Introduction
Given a \sigma-compact analytic space X, of bounded dimension, we know how to construct [10] a bounded complex \mathbf{K}_X^\bullet of {\mathscr{O}}_X-modules of coherent cohomology, and a (continuous linear) “trace” \mathbf{T}_X\colon\mathrm{H}_\mathrm{c}^0(X;\mathbf{K}_X^\bullet)\to\mathbb{C} that can happily provide a duality between the separations of the spaces \mathrm{H}_\mathrm{c}^p(X;{\mathscr{F}}) and \operatorname{Ext}^{-p}(X;{\mathscr{F}},\mathbf{K}_X^\bullet), for every integer p and every coherent {\mathscr{O}}_X-module {\mathscr{F}}, as long as we endow these space with suitable topologies; similarly, the trace induces a perfect pairing between \mathrm{H}^p(X;{\mathscr{F}}) and \operatorname{Ext}_\mathrm{c}^{-p}(X;{\mathscr{F}},\mathbf{K}_X^\bullet), suitably topologised. The purpose of this article is the study of the compatibility of this duality with direct images: if f\colon X\to Y is a morphism of analytic spaces, then we have to construct a “relative trace” \mathbf{T}_{X/Y}\in\operatorname{Ext}^0(Y;\mathrm{R}{f_!}\mathbf{K}_X^\bullet,\mathbf{K}_Y^\bullet) (where f_! denotes the direct image with proper support). We then have, for every complex {\mathscr{F}}^\bullet of {\mathscr{O}}_X-modules of coherent cohomology ({\mathscr{F}}^\bullet\in\mathbf{D}_\mathrm{coh}(X)) and for every complex {\mathscr{G}}^\bullet of bounded-below coherent cohomology ({\mathscr{G}}^\bullet\in\mathbf{D}_\mathrm{coh}^+(Y)), a canonical morphism \begin{aligned} \mathrm{R}{f_!}{\mathscr{F}}\operatorname{Hom}\big( X ; {\mathscr{F}}^\bullet, \mathrm{R}\operatorname{Hom}( \mathrm{L}f^*&\mathrm{R}\operatorname{Hom}( Y;{\mathscr{G}}^\bullet, \mathbf{K}_Y^\bullet ), \mathbf{K}_X^\bullet) \big) \\\to\, &\mathrm{R}\operatorname{Hom}(Y;\mathrm{R}{f_*}{\mathscr{F}}^\bullet,{\mathscr{G}}^\bullet) \end{aligned} \tag{1} that is, in certain cases, an isomorphism (we obtain another formula by swapping \mathrm{R}{f_!} and \mathrm{R}{f_*}, and more striking formulas by setting {\mathscr{G}}^\bullet=\mathbf{K}_Y^\bullet).
In general, direct images are not coherent, and, just like the absolute case, we can only hope to address these questions by defining a topological dualisation \mathrm{R}\operatorname{Hom}(Y;-,\mathbf{K}_Y^\bullet) on Y (the relative trace should live in a topological \operatorname{Ext}^0, and (1) only gives isomorphisms of suitable operations up to “separation”). We restrict ourselves here to the case where f is proper, and we use the Grauert direct image theorem.1 In this case, (1) is an isomorphism (the \operatorname{Hom} being the usual functors), and we have an “algebraic” statement. The proof, however, uses topology: to establish the existence of the relative trace, and then to prove that (1) is an isomorphism; we invoke, in particular, the separation of certain cohomology spaces.
Section 4, which deals with questions of separation, is independent of the previous sections. There we show the two following theorems, conjectured by Malgrange, and which form an indispensable complement to the absolute duality theorems:
Given an analytic space X, a coherent sheaf {\mathscr{F}}, and an integer p, suppose that every compact K\subset X has a compact “envelope” in K' such that every element of \mathrm{H}_K^p(X;{\mathscr{F}}) whose image in \mathrm{H}_\mathrm{c}^p(X;{\mathscr{F}}) is zero already has zero image in \mathrm{H}_{K'}^p(X;{\mathscr{F}}). Then the space \mathrm{H}_\mathrm{c}^p(X;{\mathscr{F}}), endowed with the topology constructed in [10] (the only quotient topology of the Fréchet–Schwartz dual compatible with the duality theorem), is separated.
As for the statement of Theorem 4.9, we obtain this by replacing, in the lines above, \mathrm{H}_\bullet^p(X;{\mathscr{F}}) with \operatorname{Ext}_\bullet^p(X;{\mathscr{F}},\mathbf{K}_X^\bullet).
In an appendix, we will give another proof, using Theorem 4.9, of a theorem of Siu on the vanishing of certain cohomology groups.
