Completed tensors and nuclear coherent sections

Written by GPT-6 Astra (OpenAI), at Ultra, October 2026. Independent exposition: CC0-1.0.

Read this argument with the general surface theorem and its prerequisite guide. Result identifiers link to the exact arguments in this companion family.

The supporting arguments retain linear mathematical notation: -> denotes a map, ^ an exponent or cohomological degree, _ a subscript, and direct_sum a direct sum. Parentheses specify grouping. Algebraic quotients and completed topological tensor products are distinguished explicitly; a quotient is not silently replaced by its completion.

Conventions

E_p is the Banach completion of E/ker(p), not E with an assumed complete p-norm. An increasing defining sequence (p_j) gives continuous maps E_(j+1) -> E_j.

hat_pi and hat_epsilon denote Hausdorff completions for the projective and injective tensor seminorms. hat_tensor without a suffix is used only after NS-3 proves the two topologies coincide in the case at hand.

In a short exact sequence of Frechet spaces the first map identifies a closed subspace and the last map is an open surjection. These topological properties follow from the open mapping theorem when the sequence is algebraically exact and its maps are continuous.

A map T:E -> F is nuclear when T(x)=sum_j lambda_j xi_j(x)y_j, with sum_j |lambda_j| finite, (xi_j) equicontinuous in E', and (y_j) bounded in F. This is a representation, not an asserted unique tensor for T.

E is nuclear if it has an increasing defining sequence p_j for which E_(j+1) -> E_j is a nuclear Banach-space map for every j.

F boxtimes G on X x Y means pr_X^*F tensor_(O_(X x Y)) pr_Y^*G. Pullbacks and tensor products use the full structure sheaves.

NS-0. The two completed tensor constructions and their elementary functorial properties

Statement. Projective completion preserves open surjections of Frechet spaces. Injective completion preserves closed topological embeddings. Tensoring a continuous injective map with the identity by injective completion also gives an injective map, without asserting its image is closed.

Proof.

For continuous seminorms p on E and q on F, define pi_(p,q)(t) as the infimum of sum_i p(x_i)q(y_i) over finite expressions t=sum_i x_i tensor y_i. Define epsilon_(p,q)(t) as the supremum of |(xi tensor eta)(t)| over linear functionals satisfying |xi|<=p and |eta|<=q. These are seminorms: combine finite expressions for the projective triangle inequality, and take a supremum of absolute values for the injective one. Every eligible pair of functionals bounds every projective expression, so epsilon<=pi. Hahn-Banach supplies functionals attaining p(x) and q(y) on a specified pair of vectors after passing to their seminorm quotients. Thus both seminorms equal p(x)q(y) on x tensor y, including the zero-seminorm cases.

Use all pairs from countable defining seminorm families and complete after dividing out the common zero subspace. The resulting spaces are Hausdorff Frechet spaces; a diagonal sequence of seminorm tolerances gives a complete translation-invariant metric. If a:E->E' and b:F->F' are continuous, a target pair p',q' is bounded on their images by C p and D q for some source pair. The induced algebraic tensor map then satisfies the pi estimate with factor CD. Pulling back the functionals gives the same estimate for epsilon. Both maps consequently extend to the corresponding completions.

Suppose a and b are open surjections. Given source seminorms p,q, their quotient seminorms p'(x')=inf_(a(x)=x') p(x) and q'(y')=inf_(b(y)=y') q(y) are continuous: their open unit balls are the images of the source open balls. A finite target tensor expression with small sum_i p'(x'_i)q'(y'_i) can be lifted, term by term, with the sum of source products arbitrarily close to that target sum. Approximation of a completed target tensor by algebraic tensors now shows that the closure of the image of any source projective-seminorm neighbourhood contains a target neighbourhood. A finite intersection of source constraints can be dominated by one pair after taking maxima, so this holds for every source neighbourhood.

