Proper pushforwards on analytic adic spaces

ABSTRACT

1. INTRODUCTION

Arguably the most significant early achievement of Huber’s approach to analytic geometry, via his theory of adic spaces, was that it enabled the development of a robust theory of étale cohomology with compact support for rigid analytic varieties, and in particular, the proof in [10] of a Poincaré duality theorem for smooth morphisms. Previously, van der Put in [15] constructed a theory of compactly supported cohomology and proper pushforwards for abelian sheaves on rigid analytic varieties over a non-Archimedean field and proved a version of Serre duality for smooth and proper morphisms.

Our first goal in this article is to recast van der Put’s definition using Huber’s language of adic spaces, thereby generalising it to include analytic adic spaces that are not necessarily defined over a field. In fact, the definition is very simple: if |$f\colon X\rightarrow Y$| is a partially proper morphism of analytic adic spaces, we simply define |$\mathbf{R}f_!$| to be the derived functor of sections whose support is quasi-compact (and hence proper) over Y. The important thing is to show that these compose correctly, and the proof that they do so follows the strategy of [10, Chapter 5] very closely.

The main application we envision for the formalism developed here lies in the theory of rigid cohomology, which necessitates working not just with adic spaces but also with germs of adic spaces, that is, closed subsets of adic spaces with the ‘induced’ analytic structure. (For us, the motivating example of a germ is the tube |$]X[_{\mathfrak{P}}$| of a locally closed subset X of a formal scheme |$\mathfrak{P}$|⁠.) This generalisation is similar in spirit to the ‘pseudo-adic spaces’ that Huber works with in [10], although far more modest in scope.

Our second goal is to define the trace map for smooth morphisms that are ‘partially proper in the sense of Kiehl’ (see Definition 5.4), and where the base is ‘overconvergent’ (that is, closed under generalisation inside its ambient adic space). 

Broadly speaking, the construction of |$\mathrm{Tr}_{X/Y}$| follows the same outline as in [15]. First we work with relative open unit polydiscs, then with closed subspaces of relative open unit polydiscs, and finally show that the map |$\mathrm{Tr}_{X/Y}$| constructed does not depend on the choice of embedding and therefore globalises.

The question of what form of Serre–Grothendieck duality holds, and in what generality, we do not address here at all. Our main motivation for developing a formalism of proper pushforwards, and for constructing trace morphisms, was to understand particular constructions in rigid cohomology and the theory of arithmetic |$\mathscr{D}$|-modules [1]. For us, it was enough simply to have the formalism (together with an explicit computation for relative open unit polydiscs); a detailed study of duality would therefore have distracted us rather too much from our main goal. Another natural question to ask is whether or not the formalism of |$\mathbf{R}f_!$| extends in any reasonable way beyond the partially proper case. We give an example in Section 7 to show that this cannot be done, essentially for the same reason as in the case of abelian sheaves on the Zariksi site of schemes, namely, the failure of the proper base change theorem. We thank B. Le Stum for the main idea behind this example.

Let us now give a brief summary of the contents of this article. In Section 2 we gather together various (existing) results in general topology that will be useful in the rest of the article, particularly concerning sheaf cohomology on spectral spaces. In Section 3 we introduce the notion of a germ of an adic space along a closed subset and define the category in which these live. In Section 4 we define proper pushforwards |$f_!$| and |$\mathbf{R}f_!$| for partially proper morphisms of germs of adic spaces and show that these derived proper pushforwards compose correctly. In Section 5 we prove a result on the cohomological dimension of coherent sheaves, which is then used in Section 6 to construct the trace map for smooth morphisms that are partially proper in the sense of Kiehl. Finally, in Section 7 we give an example to show that there is no satisfactory formalism for |$\mathbf{R}f_!$| beyond the partially proper case.

Notation and conventions

We will only deal with abelian sheaves. Thus if X is a topological space, a sheaf on X will always mean an abelian sheaf. The category of abelian sheaves on X will be denoted by |$\mathbf{Sh}(X)$|⁠, and its derived category by |$\textbf{D}(X)$|⁠. If |$\mathcal{A}$| is a sheaf of rings on X, the derived category of |$\mathcal{A}$|-modules will be denoted |$\textbf{D}(\mathcal{A})$|⁠.

A Huber ring will be a topological ring admitting an open adic subring R₀ with finitely generated ideal of definition. For such a ring R, we will denote by |$R^\circ \subset R$| the subset of power-bounded elements, and by |$R^{\circ\circ}\subset R^\circ$| the subset of topologically nilpotent elements. A Huber pair is a pair |$(R,R^+)$| consisting of a Huber ring R and an open, integrally closed subring |$R^+\subset R^\circ$|⁠. A Huber ring R is said to be a Tate ring if there exists some |$\varpi\in R^\times\cap R^{\circ\circ}$|⁠, such an element will be called a quasi-uniformiser. A Huber pair |$(R,R^+)$| is said to be a Tate pair if R is a Tate ring. An adic space isomorphic to |$\mathrm{Spa}\left(R,R^+\right)$|⁠, where |$(R,R^+)$| is a Tate pair, will be called a Tate affinoid.