1 Proof sketch
We denote by \iota the canonical isomorphism from \mathrm{H}_\mathrm{c}^0(Y;\mathrm{R}{f_*}\mathbf{K}_X^\bullet) to \mathrm{H}_c^0(X;\mathbf{K}_X^\bullet). The trace \mathbf{T}_Y induces a pairing between \mathrm{H}_\mathrm{c}^0(Y;\mathrm{R}{f_*}\mathbf{K}_X^\bullet) and \operatorname{Ext}^0(Y;\mathrm{R}{f_*}\mathbf{K}_X^\bullet,\mathbf{K}_Y^\bullet), and we are looking for an element \mathbf{T}_{X/Y} in this latter space such that we have, for all a\in\mathrm{H}_\mathrm{c}^0(Y;\mathrm{R}{f_*}\mathbf{K}_X^\bullet), \mathbf{T}_Y(a\cdot\mathbf{T}_{X/Y}) = \mathbf{T}_x\iota a. In other words, we are looking for a transpose {}^t\iota of \iota with respect to the pairings \mathbf{T}_Y and \mathbf{T}_X (with \mathbf{T}_{X/Y} as the image of {}^t\iota under the function 1\in\mathrm{H}^0(X;{\mathscr{O}}_X)). Since this problem is local on Y (provided that we are guaranteed the local vanishing of \operatorname{Ext}^i(Y;\mathrm{R}{f_*}\mathbf{K}_X^\bullet,\mathbf{K}_Y^\bullet) for i<0), we can suppose that Y is realisable as an analytic subset of an open polydisc V of dimension p. If we still denote by f the composite of f and an embedding of Y into V, it follows from the construction of \mathbf{K}_Y^\bullet that what we are looking for is an element of \operatorname{Ext}^p(V;\mathrm{R}{f_*},\mathbf{K}_X^\bullet,\Omega_V). We now want to proceed locally on X, supposing (being content to shrink X) that X is the union of a finite number of opens X_i, each of which is embeddable into an open U_i of \mathbb{C}^n: we can incorporate each X_i into a commutative diagram
where I is an embedding, and where F denotes the second projection of U_i\times V. We denote by f_{i_0i_1\ldots i_r} the direct image with proper support in X_{i_0}\cap X_{i_1}\cap\ldots X_{i_r} and we first search for a “trace” \mathbf{T}_i in each \operatorname{Ext}^p(V;\mathrm{R}{f_i}\mathbf{K}_X^\bullet,\Omega_V). We know that \mathbf{K}_{X_i}^\bullet is precisely \operatorname{Hom}_{{\mathscr{O}}_{U_i\times V}}({\mathscr{O}}_X,\mathbf{K}_{U_i\times V}^\bullet) where \mathbf{K}_{U_i\times V}^\bullet is (with a shift by n_i+p) a particularly beautiful resolution of the sheaf \Omega_i of holomorphic differential forms of maximum degree on U_i\times V. The function 1 on X_i thus gives a morphism from \mathbf{K}_{X_i}^\bullet to \mathbf{K}_{U_i\times V}^\bullet, whence a morphism to \operatorname{Ext}^p(V;\mathrm{R}{f_i}\mathbf{K}_X^\bullet,\Omega_V) from \operatorname{Ext}^n(V;\mathrm{R}{F_!}\Omega_i,\Omega_V), and it is in this latter space that we will find an elementary trace t_i.
The sheaf \mathrm{R}^q{F_!}\Omega_i is generated, by a Künneth formula, by the presheaf v\mapsto\mathrm{H}_\mathrm{c}^{n_i}(U_i;\Omega_{U_i})\widehat{\otimes}\mathrm{H}^{q-n}(v;\Omega_V) (since \mathrm{H}_\mathrm{c}^\bullet(U_i;\Omega_{U_i}) is zero in degrees not equal to n_i)2; it thus follows that the complex \mathrm{R}{F_!}\Omega_i is equivalent to the sheaf \mathrm{R}^{n_i}F_!\Omega_i in degree n_i, and that we are simply looking for a homomorphism from this sheaf to \Omega_V, which immediately gives the interpretation given above, and the absolute trace on U_i.