Here is the completion argument which turns this last property into actual openness and surjectivity. For a continuous linear map T:P->Q between Frechet spaces, suppose closure(T(V)) is a neighbourhood of zero whenever V is. Given a closed absolutely convex neighbourhood V, choose absolutely convex neighbourhoods V_n inside 2^(-n-1)V, and small enough that p_j(x)<=2^(-n) on V_n whenever j<=n, where p_j define P. Choose open W_n contained in closure(T(V_n)) and in the nth member of a shrinking neighbourhood basis of Q. Starting with y_0 in W_0, choose x_n in V_n such that y_(n+1)=y_n-T(x_n) belongs to W_(n+1); density in the indicated closure allows this choice. The series sum_n x_n converges in P, its sum lies in V, and y_n tends to zero. Hence T(sum_n x_n)=y_0. Thus W_0 is contained in T(V). This proves openness; a linear range containing a neighbourhood is all of Q. Apply this argument to the completed projective tensor map.

For a closed topological subspace K of E, a continuous seminorm on K is dominated by the restriction of a continuous seminorm on E. Hahn-Banach extends every functional controlled by a restricted ambient seminorm without increasing that bound. Consequently the injective tensor topology on K tensor F is exactly the topology inherited from E tensor_epsilon F; this can first be checked with the ambient defining seminorms and then with the dominated subspace seminorms. The induced map on completions is a topological embedding. Its image is closed because its domain is complete. The same reasoning applies in both factors.

For the weaker claim, let j:F->G be any continuous injective map and let t in E hat_epsilon F map to zero. Each xi in E' gives a continuous evaluation xi tensor Id_F:E hat_epsilon F->F. Applying j to this evaluation gives zero, so the evaluation itself is zero. It follows that every scalar evaluation xi tensor eta is zero on t. For a Cauchy sequence of algebraic tensors, its defining epsilon seminorm equals the supremum of these evaluations also in the limit: the convergence is uniform over each pair of seminorm-controlled dual balls. Thus all defining seminorms of t are zero. This proves injectivity, but makes no assertion of a topological embedding when j has nonclosed image.

The scalar-valued continuous dual of E hat_pi F is the space of jointly continuous bilinear forms on E x F. Indeed a continuous bilinear form is bounded by C p(x)q(y) for suitable continuous seminorms; its linearization is bounded by C pi_(p,q) and extends to the completion. Conversely such a bound gives a continuous bilinear form on pure tensors. These two constructions are inverse.

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NS-1. Nuclear maps, factorizations, and the binuclear subspace argument

Statement. Nuclear maps form an operator ideal and factor through a Banach space. A nuclear map between Banach spaces factors through a Hilbert space. A composition of two nuclear Banach-space maps can be corestricted to a closed subspace containing its range or descended through a closed subspace of its kernel while retaining nuclearity.

Proof.

Consider T(x)=sum_j lambda_j xi_j(x)y_j as defined above. There is a continuous seminorm p controlling all |xi_j|. For each target seminorm q, boundedness of the y_j supplies M_q=sup_j q(y_j)<infinity. The tails on p(x)<=1 are bounded by M_q sum_(j>N)|lambda_j|. They therefore converge in every target seminorm, uniformly on that neighbourhood; completeness gives a continuous linear map. Precomposing or postcomposing by continuous linear maps preserves respectively equicontinuity of the functionals or boundedness of the vectors. This proves the operator-ideal property. Concatenating two representations proves closure under finite sums.

A Banach factorization requires no injectivity statement about any tensor-to-operator map. Define alpha:E->ell^1 by alpha(x)=(lambda_j xi_j(x))_j; its norm is at most p(x) sum_j|lambda_j|. Define beta:ell^1->F by beta(c)=sum_j c_j y_j. For each q its norm estimate is q(beta(c))<=M_q ||c||_1, so the series converges in F and beta is continuous. Then T=beta alpha. For Banach spaces, use the nuclear norm inf sum_j ||a_j|| ||b_j|| over all representations T(x)=sum_j a_j(x)b_j. It is the quotient norm on representations; no uniqueness of the representing tensor is needed or asserted.