If X is an adic space and |$x\in X$|⁠, we will denote by |$\mathcal{O}_{X,x}$| and |$\mathcal{O}_{X,x}^+$| the stalks of the structure sheaf and integral structure sheaf of X at x, respectively. The residue field of |$\mathcal{O}_{X,x}$| will be denoted k(x), and the image of |$\mathcal{O}_{X,x}^+$| in k(x) by |$k(x)^+$|⁠. The residue field of |$k(x)^+$| will be denoted |$\widetilde{k(x)}$|⁠.

2. GENERAL TOPOLOGY

In this section we will gather together various existing definitions and results that we will need from general topology, mostly using either [6] or [10] as references.

2.1. Basic definitions

For the reader’s convenience we recall several definitions that will be used in this article.  

If X is a locally spectral space, and |$x,y\in X$| we will write |$y\succ x$| if |$x\in \overline{\{y\}}$|⁠, that is, if x is a specialisation of y. We will also write G(x) for the set of generalisations of x.  

If f is topologically proper, then preimages of quasi-compact sets are quasi-compact [4, Section 10.2, Proposition 6], and if X is Hausdorff and Y is locally compact, then the converse holds [4, Section 10.3, Proposition 7].

2.2. Sheaf cohomology on spectral spaces

The following result will be used constantly:  

2.3. Dimensions of spectral spaces

The dimension theory of locally spectral spaces works as in the case of schemes.  

We will need the following generalisation of Grothendieck vanishing:  

2.4. Separated quotients

Let X be a valuative space, and let |$[X]$| denote its set of maximal points, that is, points such that |$G(x)=\{x\}$|⁠. Since every point of a valuative space admits a maximal generalisation [6, Chapter 0, Remark 2.3.2], taking a point to its maximal generalisation induces a surjective map

and we equip |$[X]$| with the quotient topology. Recall that a topological space is T₁ if for any two distinct points, each has an open neighbourhood not containing the other.

Note that the space |$[X]$| is generally no longer valuative, since it does not admit a basis of quasi-compact opens.  

Equivalently, it is the preimage of an open (resp. closed) subset of |$[X]$| under the separation map ν. Note that the complement of an overconvergent open subset is an overconvergent closed subset, and vice versa.  

3. GERMS OF ADIC SPACES

In this section we introduce the category of germs of adic spaces. This will be the category in which we work for the rest of this article.

3.1. Standing hypotheses

We use Huber’s theory of adic spaces, see either [9] or [10, Chapter 1] for an introduction. We will assume that all adic spaces are analytic in the sense of [10, Section 1.1]. That is, each point |$x\in X$| will have an open affinoid neighbourhood |$x\in \mathrm{Spa}\left(R,R^+\right)\subset X$| such that R is Tate. This implies that all morphisms of adic spaces are adic in the sense of [10, Section 1.2]. We will let Ad denote the category of analytic adic spaces. Note in particular the standing assumption [10, (1.1.1)], which implies that for all complete Huber pairs |$(R,R^+)$| we consider, R will be Noetherian.

3.2. Germs of adic spaces

In [10, Section 1.10] Huber introduces the notion of a pseudo-adic space, which roughly speaking consists of an adic space X, together with a ‘reasonably nice’ subspace |$X\subset \boldsymbol{X}$|⁠. We will work instead with germs of adic spaces along closed subsets.  

We can construct a category Germ of germs of adic spaces in the usual way. We first consider the category of pairs |$(X,\boldsymbol{X})$| as in Definition 3.1, where morphisms are commutative squares. We then declare a morphism |$j:(X,\boldsymbol{X})\rightarrow (Y,\boldsymbol{Y})$| to be a strict neighbourhood if j is an open immersion and |$j(X)=Y$|⁠. Finally, we localise the category of pairs at the class of strict neighbourhoods (it is easy to verify that a calculus of right fractions exists). If |$(X,\boldsymbol{X})$| is a pair, we will often abuse notation and write X for |$(X,\boldsymbol{X})$|⁠, considered as an object in the category Germ.  

The assignment |$X\mapsto (X,X)$| induces a fully faithful functor from Ad to Germ . We can also consider any pair |$(X,\boldsymbol{X})$| as in Definition 3.1 as a pseudo-adic space in the sense of Huber [10, Section 1.10], thus it makes sense to consider any of the following properties of morphisms of such pairs:

1. locally of finite type, locally of ⁺weakly finite type and locally of weakly finite type;

2. quasi-compact, quasi-separated, coherent and taut;

3. an open immersion, closed immersion and locally closed immersion;

4. separated, partially proper and proper;

5. smooth and étale.