To glue the \mathbf{T}_i, we would like to be able to confirm the existence of a spectral sequence abutting to \operatorname{Ext}^{r-q}(V;\mathrm{R}{f}\mathbf{K}_X^\bullet,\Omega_V) with page E_1^{r,q} \prod \operatorname{Ext}^{-q}(V;\mathrm{R}{f_{i_0i_1\ldots i_r}}\mathbf{K}_X^\bullet,\Omega_V) and then say that the \mathbf{T}_i give an element of E_2^{0,-p}, which would be equal to \operatorname{Ext}^p(V;\mathrm{R}{f_*}\mathbf{K}_X^\bullet,\Omega_V) by the vanishing of E_1^{r,q} for q<-p. All of this is sadly unlikely to be feasible, since the usual \operatorname{Ext} groups are not well adapted to the study of complexes such as \mathrm{R}{f_i}\mathbf{K}_X^\bullet; why, for example, would \operatorname{Ext}^j(V;\mathrm{R}{F_!}\Omega_i,\Omega_i) be zero for j\geqslant1 when \mathrm{R}{F_!}\Omega_i is a sheaf of analytic functions on V with values in a \mathbf{DFS}-space? Having still not established a direct solution to this (i.e. defined for certain complexes of {\mathscr{O}}_V-modules of \mathrm{Extop}^\bullet(V;-,\Omega_V) groups that should coincide with the “algebraic” \operatorname{Ext} under the hypothesis of perfectness of the complexes), we note in Section 2 a more artificial method of construction for \mathbf{T}_{X/Y}.
Thanks to the relative trace, which is, up to homotopy, a morphism from \mathrm{R}_{f_*}\mathbf{K}_X^\bullet to \mathbf{K}_Y^\bullet, it is easy to construct the morphism (1): first, for any complex {\mathscr{F}}^\bullet of {\mathscr{O}}_X-modules of coherent cohomology, we denote by D_X{\mathscr{F}}^\bullet the complex \operatorname{RHom}({\mathscr{F}}^\bullet,\mathbf{K}_X^\bullet), which is also of coherent cohomology. We know [10] that the canonical morphism from {\mathscr{F}}^\bullet to D_XD_X{\mathscr{F}}^\bullet is a quasi-isomorphism, and we will note that, if {\mathscr{G}}^\bullet\in\mathbf{D}_\mathrm{coh}^+(X), there is a canonical isomorphism from \operatorname{RHom}({\mathscr{F}}^\bullet,{\mathscr{G}}^\bullet) to D_X({\mathscr{F}}^\bullet\otimes^\mathrm{L}D_X{\mathscr{G}}^\bullet) (where X denotes an arbitrary analytic space, for example Y). Indeed, \begin{aligned} \operatorname{RHom}({\mathscr{F}}^\bullet,{\mathscr{G}}^\bullet) &= \operatorname{RHom}({\mathscr{F}}^\bullet,D_XD_X{\mathscr{G}}^\bullet) \\&= \operatorname{RHom}({\mathscr{F}}^\bullet,\operatorname{RHom}(D_X{\mathscr{G}}^\bullet,\mathbf{K}_X^\bullet)) \\&= \operatorname{RHom}({\mathscr{F}}^\bullet\otimes^\mathrm{L}D_X{\mathscr{G}}^\bullet,\mathbf{K}_X^\bullet) \\&= D_X({\mathscr{F}}^\bullet\otimes^\mathrm{L}D_X{\mathscr{G}}^\bullet). \end{aligned} Finding the morphism (1) thus reduces to exhibiting a morphism \tau\colon \mathrm{R}{f_*}D_X({\mathscr{F}}^\bullet\otimes^\mathrm{L}\mathrm{L}{f^*}D_Y{\mathscr{G}}^\bullet) \to D_Y(\mathrm{R}{f_*}{\mathscr{F}}^\bullet\otimes^\mathrm{L}D_Y{\mathscr{G}}^\bullet) and this comes from the canonical homomorphism from \mathrm{R}{f_*}\operatorname{RHom}({\mathscr{F}}^\bullet\otimes^\mathrm{L}\mathrm{L}{f^*}D_Y{\mathscr{G}}^\bullet,\mathbf{K}_X^\bullet) to \operatorname{RHom}(\mathrm{R}{f_*}({\mathscr{F}}^\bullet\otimes^\mathrm{L}\mathrm{L}{f^*}D_Y{\mathscr{G}}^\bullet), \mathrm{R}{f_*}\mathbf{K}_X^\bullet) along with the isomorphism given by the projection formula \mathrm{R}{f_*}({\mathscr{F}}^\bullet\otimes^\mathrm{L}\mathrm{L}{f^*}D_Y{\mathscr{G}}^\bullet) \to \mathrm{R}{f_*}{\mathscr{F}}^\bullet\otimes^\mathrm{L}D_Y{\mathscr{G}}^\bullet and the trace \mathbf{T}_{X/Y}.