If T:E->F is nuclear between Banach spaces, choose a representation with sum_j ||a_j||||b_j|| finite. Discard zero terms and rescale each pair so that ||a_j||=||b_j||=sqrt(||a_j||||b_j||). The maps u:E->ell^2, u(x)=(a_j(x))_j, and v:ell^2->F, v(c)=sum_j c_j b_j, are bounded: both bounds follow from Cauchy-Schwarz and the summability of the squared rescaled norms. The last series converges absolutely in F for every c in ell^2. Thus T=v u factors through a Hilbert space.

For clarity, a closed subspace H_1 of a Hilbert space H has a bounded orthogonal projection. For x in H choose a sequence h_n in H_1 whose distances to x approach the infimum. The parallelogram identity shows h_n is Cauchy, since (h_n+h_m)/2 is another candidate. Its limit h realizes the infimum. Varying h along any real and imaginary scalar multiple of a vector in H_1 shows x-h is orthogonal to H_1. This decomposition is unique, linear in x, and its H_1 component has norm at most ||x||. It also identifies H/H_1 isometrically with H_1's orthogonal complement.

Now let E --f--> F --g--> G be nuclear Banach-space maps and suppose gf(E) is contained in a closed T contained in G. Factor g=v u through a Hilbert space H and put H_1=v^(-1)(T). This is closed, and u f(E) is contained in H_1. Therefore gf, with codomain T, is the composition E --f--> F --u--> H --P_(H_1)--> H_1 --v|_(H_1)--> T. All maps other than f are bounded; the last map is bounded for T's subspace norm. The ideal property proves nuclearity into T.

For descent, assume S is a closed subspace of E killed by gf. Factor f=v u through H and let H_1 be the closure of u(S). The bounded map gv kills H_1. The map x+S -> u(x)+H_1 from E/S to H/H_1 is well-defined and bounded, since one can take the infimum of ||u(x+s)|| over s in S. Identify this quotient with H_1's orthogonal complement. Composing with v restricted to that complement and then with the nuclear map g gives precisely the descended map E/S->G. Hence it is nuclear. The closure in the definition of H_1 is essential.

A nuclear map is compact in the locally convex sense that some neighbourhood has relatively compact image. Use p(x)<=1 as above. Every finite partial sum has totally bounded image, because its coefficients are bounded and its range is finite dimensional. For any finite list of target seminorms, the common tail bound tends uniformly to zero. The full image is therefore totally bounded. Its closure is compact in the complete metrizable target. Finally, tensoring two nuclear Banach-space maps gives a nuclear map for either completed tensor norm: multiply their scalar coefficients, use the product functionals, and use the tensor products of their bounded vectors. The absolute coefficient sum is the product of the two absolute sums.

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NS-2. Stability and the Banach-factor test for nuclear Frechet spaces

Statement. Closed subspaces, separated quotients, countable products and countable inverse limits of nuclear Frechet spaces are nuclear; so is the completed projective tensor of two such spaces. For a nuclear domain, a continuous map is nuclear precisely when it factors continuously through a Banach space. Closed corestriction and closed-kernel descent of a nuclear map with nuclear domain are nuclear.

Proof.

Write E_j for the p_j completion in the definition of a nuclear space. If a:E->M is continuous and M is Banach, its norm is bounded by C p_j for some j. Extend it to a bounded map E_j->M and factor a as E->E_(j+1)->E_j->M. The middle map is nuclear. Composing further with any continuous M->F is nuclear by NS-1. Conversely every nuclear map has the ell^1 factorization of NS-1. This proves the Banach-factor test.

Let K be closed in E. For every j, its completion for the restricted seminorm p_j is the closed subspace K_j=closure(image(K->E_j)). The map K_(j+2)->K_j comes from the composition E_(j+2)->E_(j+1)->E_j restricted on its domain to K_(j+2) and corestricted on its codomain to K_j. The first restricted map and the second map are nuclear by the ideal property. The binuclear corestriction argument of NS-1 proves the required nuclearity. The even subsequence of the restricted seminorms still defines the topology of K. Completeness follows from K being closed; hence K is nuclear.

