are closed by continuity; the product in the finiteness criterion has two factors.

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Complete Fréchet quotients

AN03-Frechet-closed-quotient

Registered conclusion. E/N with its quotient topology is Hausdorff, locally convex, metrizable and complete; the quotient map is continuous, linear and open.

Statement and argument in Banach estimates, quotient spaces and compact parameter arguments

Connections

• Fréchet spaces of sections and Schwartz's theorem

  Location: Lemma 1.3 and the quotient reductions in Theorem 2.1

  Result used: Section 14.8

  Connection: Required input

  Conclusion: E/N with its quotient topology is Hausdorff, locally convex, metrizable and complete; the quotient map is continuous, linear and open.

  Hypotheses for this use: A real or complex Frechet space E with its countable separating seminorms, and a closed linear subspace N.

  For Lemma 1.3, N is the kernel of a continuous linear map into a Hausdorff space. The result retains the nonnormable scope and applies also to the closed finite-dimensional quotients used in Section 2.

All cohomology connections

An alternative Fréchet open-mapping proof

AN03-Frechet-open-mapping-alternative

Registered conclusion. The map is open; in the bijective case the inverse is continuous.

Statement and argument in Banach estimates, quotient spaces and compact parameter arguments

The local open-mapping theorem and its proof

Connections

• Fréchet spaces of sections and Schwartz's theorem

  Location: Theorem 1.1

  Result used: Section 14.4

  Connection: Alternative proof

  Conclusion: The map is open; in the bijective case the inverse is continuous.

  Hypotheses for this use: A continuous surjective linear map between real or complex Frechet spaces, with its given family of seminorms.

  Exactly the consumer Theorem 1.1 scope, without a normability or separability assumption. This is an alternative proof, not an additional required dependency.

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Completed Euclidean Lebesgue measure

completed-Lebesgue-measure

Registered conclusion and application. Volume, countable additivity, Borel measurability and null completion; used for the scalar L2 spaces.

Statement and argument in Banach estimates, quotient spaces and compact parameter arguments

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Measure and distribution inputs, at the weights used here (Sections 2–3)

  Result used: Section 15.0

  Connection: Required input

  Hypotheses for this use: Completed Euclidean Lebesgue measure constructed from boxes.

  Volume, countable additivity, Borel measurability and null completion; used for the scalar L2 spaces.

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Convergence and Euclidean product integration

dominated-convergence-product-integration

Registered conclusion and application. Monotone and dominated convergence and Euclidean Tonelli-Fubini; weighted cutoff tails use the integrable function |f|^2 exp(-phi).

Statement and argument in Banach estimates, quotient spaces and compact parameter arguments

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Measure and distribution inputs, at the weights used here (Sections 2–3)

  Result used: Section 15.1

  Connection: Required input

  Hypotheses for this use: Measurable functions, with an integrable dominator for dominated convergence and the stated nonnegative or absolute-integrability condition for product integration.

  Monotone and dominated convergence and Euclidean Tonelli-Fubini; weighted cutoff tails use the integrable function |f|^2 exp(-phi).

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Hölder, Minkowski and Young inequalities

holder-minkowski-young

Registered conclusion and application. Cauchy-Schwarz for local test pairings and the L1-L2 convolution bound needed for the uniform commutator estimate.

Statement and argument in Banach estimates, quotient spaces and compact parameter arguments

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Measure and distribution inputs, at the weights used here (Sections 2–3)

  Result used: Section 15.2

  Connection: Required input

  Hypotheses for this use: Conjugate Holder exponents and Young exponents in the proved range, including L1 convolution against L2.

  Cauchy-Schwarz for local test pairings and the L1-L2 convolution bound needed for the uniform commutator estimate.

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Lebesgue completeness and finite-exponent density

Lp-completeness-density

Registered conclusion and application. Unweighted L2 completeness; multiplication by exp(-phi/2) transfers it isometrically to each continuous-weight form space. No L-infinity translation-density assertion.

Statement and argument in Banach estimates, quotient spaces and compact parameter arguments

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Measure and distribution inputs, at the weights used here (Sections 2–3)

  Result used: Section 15.3

  Connection: Required input

  Hypotheses for this use: Lp completeness on a measure space; compact smooth density and translation continuity on Euclidean space for 1 <= p < infinity.

  Unweighted L2 completeness; multiplication by exp(-phi/2) transfers it isometrically to each continuous-weight form space. No L-infinity translation-density assertion.

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Scaled smooth approximation

scaled-mollifier-approximation

Registered conclusion and application. Smooth convolution, its derivative formulas and L2 convergence. A fixed compact upper weight bound gives the weighted convergence used here.