To prove that \tau is an isomorphism, it suffices, by substituting {\mathscr{F}}^\bullet\otimes^\mathrm{L}\mathrm{L}^{f^*}D_Y{\mathscr{G}}^\bullet for {\mathscr{F}}^\bullet, to consider the case where {\mathscr{G}}^\bullet=\mathbf{K}_Y^\bullet, i.e. to prove that \tau\colon \mathrm{R}{f_*}\operatorname{RHom}({\mathscr{F}}^\bullet,\mathbf{K}_X^\bullet) \to \operatorname{RHom}(\mathrm{R}{f_*}{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet) is a quasi-isomorphism; for this, we show that \tau induces a quasi-isomorphism between the above complexes of sections of the above complexes over every sufficiently nice open of Y: in plain words, we show that, if Y is a Stein space, then \tau induces, for every i, an isomorphism from \operatorname{Ext}^{-i}(X;{\mathscr{F}}^\bullet,\mathbf{K}_X^\bullet) to \operatorname{Ext}^{-i}(Y;\mathrm{R}{f_*}{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet). By a lemma on \partial-functors [3], it suffices to prove this in the case where {\mathscr{F}}^\bullet consists of a single coherent sheaf {\mathscr{F}}; in this case, \operatorname{Ext}^{-i}(X;{\mathscr{F}},\mathbf{K}_X^\bullet) is in duality with \mathrm{H}_\mathrm{c}^i(X;{\mathscr{F}}), and we will show in Section 3 that \operatorname{Ext}^{-i}(X;\mathrm{R}{f_*}{\mathscr{F}},\mathbf{K}_Y^\bullet) is in duality (in a topological sense) with \mathrm{H}_\mathrm{c}^i(Y;\mathrm{R}{f_*}{\mathscr{F}}), and, by the same construction, the transpose of \tau with respect to these pairings is the natural algebraic isomorphism from \mathrm{H}_\mathrm{c}^i(Y;\mathrm{R}{f_*}{\mathscr{F}}) to \mathrm{H}_\mathrm{c}^i(X;{\mathscr{F}}). The fact that \tau is an isomorphism then follows from the fact that the four spaces in question are separated, the proof of which is completed in Section 4.
The vanishing of \operatorname{Ext}^{-i}(Y;\mathrm{R}{f_*}\mathbf{K}_X^\bullet,\mathbf{K}_X^\bullet) when Y is a Stein space and i>0, to which we alluded in the start of this section, can be proved in the same way: this space is in duality with \mathrm{H}_\mathrm{c}^i(Y;\mathrm{R}{f_*}\mathbf{K}_X^\bullet), which is algebraically isomorphic to \mathrm{H}_\mathrm{c}^i(X;\mathbf{K}_X^\bullet)=\operatorname{Ext}_\mathrm{c}^i(X;{\mathscr{O}}_X,\mathbf{K}_X^\bullet), which is in duality with \mathrm{H}^{-i}(X;{\mathscr{O}}_X), which is zero, even though all these spaces were a priori separated (the first two by construction of even their topologies — cf. Section 3; as for \operatorname{Ext}_\mathrm{c}^i(X;{\mathscr{O}}_X,\mathbf{K}_X^\bullet), we know that its separation is equivalent to that of \mathrm{H}^{-i+1}(X;{\mathscr{O}}_X)).
The argument above strongly encourages us to define the trace \mathbf{T}_{X/Y} as the image of the function 1\in\mathrm{H}^0(X;{\mathscr{O}}_X) under the transpose of the natural algebraic isomorphism \iota from \mathrm{H}_\mathrm{c}^0(Y;\mathrm{R}{f_*}\mathbf{K}_X^\bullet) to \mathrm{H}_\mathrm{c}^0(X;\mathbf{K}_X^\bullet); that is, we need to prove the continuity of \iota. We can even conclude, using the separation theorem Theorem 4.2 and Lemma 3.5, that \mathrm{H}_\mathrm{c}^0(X;\mathbf{K}_X^\bullet) is separated whenever Y is a Stein space, which implies that, in this case, \mathrm{H}^1(X;{\mathscr{O}}_X) is separated, and \operatorname{Ext}^0(Y;\mathrm{R}{f_*}\mathbf{K}_X^\bullet,\mathbf{K}_Y^\bullet) is isomorphic to \mathrm{H}^0(X;{\mathscr{O}}_X).
2 The relative trace
From now on, let f\colon X\to Y be a proper morphism, where Y is an open polydisc of \mathbb{C}^p. Let Y' be an open sub-polydisc of Y, with \overline{Y'}\subset Y. We set X'=f^{-1}(Y').