Equip E/K with the quotient seminorms bar_p_j(x+K)=inf_(k in K) p_j(x+k). They define its quotient topology, and its bar_p_j completion is Q_j=E_j/K_j. To check the last assertion, the displayed quotient norm agrees on the dense image of E/K with the distance to K_j in E_j, and the latter quotient is Banach. The composition E_(j+2)->E_(j+1)->E_j->Q_j is binuclear and kills K_(j+2), since it kills the dense image of K there. NS-1 therefore gives a nuclear map Q_(j+2)->Q_j. The quotient is Hausdorff because K is closed; the images of a countable neighbourhood basis give metrizability. To see completeness, let (z_n) be a quotient Cauchy sequence. Choose a subsequence so that its jth successive difference belongs to the image of {x:p_k(x)<2^(-j) for k<=j}; this is possible because those images are neighbourhoods. Lift each difference to a vector in that set and add the successive lifts to a lift of the initial term. The resulting representatives are Cauchy in every p_k, so converge in E. Their quotient limit is the limit of the original Cauchy sequence, since it is the limit of a subsequence. Thus the even quotient seminorm sequence proves nuclearity of E/K.

For a sequence of nuclear spaces E^(k), fix a nuclear defining family p_j^(k) on each. On their product use r_j(x)=max_(k<=j) p_j^(k)(x_k). Its Banach completion is the finite product of the corresponding E_j^(k). The transition for j+1 to j has finitely many nuclear diagonal blocks; its additional coordinate maps to zero. Finite sums and bounded coordinate maps preserve nuclearity, so the product is nuclear. An inverse limit of a countable system of such spaces is the closed subspace of the product defined by the continuous compatibility equations. The subspace case therefore gives the inverse-limit claim without assuming surjectivity of the transition maps.

For two nuclear spaces E,F, use diagonal projective tensor seminorms pi_(p_j,q_j); these are cofinal among all pairs. Their Banach completions identify with E_j hat_pi F_j. Indeed the seminorm first factors through E/ker(p_j) tensor F/ker(q_j), where it is the projective norm; passing to dense Banach completions preserves that norm, as is seen either by approximating each factor in finite expressions or by extending bounded bilinear forms and applying Hahn-Banach. The transition map is the tensor of the two nuclear transition maps, hence nuclear by NS-1. Thus E hat_pi F is nuclear.

Suppose T:E->F is nuclear, E is nuclear, and T(E) is contained in a closed subspace L of F. Factor T as E --a--> M --b--> F with M Banach. Then M_1=b^(-1)(L) is a closed Banach subspace, and a:E->M_1 and b|_(M_1):M_1->L are continuous. The Banach-factor test makes the corestricted map nuclear. If instead a closed K contained in E is killed by T, let M_0=closure(a(K)). The map b kills M_0, so the descended map factors as E/K -> M/M_0 -> F. Its domain is nuclear by the quotient result, and its middle space is Banach. Apply the same test. Neither argument licenses a corresponding assertion for a lone nuclear map with an arbitrary nonnuclear domain.

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NS-3. Exact tensors, kernels of nonclosed-image maps, and countable products

Statement. If one Frechet factor is nuclear, pi and epsilon give the same completed tensor. Tensoring with a nuclear Frechet space is exact on short exact Frechet sequences, preserves the kernel of any continuous map between Frechet spaces, and commutes with countable products.

Proof.

First let T:E->F be a nuclear map of Banach spaces and let G be Banach. In a nuclear representation of T, contraction of a tensor t by a functional xi produces an element (xi tensor Id_G)t with norm at most ||xi|| times ||t||_epsilon: apply every norm-one functional of G and then Hahn-Banach. Applying this estimate to all terms of T proves ||(T tensor Id_G)t||_pi <= ||T||_nuclear ||t||_epsilon after taking the infimum over representations. The series giving this map converges in F hat_pi G. Consequently it extends continuously from E hat_epsilon G to F hat_pi G.

For nuclear E, apply that estimate to E_(j+1)->E_j and to each Banach seminorm completion F_q. On E tensor F it yields pi_(p_j,q) <= C_j epsilon_(p_(j+1),q). Together with epsilon<=pi, these inequalities show that the two locally convex tensor topologies agree. Their Hausdorff completions are therefore canonically the same. Symmetry gives the assertion if F rather than E is nuclear.