Statement and argument in Banach estimates, quotient spaces and compact parameter arguments

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Measure and distribution inputs, at the weights used here (Sections 2–3)

  Result used: Section 15.4

  Connection: Required input

  Hypotheses for this use: A smooth compactly supported kernel of integral one and finite-p Euclidean data; the zero-extended data here have compact support inside the open domain.

  Smooth convolution, its derivative formulas and L2 convergence. A fixed compact upper weight bound gives the weighted convergence used here.

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Weak-derivative limits and local approximation

weak-derivative-limits

Registered conclusion and application. Distributional limit continuity, cutoff Leibniz rule and derivative-convolution commutation. Provider convention D=-i partial is converted explicitly.

Statement and argument in Banach estimates, quotient spaces and compact parameter arguments

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Measure and distribution inputs, at the weights used here (Sections 2–3)

  Result used: Section 15.5

  Connection: Required input

  Hypotheses for this use: Local Lp convergence and weak derivatives on an open set; zero extension only after a compactly supported cutoff.

  Distributional limit continuity, cutoff Leibniz rule and derivative-convolution commutation. Provider convention D=-i partial is converted explicitly.

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Explicit cutoffs on coordinate balls

coordinate-ball-cutoff-alternative

Section 13.10: construction and proof

Keep the function from this lesson:

\[\begin{gathered} \eta(s)=\begin{cases}0,&s\le0,\\ E(-1/s),&s>0.\end{cases} \\ \chi(s)=\frac{\eta(3/2-s)} {\eta(3/2-s)+\eta(s-1)}. \end{gathered}\tag{OC38} \]

For \(s>0\), chain and product differentiation give

\[\begin{gathered} \eta^{(r)}(s)=E(-1/s)P_r(1/s),\\ P_0(u)=1,\\ P_{r+1}(u)=u^2[P_r(u)-P_r'(u)]. \end{gathered}\tag{OC39} \]

This exact recursion retains every polynomial coefficient, sign and power. If \(u>0\) and \(m\ge0\) is an integer, the positive series gives \(E(u)\ge u^{m+1}/(m+1)!\), whence

\[\begin{gathered} 0\le u^mE(-u) \\\le\frac{(m+1)!}{u}\longrightarrow0 \\(u\longrightarrow+\infty). \end{gathered}\tag{OC40} \]

Every term of \(P_r(u)\), and of \(uP_r(u)\), therefore tends to zero after multiplication by \(E(-u)\). It follows first that \(\eta\) is continuous at zero. Inductively, extend the positive-side \(r\)-th derivative by zero for \(s\le0\). It is continuous at zero, and its derivative there is the limit of \(E(-1/s)P_r(1/s)/s\), which is zero by the just-proved estimate. Its derivative away from zero is precisely the next recursion. This proves \(\eta\in C^\infty(\mathbb R)\) and every derivative zero at the original endpoint.

The denominator in \(\chi\) is strictly positive for every real \(s\). Otherwise both terms would vanish, requiring simultaneously \(s\ge3/2\) and \(s\le1\). The full reciprocal and product rules show that \(\chi\) is smooth, with \(0\le\chi\le1\), \(\chi=1\) for \(s\le1\), and \(\chi=0\) for \(s\ge3/2\). The function \(\chi\) itself is not asserted to have compact support on the whole real line: its negative half-line is one. On \([0,\infty)\) its support is exactly \([0,3/2]\).

The actual Euclidean cutoff is \(\theta(x)=\chi(\sum_{j=1}^d x_j^2)\). The inner function is a polynomial in every original coordinate. The proved full chain rules therefore make \(\theta\) smooth; it equals one when \(\sum_jx_j^2\le1\) and vanishes when \(\sum_jx_j^2\ge3/2\). Its support is the full closed ball \(\sum_jx_j^2\le3/2\), compact by the proved original topology. Every derivative is continuous on that support and zero outside it; compact extrema give its actual finite derivative bounds. In dimension zero the entire space is a point, the value is one and its support is compact; no nonexistent coordinate derivative is assigned.

For any specified original center \(x_0\) and radii \(0\le r<R\), the map \[\begin{gathered} x\longmapsto\\ \chi\left(\begin{gathered}1+\\\frac{\sum_{j=1}^d(x_j-x_{0j})^2-r^2}{2(R^2-r^2)}\end{gathered}\right) \end{gathered}\tag{OC41} \] retains every translation, square and denominator. It equals one on the closed radius-\(r\) ball. Its nonzero set is exactly the open radius-\(R\) ball, because the full cutoff is positive precisely when its scalar input is below \(3/2\). Its support is therefore the full closed radius-\(R\) ball, including the outer sphere where the value itself is zero. In dimension zero the domain is its single point and these sets have that same one-point meaning. This is a proved coordinate construction, not a change to the original cutoff in (OC38). For an open domain and any one of its points, choose an outer ball whose closure lies inside that domain; the construction gives the required local compact cutoff.