We will define “in a natural way” a complex of \mathbf{DFS}-spaces whose cohomology will be the hypercohomology with compact support of Y with values in \mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet. Since \mathrm{R}{f_*}\mathbf{K}_X^\bullet is bounded and of coherent cohomology, we can find [4], in a neighbourhood of Y', a representative {\mathscr{L}}^\bullet that is perfect (i.e. bounded, and whose objects are free of finite type). Let {\mathfrak{L}} be a locally finite cover of Y' by closed polydiscs. Since the groups \mathrm{H}_\mathrm{c}^i(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) are the cohomology groups of the total complex of the double complex \mathrm{C}^\bullet({\mathfrak{L}};\mathrm{H}^0({\mathfrak{L}};{\mathscr{L}}^\bullet)) of \mathbf{DFS}-spaces (when there is no ambiguity in the choice of cover, we denote this complex by \mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet)), which allows us to topologise them, in a way that is independent of the choice of resolution {\mathscr{L}}^\bullet, since if {\mathscr{L}}_1^\bullet\to and {\mathscr{L}}_2^\bullet\to are perfect resolutions of \mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet then there exists another perfect resolution {\mathscr{L}}_3^\bullet that dominates them both:
With the group \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) being thus topologised3, we endow \mathrm{H}_\mathrm{c}^0(X';\mathbf{K}_{X'}^\bullet) with its usual \mathbf{DFS} topology4, and we propose the following proposition.
The natural algebraic isomorphism \iota from \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) to \mathrm{H}_\mathrm{c}^0(X';\mathbf{K}_{X'}^\bullet) is continuous.
Before proving this, we note that there exists on \mathrm{H}_\mathrm{c}^i(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) = \mathrm{H}_\mathrm{c}^i(Y';{\mathscr{L}}^\bullet) a unique \mathbf{QDFS} topology, and on \operatorname{Ext}^{-i}(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet,\mathbf{K}_{Y'}^\bullet) a unique \mathbf{QFS} topology, such that the trace \mathbf{T}_{Y'} places in duality the separations of these spaces. Furthermore, these spaces are separated, and the topology on \mathrm{H}_\mathrm{c}^i(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) coincides with that described above (we proceed as in [10, Theorem 2], by using the fact that {\mathscr{L}}^\bullet is perfect, and thus has coherent objects, and, in parallel, we use Proposition 3.2; the topologies are the same by uniqueness, and thus separated).
We recall that Proposition 2.1 allows us to construct the relative trace \mathbf{T}_{X'/Y'}; it is the image of the function 1\in\mathrm{H}^0(X';{\mathscr{O}}_{X'}) under the transpose5 of \iota (note that \operatorname{Ext}^0(Y';{\mathscr{L}}^\bullet,\mathbf{K}_{Y'}^\bullet)=\operatorname{Ext}^0(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet,\mathbf{K}_{Y'}^\bullet)).
It remains to prove the proposition. For this, we introduce the diagram \begin{CD} \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) @>{\iota}>> \mathrm{H}_\mathrm{c}^0(X';\mathbf{K}_{X'}^\bullet) \\@V{\alpha}VV @AA{\gamma}A \\\mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) @>>{\beta}> \mathrm{H}_\mathrm{c}^0(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) \end{CD} where \beta is the natural map, \alpha is the morphism induced by the natural morphism \rho\colon {\mathscr{F}}^\bullet \to \mathrm{R}{f_*}\mathrm{L}{f^*}{\mathscr{F}}^\bullet (cf. [3, II, Proposition 5.10]), and where \gamma is a map (that makes the diagram commute) that we will define later on.
To show the continuity of \iota, we will topologise \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) \quad\text{and}\quad \mathrm{H}_\mathrm{c}^0(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet). We then prove the continuity of the maps \alpha, \beta, and \gamma.
Let K be a Stein compact of X', and {\mathscr{F}} a coherent {\mathscr{O}}_{X'}-module. Let {\mathscr{F}}_K be the sheaf whose germs of sections at the points of K are those of {\mathscr{F}}, and 0 elsewhere. Then the sheaf {\mathscr{F}}_K is f_*-acyclic, and the direct image f_*{\mathscr{F}}_K is \Gamma_\mathrm{c}(Y';-)-acyclic.
If f_! denotes the direct image with proper support functor, then \mathrm{R}{f_*}{\mathscr{F}}_K=\mathrm{R}{f_!}{\mathscr{F}}_K and we are reduced to proving the f_!-acyclicity of {\mathscr{F}}_K. This simply follows from the fact that the intersection of a Stein open of X' with the preimage of a Stein open of Y' is a Stein open.