Consider a short exact sequence 0->K->E --a--> Q->0. If F is nuclear, all three tensor products with F have the common topology just established. If instead E is nuclear, NS-2 makes K and Q nuclear, so the same conclusion holds for arbitrary Frechet F. NS-0 gives a closed embedding K hat_tensor F->E hat_tensor F and an open surjection E hat_tensor F->Q hat_tensor F. To identify the kernel, let a continuous scalar functional on E hat_tensor F annihilate the former image. By the projective universal property it is a continuous bilinear form B on E x F vanishing on K x F. It descends to a jointly continuous bilinear form on Q x F: if |B(x,y)|<=C p(x)q(y), take the infimum over representatives of x+K, and the quotient seminorm of p is continuous because a is open. Its linear extension factors through Q hat_tensor F and hence annihilates the kernel of a hat_tensor Id_F. Hahn-Banach separation by continuous functionals, applied to the closed image of K hat_tensor F, shows that this image equals that kernel.

For a general continuous a:E->G, its kernel K is closed, while a(E) need not be. Factor a into the open quotient E->E/K and the continuous injective map j:E/K->G. With F nuclear, short exactness proves that the kernel of the first tensor map is F hat_tensor K. The second tensor map is injective by the final injectivity assertion of NS-0, applied with the epsilon topology. Therefore ker(Id_F hat_tensor a)=F hat_tensor K. This step does not put a quotient topology on a(E), does not complete it as a replacement codomain, and does not assume its image is closed.

Let P=product_i E_i. A cofinal family of its seminorms is r(x)=max_(i in I) q_i(x_i) for finite I. For any p on F and algebraic t in F tensor P, one has epsilon_(p,r)(t)=max_(i in I) epsilon_(p,q_i)(t_i), where t_i is its ith coordinate tensor. Indeed a functional bounded by r depends only on the coordinates in I and decomposes into coordinate functionals with the sum of their dual bounds at most one, by the dual norm of a finite maximum product. This gives the inequality <=; evaluation on a single coordinate gives >=. Thus F tensor_epsilon P maps topologically into product_i(F hat_epsilon E_i). Its range is dense: approximate finitely many prescribed coordinate tensors by finite algebraic sums and insert each into the corresponding coordinate of P, zero elsewhere. Completing gives the canonical topological isomorphism F hat_epsilon P -> product_i(F hat_epsilon E_i). When F is nuclear these are the unsuffixed tensors claimed. Finite products and tensoring with a finite-dimensional free vector space are included.

The symmetry and associativity maps on algebraic projective tensors extend to completions. In the triple case both groupings have the topology defined by inf sum_i p(x_i)q(y_i)r(z_i); expanding finite expressions and taking infima gives the two inequalities needed for equality. A tensor involving a completed factor is approximated by finite algebraic expressions in that factor for every finite list of seminorms. This proves density of the common algebraic triple tensor and hence the asserted completed identification. In the nuclear cases used below the same maps apply to the common tensor topology. No cohomological Kunneth formula is being inferred when a complex has non-Hausdorff cohomology.

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NS-4. Nuclearity of section spaces without discarding nilpotents

Statement. For every coherent sheaf F on a second-countable full complex space X, Gamma(X,F), with its canonical coherent-section Frechet topology, is nuclear.

Proof.

Begin with a polydisc D=product_(i=1)^N { |z_i-a_i|<R_i }. Give O(D) the seminorm p_r(h)=sup_(|z_i-a_i|<=rR_i)|h(z)| for 0<r<1. Choose 0<r<s<1. If a_alpha(h) is the Taylor coefficient at a, Cauchy's estimate gives |a_alpha(h)|<=p_s(h) product_i(sR_i)^(-alpha_i). It extends to a bounded functional on the p_s completion. The monomial m_alpha(z)=(z-a)^alpha has p_r norm product_i(rR_i)^(alpha_i). The restriction map from the p_s completion to the p_r completion consequently has nuclear expansion with lambda_alpha=(r/s)^(|alpha|), xi_alpha=product_i(sR_i)^(alpha_i) a_alpha, and y_alpha=m_alpha/product_i(rR_i)^(alpha_i). Both the functional and vector norms are at most one.