For a locally finite family of supports, every point has a neighborhood meeting only finitely many of them. All other functions and all their derivatives vanish on that neighborhood. Its full sum and every derivative are therefore the corresponding finite sums there. The same argument applies to a compact set by its finite neighborhood subcover. No interchange of an uncontrolled infinite derivative sum is asserted.

Construction and proof in Metric and topological foundations

The general partition and cutoff theorems

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Exact cutoff and partition inputs (Section 3)

  Result used: Section 13.10, OC38-OC41

  Connection: Alternative local construction

  Hypotheses for this use: Specified Euclidean center and radii 0 <= r < R; the closed outer ball must lie in the open domain.

  Smooth explicit cutoff equal to one on the inner closed ball and supported in the full closed outer ball; an alternative local construction, with the scope of a local cutoff.

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Baire category for complete metric spaces

complete-metric-baire

Theorem 3.1 (Baire). Let \((X,d)\) be a nonempty complete metric space, and \((U_n)_{n\ge1}\) a sequence of dense open subsets. Then \(\bigcap_nU_n\) is dense. Equivalently, \(X\) is not a countable union of closed sets with empty interior.

Statement and argument in Hahn–Banach, Baire and the basic theorems on Banach spaces

Connections

• Fréchet spaces of sections and Schwartz's theorem

  Location: Theorem 1.1

  Result used: Theorem 3.1 (Baire)

  Connection: Required input

  Conclusion: A countable union of closed sets with empty interior cannot cover the space.

  Hypotheses for this use: A nonempty complete metric space; no normability or separability hypothesis.

  The codomain Frechet space has a complete compatible metric and contains zero. The closed dilates of the closure of the image of a zero-ball form a countable cover. Baire supplies interior, as required by the local Frechet open-mapping proof.

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Cauchy–Schwarz for positive semidefinite forms

hilbert-cauchy-schwarz

A sesquilinear form on a complex vector space \(V\) is a map \(B:V\times V\to\mathbb C\), linear in the first variable and conjugate-linear in the second. It is Hermitian if \(B(\eta,\xi)=\overline{B(\xi,\eta)}\), and positive semidefinite if \(B(\xi,\xi)\ge0\) for all \(\xi\).

Proposition 1.1.

1. (Polarization) \(4B(\xi,\eta)=\sum_{k=0}^3i^kB(\xi+i^k\eta,\xi+i^k\eta)\).
2. A form with \(B(\xi,\xi)\in\mathbb R\) for all \(\xi\) is Hermitian.
3. (Cauchy–Schwarz) If \(B\) is positive semidefinite, then \(|B(\xi,\eta)|^2\le B(\xi,\xi)B(\eta,\eta)\). Consequently \(\{\xi:B(\xi,\xi)=0\}\) is a subspace, and \(\xi\mapsto B(\xi,\xi)^{1/2}\) is a seminorm.

Statement and argument in Hilbert spaces and compact operators

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Lemma 1.1

  Result used: Proposition 1.1

  Connection: Required input

  Conclusion: The Cauchy-Schwarz bound holds, including the zero-diagonal case.

  Hypotheses for this use: A positive semidefinite complex sesquilinear form, linear in its first variable.

  Apply it to g and the orthogonal projection f1; combine with the coercive inequality to bound the functional by C^(-1/2) times the norm of g.

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Completions and bounded extensions

bounded-functional-extension

Lemma (Completion). A normed space \(E\) embeds linearly and isometrically as a dense subspace of a Banach space \(\widehat E\). If its norm comes from an inner product, that product extends to \(\widehat E\), which is then a Hilbert space. Every bounded linear map \(T:E\to F\), with \(F\) a Banach space, has a unique bounded linear extension to \(\widehat E\), of the same norm. In particular the Hilbert construction applies to a positive semidefinite form after taking its quotient by the null subspace.

Statement and argument in Hilbert spaces and compact operators

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Adjoint definition and Lemma 1.1

  Result used: Lemma (Completion), Section 1

  Connection: Required input

  Conclusion: There is a unique continuous extension to the completion with the same norm.

  Hypotheses for this use: A bounded linear map from a normed space into a Banach space; the functional codomain here is the scalar field.

  For the adjoint definition extend the bounded functional on the dense domain of T. For Lemma 1.1 extend the bounded functional on the image of T* to its closure, identified with its completion in H1; then use orthogonal projection for the zero extension.

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Hilbert projection and Riesz–Fréchet representation

hilbert-projection-and-riesz

Theorem 2.1. Let \(C\) be a nonempty closed convex subset of a real or complex Hilbert space \(H\), and \(\xi\in H\). There is exactly one \(c_0\in C\) with \(\|\xi-c_0\|=\operatorname{dist}(\xi,C)\).