To establish the \Gamma_\mathrm{c}(Y';-)-acyclicity of f_*{\mathscr{F}}_K=f_!{\mathscr{F}}_K, we note that \Gamma_\mathrm{c}(X';-)=\Gamma_\mathrm{c}(Y';-)f_!, and we use the spectral sequence for composite functors, which gives \mathrm{H}_cc^p(Y';\mathrm{R}^qf_!{\mathscr{F}}_K)\Rightarrow\mathrm{H}_\mathrm{c}^{p+q}(X';{\mathscr{F}}_K).
Let {\mathfrak{M}} be a finite cover of \overline{X'} by Stein compacts. The intersection of the compacts of {\mathfrak{M}} with the preimages of compacts of {\mathfrak{L}} are Stein compacts, and form a locally finite cover {\mathfrak{R}} of X'.
Let {\mathscr{L}}^i be the i-th component of the complex {\mathscr{L}}^i. If L is a compact of Y', we denote by {\mathscr{L}}_L^i the sheaf that is equal to {\mathscr{L}}^i on L and zero elsewhere; {\mathscr{L}}^i is a free {\mathscr{O}}_{Y'}-module, and so {\mathscr{L}}_L^i is an {\mathscr{O}}_{Y'}-module with projective fibres, and thus f^*-acyclic. We evidently have f^*{\mathscr{L}}_L^i=(f^*{\mathscr{L}}^i)_{f^{-1}(L)}, and f^*{\mathscr{L}}^i is a free {\mathscr{O}}_{X'}-module of finite type. Thus f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet) is a representative of \mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet. We denote by \mathrm{C}_{X'}^\bullet(f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet)) the total complex of the double complex \mathrm{C}^\bullet\big({\mathfrak{R}};\mathrm{H}^0({\mathfrak{R}};f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet))\big). We evidently have a morphism f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet) \to \mathrm{C}_{X'}^\bullet(f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet)), and \mathrm{C}_{X'}^\bullet(f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet)) is a representative of \mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet. If K_{k_0\ldots k_n} is an n-simplex of {\mathfrak{R}}, and L_{l_0\ldots l_m} an m-simplex of {\mathfrak{L}}, it is clear that the sheaf (f^*{\mathscr{L}}_{L_{l_0\ldots l_m}}^i)_{K_{k_0\ldots k_n}} is f^*-acyclic and that its direct image is \Gamma_\mathrm{c}(Y';-)-acyclic (Lemma 2.2). Thus f_*\mathrm{C}_{X'}^\bullet(f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet)) is a representative of \mathrm{R}{f_*}\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet. Finally, the natural morphism from \mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet) to f_*\mathrm{C}_{X'}^\bullet(f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet)) is a representative of the morphism \rho\colon\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet\to\mathrm{R}{f_*}\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet.
We will now clearly see how to topologise our spaces, and how to prove the continuity of \alpha and \beta, since, if {\mathfrak{R}}' denotes the Stein cover of Y' consisting of the K'=f(K), where K runs over {\mathfrak{R}}, then \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) is the 0-th cohomology group of the complex of \mathbf{DFS}-spaces given by the total complex of \mathrm{C}^\bullet({\mathfrak{L}};\mathrm{H}^0({\mathfrak{L}};{\mathscr{L}}^\bullet)), or even of \mathrm{C}^\bullet({\mathfrak{R}}';\mathrm{H}^0({\mathfrak{R}}';{\mathscr{L}}^\bullet)); \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) is the 0-th cohomology group of the complex of \mathbf{DFS}-spaces given by the total complex of \mathrm{C}^\bullet({\mathfrak{R}}';\mathrm{H}^0({\mathfrak{R}};f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet))); and \mathrm{H}_\mathrm{c}^0(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) similarly comes from \mathrm{C}^\bullet({\mathfrak{R}};\mathrm{H}^0({\mathfrak{R}};f^*\mathrm{C}_{Y'}^\bullet({\mathscr{L}}^\bullet))).
It remains to construct a suitable morphism \gamma.
For this, we introduce the morphisms
where l and n are the natural morphisms, and where m in induced by \rho; by [3, Proposition 5.10], mn is an isomorphism.
We thus obtain the diagram (where \tau=\mu\nu)
The identity morphism \mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet\to\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet induces an element of \operatorname{Ext}^0(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet,\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet), which comes, by \tau, from an element e of \operatorname{Ext}^0(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet,\mathbf{K}_{X'}^\bullet); we set \nu(e)=\varepsilon.