The sum of the absolute coefficients is product_(i=1)^N sum_(k>=0)(r/s)^k=(1-r/s)^(-N). The series converges uniformly on the smaller closed polydisc and agrees with restriction on O(D), hence on its p_s completion by density. For N=0 the space is C and the assertion is finite dimensional. An increasing sequence r_j tending to one defines the topology of O(D), and its transition maps go from radius r_(j+1) to radius r_j. They are nuclear by the estimate just proved. In particular the direction is larger radius to smaller radius; no holomorphic extension from a smaller disc is asserted.

Locally embed a full analytic chart A of X as a closed subspace of an ambient open subset Omega of C^N. The pushed-forward sheaf i_*F is coherent. After shrinking around a point, choose a finite presentation O_Omega^r -> O_Omega^s -> i_*F -> 0 and a polydisc D contained in that neighbourhood. The kernels and images are coherent. Cartan B on the ambient D, applied to the two short exact sequences of this presentation, gives the section sequence O(D)^r -> O(D)^s -> Gamma(A intersection D,F) -> 0. The last arrow is a continuous surjection between the already established coherent-section Frechet spaces, so it is open. Its kernel is closed. Thus Gamma(A intersection D,F) is a topological quotient of the nuclear space O(D)^s and is nuclear by NS-2.

Choose a countable cover (U_i) of X by such small charts. The sheaf gluing axiom identifies Gamma(X,F) with the compatible families in product_i Gamma(U_i,F). This is a closed subspace: compatibility is the vanishing of continuous differences of restrictions into the Hausdorff spaces Gamma(U_i intersection U_j,F). The canonical topology on sections is the topology induced by these local presentations and restriction seminorms, as in the bound coherent-section provider. Equivalently the continuous bijection to the closed compatible-family subspace is a homeomorphism by the Frechet open mapping theorem. NS-2 now proves nuclearity. At no point was a defining ideal replaced by its radical; the local modules, presentations and kernels are those of the full structure sheaf, including all nilpotent coefficients.

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NS-5. Canonical completed product-section comparison by refined-overlap gluing

Statement. For coherent F on X and G on Y there is a canonical topological isomorphism Gamma(X,F) hat_tensor Gamma(Y,G) -> Gamma(X x Y,F boxtimes G), taking s tensor t to pr_X^*s times pr_Y^*t. Both spaces retain their full, possibly nonreduced, structures.

Proof.

For polydiscs D,D' and their structure sheaves, take compact concentric subpolydiscs K,L and the corresponding supremum seminorms p,q. For t=sum_i f_i tensor g_i the injective tensor seminorm equals sup_(z,w in K x L)|sum_i f_i(z)g_i(w)|. Point evaluations give one inequality. For the other, fix a functional xi controlled by p. The function w -> sum_i xi(f_i)g_i(w) has q-seminorm at most the displayed supremum, since xi is bounded by the supremum on K; applying a functional eta controlled by q gives the other inequality. Taylor polynomials on the product approximate every holomorphic function uniformly on each smaller product polydisc, and each polynomial is a finite sum of separated monomials. Completion, followed by NS-3, therefore identifies O(D) hat_tensor O(D') with O(D x D').

Next let X be closed in a polydisc D and suppose i_*F has a finite presentation O_D^r -> O_D^s -> i_*F -> 0 there. Let Y=D' and G=O_D'. Taking its external product with O_D' is right exact; equivalently its cokernel is computed by pulling the matrix to D x D'. The modules and image on the ambient product are coherent. Cartan B gives the right-exact sequence on global sections O(D x D')^r -> O(D x D')^s -> Gamma(X x D',F boxtimes O_D') -> 0. On the other hand the section presentation on D has closed image at O(D)^s and is a quotient presentation of Gamma(X,F). NS-3, factoring that presentation through its closed image, yields the right-exact sequence O(D)^r hat_tensor O(D') -> O(D)^s hat_tensor O(D') -> Gamma(X,F) hat_tensor O(D') -> 0. The free-space identifications in the preceding paragraph intertwine the matrices, hence induce the desired isomorphism on the two cokernels. All spaces are Frechet, so the continuous bijection is a homeomorphism.