For example, a morphism of pairs |$f:(X,\boldsymbol{X}) \rightarrow (Y,\boldsymbol{Y})$| is smooth if and only if |$f:\boldsymbol{X}\rightarrow \boldsymbol{Y}$| is smooth in the sense of [10, Section 1.6], and X is open in |$f^{-1}(Y)$|⁠. It is easily checked that all of these properties descend to the category Germ of germs. While ‘analytic’ properties of a germ X are generally defined via the ambient adic space X, ‘topological’ properties are generally defined using the topological space X itself. In particular, a point of a germ will be a point of X, and a sheaf on a germ will be a sheaf on X. Note that the underlying topological space of any germ X is locally spectral, and morphism of germs induces a spectral map between the underlying topological spaces.  

It is perhaps worth carefully recalling the definitions of the different types of immersions for adic spaces and germs. Following [10, (1.4.1)] a closed analytic subspace of an adic space X is one defined by a coherent sheaf of ideals |$\mathcal{I}\subset \mathcal{O}_X$|⁠. A morphism |$f:X\rightarrow Y$| of adic spaces is a closed immersion if it is isomorphic to the inclusion of a closed analytic subspace, and a locally closed immersion if it factors as the composition of a closed immersion followed by an open immersion.

A (locally) closed immersion of germs is one that has a representative |$f:(X,\boldsymbol{X}) \rightarrow (Y,\boldsymbol{Y})$| as a morphism of pairs such that |$f:\boldsymbol{X}\rightarrow \boldsymbol{Y}$| is a locally closed immersion of adic spaces, and X is (locally) closed in Y. Finally, an open immersion of germs is one that has representative |$f:(X,\boldsymbol{X}) \rightarrow (Y,\boldsymbol{Y})$| a morphism of pairs such that |$f:\boldsymbol{X}\rightarrow \boldsymbol{Y}$| is an open immersion of adic spaces, and X is open in Y. Note that a locally closed immersion of germs, which is an open immersion on the underlying topological spaces, need not be an open immersion.

We will also use the following elementary fact constantly:

As with adic spaces, or pseudo-adic spaces, fibre products in general are not representable in Germ. However, they will be representable if at least one of the morphisms is locally of weakly finite type. If |$X\overset{f}{\rightarrow} Z \leftarrow Y$| is a diagram of germs, represented by a diagram

of pairs, with f, say, locally of weakly finite type, then the fibre product |$X\times_Y Z$| is represented by the pair

as germs over κ.

If X is a germ with ambient adic space X we will write |$\mathcal{O}_X:=\mathcal{O}_{\boldsymbol{X}}\!\!\mid_X$|⁠. We can similarly extend the notions of local rings, residue fields, et cetera, to germs of spaces. For example, if |$x\in X$| is a point of a germ, then we may speak of the local ring |$\mathcal{O}_{X,x}$|⁠, the residue field k(x) and the residue valuation ring |$k(x)^+\subset k(x)$|⁠.

3.3. Local germs

We recall from [10] the definition of an (analytic) affinoid field. In the theory of analytic adic space, these play a role roughly analogous to that played by local rings in algebraic geometry.

Note that the condition on the topology of k is equivalent to requiring that |$(k,k^+)$| is a Tate pair.

Note that the points of a local germ are totally ordered by generalisation.

Let |$f:X\rightarrow Y$| be a morphism of germs, locally of weakly finite type. We may therefore take the fibre product of f with any other morphism g. For any point |$y\in Y$|⁠, we define

which is locally of weakly finite type over the local germ |$\mathrm{Spa}\left(\kappa(y)\right)\cap Y$|⁠. Note that the underlying topological space of |$X_{(y)}$| is equal to |$f^{-1}(G(y))$|⁠, and it contains the space |$f^{-1}(y)$| as a closed subspace, equal to the closed fibre of the natural map

The following is then just a rephrasing of Corollary 2.6:

is an isomorphism.

4. PROPER PUSHFORWARDS ON GERMS OF ADIC SPACES

We can now define proper pushforwards for adic spaces, following Huber.

4.1. Sections with proper support

Let |$f:X \rightarrow Y$| be a morphism of germs, separated and locally of ⁺weakly finite type. Let |$\mathscr{F}$| be a sheaf on X, |$V\subset Y$| an open subset and |$s\in \Gamma(f^{-1}(V),\mathscr{F})$| a section. Then the support

is a germ of an adic space (as it is a closed subset of |$f^{-1}(V)$|⁠), and it therefore makes sense to ask whether or not the natural map |$\mathrm{supp}(s)\rightarrow V$| is proper.

to be the subsheaf consisting of sections |$s\in \Gamma(V,f_*\mathscr{F})=\Gamma(f^{-1}(V),\mathscr{F})$| whose support is proper over V.