The composition product of the \operatorname{Ext} gives \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) \times \operatorname{Ext}^0(Y';\mathrm{R}{f_*}\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet, \mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) \longrightarrow \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) and \mathrm{H}_\mathrm{c}^0(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) \times \operatorname{Ext}^0(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet,\mathbf{K}_{X'}^\bullet) \longrightarrow \mathrm{H}_\mathrm{c}^0(X';\mathbf{K}_{X'}^\bullet). We can thus build, corresponding to \varepsilon, a linear map \delta\colon \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) \longrightarrow \mathrm{H}_\mathrm{c}^0(Y';\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) and, to e, a linear map \gamma\colon \mathrm{H}_\mathrm{c}^0(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) \longrightarrow \mathrm{H}_\mathrm{c}^0(X';\mathbf{K}_{X'}^\bullet).
By construction, \delta\alpha is equal to the identity. The commutativity of diagram (2) then follows.
The trace \mathbf{T}_{X'} puts into duality (the separation of) \mathrm{H}_\mathrm{c}^0(X;\mathbf{K}_{X'}^\bullet) and \Gamma(X';{\mathscr{O}}_{X'}). We can easily show (as in [10]) that there exists on \mathrm{H}_\mathrm{c}^i(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet)=\mathrm{H}_\mathrm{c}^i(X';\mathrm{L}{f^*}{\mathscr{L}}^\bullet) a unique \mathbf{QDFS} structure, and on \operatorname{Ext}^{-i}(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet,\mathbf{K}_{X'}^\bullet)=\operatorname{Ext}^{-i}(X';\mathrm{L}{f^*}{\mathscr{L}}^\bullet,\mathbf{K}_{X'}^\bullet) a unique \mathbf{QFS} structure, such that the trace \mathbf{T}_{X'} puts the separations[^6] of these two spaces into duality (since \mathrm{L}{f^*}{\mathscr{L}}^\bullet is a perfect complex of {\mathscr{O}}_{X'}-modules; the \mathbf{QDFS} structure thus obtained on \mathrm{H}_\mathrm{c}^0(X';\mathrm{L}{f^*}\mathrm{R}{f_*}\mathbf{K}_{X'}^\bullet) coincides with that with which we have already endowed this space).
[^6] We can in fact show that these spaces are separated.
Denote by \theta the morphism given by the composite of the natural map \Gamma(X';{\mathscr{O}}_{X'})\to\operatorname{Ext}^0(X';\mathbf{K}_{X'}^\bullet,\mathbf{K}_{X'}^\bullet) and \tau^{-1}\lambda. We have that \theta(1)=e. It thus follows that the map \beta admits \theta as transpose, and is thus continuous. The proof of Proposition 2.1 is thus complete.
3 Complexes of coherent cohomology on a Stein space
All throughout this section, Y denotes a Stein space, and {\mathscr{F}}^\bullet a complex of {\mathscr{O}}_Y-modules of coherent cohomology. Before endowing the spaces \mathrm{H}^i(Y;{\mathscr{F}}^\bullet) with (Fréchet–Schwartz) topologies, recall the following lemma.
Let A and B be abelian categories, with A having enough injectives, and let F be an additive functor from A to B, left exact and of finite cohomological dimension. Let I^\bullet be a complex in A whose objects, as well as its cohomology objects, are F-acyclic. Then we have, for all i, \mathrm{H}^i F(I^\bullet) = F(\mathrm{H}^i I^\bullet).
Proof. This is particularly evident when I^\bullet has bounded-below cohomology: we can show, by induction from the left, that all the boundary and cycle objects of I^\bullet are F-acyclic, by cutting I^\bullet into short exact sequences to which we then apply F. In the general case, one should use a Cartan–Eilenberg resolution of I^\bullet.
We apply this lemma to the category A of {\mathscr{O}}_Y-modules, and to the functor F of global sections \Gamma, taking I^\bullet to be an injective resolution of the complex {\mathscr{F}}^\bullet of coherent cohomology. By Cartan’s Theorem B, all the hypotheses are satisfied, and we have, for all i, \mathrm{H}^i(Y;{\mathscr{F}}^\bullet) = \Gamma(Y;\mathrm{H}^i{\mathscr{F}}^\bullet) which allows us to topologise the left-hand side.
We can now prove a first duality theorem for complexes of coherent cohomology on Stein spaces:
For all i, the trace \mathbf{T}_Y gives a pairing \tau\colon \mathrm{H}^i(Y;{\mathscr{F}}^\bullet) \times \operatorname{Ext}_\mathrm{c}^{-i}(Y;{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet) \to \mathbb{C}. There exists a (unique) pair of (\mathbf{FS} and \mathbf{DFS}, respectively) topologies on these spaces such that \tau makes each one the strong dual of the other.