If also Y is closed in a polydisc D' and j_*G has a finite presentation there, external product of that presentation with i_*F gives a right-exact coherent sequence on D x D'. Cartan B again gives its exact section cokernel. The corresponding sequence tensoring the section presentation of G by Gamma(X,F) is exact by NS-3 and nuclearity from NS-4. The first two vertical maps are the isomorphisms already established for F with a free structure sheaf. Their compatibility with the presentation matrix identifies the two cokernels. This proves the assertion when both pairs have such finite ambient presentations. This argument only used right exactness of pullback and tensor for the presentations; it did not silently assume either coherent sheaf is locally free.

Fix now a pair (X,F) of that local presented kind, but allow Y,G to be arbitrary. Cover Y by countably many small charts U_i on which G has a finite ambient presentation. For each pair i,j cover U_i intersection U_j by countably many further small presented charts V_(ij,k). Such covers exist by coherence and second countability. Put P=product_i Gamma(U_i,G) and Q=product_(i,j,k) Gamma(V_(ij,k),G). Define delta:P->Q by restricting the difference of the ith and jth entries. The sheaf gluing axiom says Gamma(Y,G)=ker(delta). This description needs neither a finite presentation on the whole overlap nor any closed-image property of delta.

Let A=Gamma(X,F), which is nuclear by NS-4. NS-3 identifies A hat_tensor Gamma(Y,G) with the kernel of A hat_tensor P -> A hat_tensor Q and identifies those tensor products with the products of the individual tensor factors. For every U_i and every V_(ij,k), the already proved local comparison turns those factors into the corresponding section spaces on X x U_i or X x V_(ij,k). The isomorphisms commute with restrictions because on pure tensors they are the actual pullback-product maps, and pure tensors are dense. The resulting kernel is exactly Gamma(X x Y,F boxtimes G), by gluing on the cover (X x U_i), tested on its refined overlaps (X x V_(ij,k)). This is a topological identification of closed kernels in products of Frechet spaces. Notice that no tensor/cohomology interchange or Hausdorffness of any positive-degree cohomology was used.

Finally take arbitrary X,F as well. Cover X by presented charts and their pairwise overlaps by further presented charts exactly as above. For each of these charts the comparison with the arbitrary fixed (Y,G) was just proved. Repeat the kernel-and-product argument, now tensoring by the nuclear space Gamma(Y,G). It gives the desired comparison globally. Every stage agrees on pure sections with s tensor t -> pr_X^*s times pr_Y^*t, which makes the global map canonical and compatible with restrictions in either variable and with morphisms of coherent sheaves. All local presentations and external products used full analytic modules, so the assertion includes nonreduced X and Y.

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NS-6. Nuclear coherent restriction and its explicit free relative form

Statement. If V is relatively compact in U and F is coherent on U, restriction Gamma(U,F)->Gamma(V,F) is nuclear. Its valid closed corestrictions and closed-kernel descents are nuclear. For a nuclear unital Frechet algebra A and a nuclear map r:V_0->W_0 of nuclear Frechet spaces, Id_A hat_tensor r is A-nuclear between the free modules A hat_tensor V_0 and A hat_tensor W_0 in the explicit sense stated below.

Proof.

First note why a compact map T:E->F of Frechet spaces factors through a Banach space. Choose a continuous seminorm p whose open unit ball lies in a neighbourhood with relatively compact image. For every target seminorm q there is C_q such that q(Tx)<=C_q on p(x)<1. Scaling gives q(Tx)<=C_q p(x) for every x, and if p(x)=0 the same conclusion follows by scaling x arbitrarily. Thus T vanishes on ker(p), is uniformly continuous from the normed quotient E/ker(p) for each q, and extends uniquely to a continuous map E_p->F, since F is complete. This provides the Banach factorization E->E_p->F.