Theorem 2.2 (projection theorem). Let \(L\) be a closed subspace of a real or complex Hilbert space \(H\). Then:

1. \(H=L\oplus L^\perp\);
2. the map \(P:\xi\mapsto c_0\) of Theorem 2.1 is the orthogonal projection onto \(L\), a bounded self-adjoint idempotent with \(\|P\|\le1\);
3. for every subspace \(Y\), \(\overline Y=(Y^\perp)^\perp\).

Theorem 2.3 (Riesz–Fréchet). Every bounded linear functional \(f\) on \(H\) is \(f(\xi)=\langle\xi,\eta\rangle\) for exactly one \(\eta\in H\), and \(\|f\|=\|\eta\|\).

Statement and argument in Hilbert spaces and compact operators

Connections

• Hörmander's L² estimates on pseudoconvex domains

  Location: Adjoint definition and Lemma 1.1

  Result used: Theorems 2.1-2.3, especially projection 2.2 and Riesz-Frechet 2.3

  Connection: Required input

  Conclusion: Orthogonal decomposition and contractive projection; every bounded linear functional is inner product with a unique vector of equal norm.

  Hypotheses for this use: A Hilbert space with inner product linear in the first variable; the projected subspace is closed. No separability is assumed.

  Project onto the closed kernel of S and onto the closure of the image of T*. Represent the extended functional T*f -> <f,g>, obtaining <T*f,u>=<f,g>; conjugation gives the exact weak identity in Lemma 1.1.

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Finite-dimensional Hausdorff vector spaces

HB-finite-dimensional-Hausdorff-vector-spaces

Theorem 7.1. Let \(X\) be a Hausdorff topological vector space of finite dimension \(n\). Then every linear bijection \(f:\mathbb K^n\to X\) is a homeomorphism, where \(\mathbb K^n\) has its Euclidean topology. Consequently:

1. all Hausdorff vector topologies on a finite-dimensional space coincide;
2. every finite-dimensional subspace of a Hausdorff topological vector space is closed;
3. every linear map from a finite-dimensional Hausdorff topological vector space to a topological vector space is continuous.

Statement and argument in Hahn–Banach, Baire and the basic theorems on Banach spaces

Connections

• Fréchet spaces of sections and Schwartz's theorem

  Location: Lemma 1.3 and finite-dimensional quotient reductions in Section 2

  Result used: Finite-dimensional Hausdorff vector spaces

  Connection: Required input

  Conclusion: The subspace is closed, has its Euclidean vector topology, and is complete; linear maps out of it are continuous.

  Hypotheses for this use: Finite-dimensional subspace of a real or complex Hausdorff topological vector space.

  The Frechet ambient space is Hausdorff; no ambient normability or local compactness is imposed.

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The complete product metric within the Polish-space lemma

EF-countable-product-metric-component

Containing lemma: full statement. The application below uses the complete-metric argument in part (3). The Polish-space conclusion also assumes separability.

Lemma 1.3.

1. (Testing on generators) Suppose \(\mathcal B=\sigma(f_i:i\in I)\). A map \(g:\Gamma\to X\) is measurable if and only if every \(f_i\circ g\) is measurable.
2. (Countable bases) If a topological space has a countable base \(\mathcal U\), every open set is a countable union of members of \(\mathcal U\), so the Borel sets are \(\sigma(\mathcal U)\). The Borel structure of a product of countably many second countable spaces is generated by the coordinate maps. In particular, for a countable set \(D\), the Borel structure of \(\mathbb R^D\) is generated by the coordinates.
3. (A stock of Polish spaces) Closed subsets and countable products of Polish spaces are Polish, and so is the disjoint union of two Polish spaces, each piece open in the union. The space \(\mathbb N^{\mathbb N}\) is Polish. A separable metrizable space, and hence every standard Borel space, has at most \(2^{\aleph_0}\) points.
4. (Pieces) Let \(\Gamma\) be the union of countably many sets \(\Gamma_d\in\Sigma\). A map \(g\) on \(\Gamma\) whose restriction to every \(\Gamma_d\) is measurable for the relative \(\sigma\)-algebra is measurable.

Statement and argument in Polish spaces and the Effros Borel structure

The complete local product proof without separability

Connections

• Fréchet spaces of sections and Schwartz's theorem

  Location: Section 1, frechet-countable-product-input

  Result used: The complete product metric within the Polish-space lemma

  Connection: Argument component; scope stated below

  Conclusion: The weighted bounded product metric has the product topology and is complete. Separability belongs only to the separate countable-density conclusion.

  Hypotheses for this use: The reused metric component requires countably many complete metric factors. The containing Polish-space lemma additionally assumes separability.

  The full consumer argument explicitly proves completeness and locally convex product topology without separability. It does not misapply the Polish-space statement.

All cohomology connections