Having already endowed \mathrm{H}^i(Y;{\mathscr{F}}^\bullet) with a topology, we can consider the functor that associates to the complex {\mathscr{F}}^\bullet the collection of topological duals of \mathrm{H}^i(Y;{\mathscr{F}}^\bullet). Putting together everything that we have said, this is a \partial-functor, since the transpose of an exact complex of \mathbf{FS}-spaces is an exact complex. It remains to compare this \partial-functor with the \partial-functor \operatorname{Ext}_\mathrm{c}^{-i}(Y;-,\mathbf{K}_Y^\bullet), knowing, by the first absolute duality theorem, that these functors agree whenever they are applied to a coherent sheaf. By a known lemma, it suffices to show that there exists a morphism from the second functor the the first, i.e. that the pairing \tau is continuous with respect to the first variable.
So let e\in\operatorname{Ext}_\mathrm{c}^{-i}(Y;{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet), actually coming from \operatorname{Ext}_L^{-i}(Y;{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet) where L is a compact of Y, and let U be an open neighbourhood of L over which the sheaf \mathrm{H}^i\mathscr{F}^\bullet can be written as the quotient of a free sheaf {\mathscr{L}} of finite type. We can factor the linear for -\cup e on \mathrm{H}^i(Y;{\mathscr{F}}^\bullet) as follows:
The only thing that poses a problem is the continuity of \alpha, but the topology of its source is the quotient of the evident topology on \Gamma(U;{\mathscr{L}}), and, by freeness, \alpha can be lifted to an {\mathscr{O}}(U)-linear map \beta, which is necessary continuous by finiteness.
Applying this first duality theorem to D_Y{\mathscr{F}}^\bullet, we see the following:
There exists a \mathbf{DFS} topology on \mathrm{H}_\mathrm{c}^i(Y;{\mathscr{F}}^\bullet), and an \mathbf{FS} topology on \operatorname{Ext}^{-i}(Y;{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet), such that \mathbf{T}_Y puts these spaces in duality with one another.
Adding to the diagram an analytic polyhedron L of Y, we can also prove:
There exists an \mathbf{FS} topology on \mathrm{H}_L^i(Y;{\mathscr{F}}^\bullet), and a \mathbf{DFS} topology on \operatorname{Ext}^{-i}(L;{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet), such that \mathbf{T}_Y puts these spaces in duality with one another.
It is this time that we have a natural topology on \operatorname{Ext}^{-i}(L;{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet), since this space is isomorphic to \Gamma(L;\operatorname{Ext}^{-i}({\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet)). As in the first proposition, we can reduce to the case where {\mathscr{F}}^\bullet consists of a single coherent sheaf {\mathscr{F}}, and then, by embedding, to the case where Y is a polydisc of \mathbb{C}^n. The result can then be found in [2], since we know that the “Fréchet limit” topology on the sections of a coherent sheaf in the neighbourhood of a compact is in fact a \mathbf{DFS} topology.
Under the hypotheses of Lemma 3.4, we can finally prove the following:
The natural map from \mathrm{H}_L^i(Y;{\mathscr{F}}^\bullet) to \mathrm{H}_\mathrm{c}^i(Y;{\mathscr{F}}^\bullet) is injective.
Proof. By duality, it suffices to show that the natural map from \operatorname{Ext}^{-i}(Y;{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet) to \operatorname{Ext}^{-i}(L;{\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet) has dense image, but, by the definition of the topologies in question, this map is the restriction from Y to L of the sections of the coherent sheaf \mathscr{E}\mathcal{xt}^{-i}({\mathscr{F}}^\bullet,\mathbf{K}_Y^\bullet), and this restriction does indeed have dense image, since L is holomorphically convex.
4 Separation criteria
Below, we give two separation criteria (one concerning the \mathrm{H}_\mathrm{c}^p(X;{\mathscr{F}}), and the other the \operatorname{Ext}_\mathrm{c}^p(X;{\mathscr{F}}^,\mathbf{K}_X^\bullet)).
Bibliography
Footnotes
In another article, J.P. Ramis and G. Ruget hope to treat the general case, once there is a generalisation to the relative case of the theorems of Andreotti–Grauret on p-convex spaces. The above theorem would then follow from the conjunction of the general duality theorem and the Grauert direct image theorem.↩︎
We only deal with nuclear spaces in these questions.↩︎
We can show that this topology also does not depend on the choice of {\mathscr{L}}^\bullet.↩︎
But not natural: when we know how to topologise the hypercohomology of a complex of coherent cohomology on a general space [5], the lemma becomes evident.↩︎
\mathrm{H}^0(X';{\mathscr{O}}_{X'}) is in duality with the separation of \mathrm{H}_\mathrm{c}^0(X';\mathbf{K}_{X'}^\bullet) (in fact, we can prove that this space is separated).↩︎