Use the compact restriction result of the pinned coherent-section provider with its maps explicitly oriented as follows. On nested standard charts A intersection D' inside A intersection D, with D' relatively compact in D and a fixed finite ambient presentation, Montel compactness gives a compact free restriction O(D)^s->O(D')^s. Composing with the target section quotient remains compact. The source quotient O(D)^s->Gamma(A intersection D,F) is open: the image of a neighbourhood on which the free map is relatively compact is a neighbourhood in the source section space. The induced coherent restriction is therefore compact. For general V relatively compact in U, cover closure(V) by finitely many smaller such charts U'_i with larger charts U_i contained in U. Map Gamma(U,F) to the finite product of Gamma(U_i,F), then restrict compactly to the product of Gamma(U'_i,F), then continuously to the product of Gamma(V intersection U'_i,F). This composite is compact. The section space Gamma(V,F) embeds as the closed subspace of compatible families in that last product. The composite takes values there; its relatively compact closure remains there because that subspace is closed. It follows that Gamma(U,F)->Gamma(V,F) itself is compact. This uses the embedding into the product, not a supposed map from an arbitrary family back to a glued section.

Both section spaces are Frechet and the domain is nuclear by NS-4. The Banach factorization proved above and NS-2 now make restriction nuclear. If its range lies in a specified closed subspace of the target, or it kills a specified closed subspace of its domain, the corresponding map is nuclear by the last part of NS-2. These conclusions apply only with those stated closedness conditions; no arbitrary quotient or nonclosed submodule is substituted.

For the relative assertion, assume multiplication in A is jointly continuous and use the free A-module action on A hat_tensor V_0 through its first factor. Say an A-linear map T:M->N is A-nuclear if it has an expansion T(x)=sum_j lambda_j phi_j(x)n_j, where sum_j|lambda_j| is finite, phi_j:M->A are continuous A-linear maps forming an equicontinuous family, and n_j form a bounded sequence in N. Equicontinuity here means that for each continuous seminorm p on A there is a continuous seminorm s on M such that p(phi_j(x))<=s(x) for every j,x. This is not the scalar nuclear-map definition.

Write r(v)=sum_j lambda_j xi_j(v)w_j with equicontinuous scalar xi_j and bounded w_j. Set phi_j=Id_A hat_tensor xi_j:A hat_tensor V_0->A and n_j=1_A tensor w_j in A hat_tensor W_0. These maps are A-linear. Choose a continuous q on V_0 with |xi_j(v)|<=q(v) for all j. For every seminorm p on A, the projective tensor estimate gives p(phi_j(x))<=pi_(p,q)(x), proving the stated equicontinuity. The n_j are bounded because each tensor seminorm is bounded by p(1_A) times the corresponding bounded seminorm of w_j. The series sum_j lambda_j phi_j(x)n_j converges in A hat_tensor W_0 and defines a continuous map: on elementary expressions its target (p,q')-seminorm is bounded by pi_(p,q)(x) times sup_j q'(w_j) sum_j|lambda_j|, and the same bound passes to the completion. On pure tensors the series is Id_A tensor r, so by density it is its completed extension. This proves A-nuclearity.

In particular take A=O(S) for a full complex space S. Its multiplication is jointly continuous, as follows locally from the multiplication of holomorphic representatives and then from the quotient presentations; its scalar section space is nuclear by NS-4. NS-5 identifies A hat_tensor O(U) with O(S x U). A relatively compact vertical restriction U->V therefore gives an A-nuclear map O(S x U)->O(S x V), and the same holds for finite free sums. This is the free vertical input for the relative construction. It is not a proof of A-subnuclear corestriction for general coherent modules, of transverse analytic base restriction, or of the finite-complex theorem. For infinite-dimensional A an A-nuclear map of this form need not be a scalar compact map; no such implication has been used.

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Sources

Jean-Pierre Demailly, Complex Analytic and Differential Geometry, author edition of 21 June 2012. . The independently written arguments above follow the cited mathematics, with the explicit domains, signs and comparisons given here.