We will sometimes denote |${\rm H}^0(Y,f_!(-))$| by either |${\rm H}^0_c(X/Y,-)$| or |$\Gamma_c(X/Y,-)$|⁠. If |$f:X\rightarrow Y$| is partially proper, then the support of |$s\in \Gamma(V,f_*\mathscr{F})$| is proper over V if and only if it is quasi-compact over V.

As a first example, we can show that this definition recovers the usual extension by zero functor for open immersions.  

4.2. Comparison with van der Put’s definition

Whenever |$f:X\rightarrow Y$| is a morphism of adic spaces of finite type over a discretely valued affinoid field |$(k,k^\circ)$|⁠, and the base Y is affinoid, a definition of |${\rm H}^0_c(X/Y,-)$| has already been given in [15, Definitions 1.4]. In fact, van der Put worked with rigid analytic spaces rather than adic spaces, but since the underlying topoi are the same [10, (1.1.11)] we can transport his definition to the adic context.

Recall that if |$U,V\subset X$| are open affinoids, we write

if there exists a closed immersion |$V\rightarrow {\mathbb D}^n_Y(0;1)$| over Y such that |$U\subset {\mathbb D}^n_Y(0;1^-)$|⁠.

(van der Put)

4.3. Basic properties of proper pushforwards

The following properties of |$f_!$| and |$\Gamma_c(X/Y,-)$| can be verified exactly as in [10, Proposition 5.2.2]:  

4.4. Derived proper pushforwards for partially proper morphisms

For partially proper morphisms only, we now define |$\mathbf{R}f_!$| as the derived functor of |$f_!$|⁠.  

We will also write |$\mathbf{R}\Gamma_c(X/Y,-)$| for the total derived functor of |${\rm H}^0_c(X/Y,-)$|⁠, and |${\rm H}^q_c(X/Y,-)$| for the cohomology groups of this complex.

To show that these derived proper pushforwards compose correctly, we can relate them to ordinary pushforwards as in [10, Section 5.3].  

The first claim was proved in [10, Lemma 5.3.3], and the second part is shown in exactly the same way as Huber does in the étale case, using the following lemma:  

The following two corollaries of Lemma 4.8 are proved word for word as in their étale counterparts [10, Propositions 5.3.7 and 5.3.8]:  

4.5. Base change theorems

We can also use Lemma 4.8 to describe the fibres of |$\mathbf{R}f_!$| by combining it with Corollary 3.9.

In particular, this says that whenever y is a maximal point, the natural map

is an isomorphism. This is not true in general if y is not maximal; we shall give a counterexample in Section 7. Nonetheless, we do have the following base change result:

is an isomorphism.

4.6. Cohomological amplitude

If |$f:X\rightarrow Y$| is a partially proper morphism between germs, then we have defined the functor

Moreover, if X and Y are finite-dimensional, then this will extend to a functor on the unbounded derived categories.

has cohomological amplitude contained in |$[0,\dim X]$|⁠.

4.7. Mayer–Vietoris for proper pushforwards

Let |$f:X\rightarrow Y$| be a partially proper morphism between finite-dimensional germs, and consider an open hypercover

of X by overconvergent open subsets. Thus each U_n is partially proper over Y by Lemma 3.4. Let |$j_n:U_n\rightarrow X$| denote the given morphism (which is a disjoint union of the inclusion of overconvergent open subsets of X), and |$f_n:U_n \rightarrow Y$| the composition |$f\circ j_n$|⁠. Suppose that we have a sheaf |$\mathscr{F}$| on X. Then there is a resolution

of |$\mathscr{F}$|⁠, coming from the fact that |$U_\bullet\rightarrow X$| is a hypercover. By Corollary 4.16 we can apply |$\mathbf{R}f_!$| to this resolution, and by Corollary 4.12 we know that |$\mathbf{R}f_!\circ j_{n!}=\mathbf{R}f_{n!}$| (note that |$j_{n!}$| is exact as it is the extension by zero along an open immersion). By Proposition 4.15 the cohomological dimension of |$\mathbf{R}f_{n!}$| is bounded independently of n, and we therefore obtain a convergent second quadrant spectral sequence

in the category |$\mathbf{Sh}(Y)$| of abelian sheaves on Y. The terms |$\mathbf{R}^qf_{n!}\mathscr{F}|_{U_{n}} $| can also be made slightly more explicit: if |$U_n=\coprod_{m} U_{n,m}$| with each |$U_{n,m}$| an open subset of X, and |$f_{n,m}:U_{n,m}\rightarrow Y$| is the restriction of f to |$U_{n,m}$|⁠, then

by Corollary 4.11.

4.8. Module structures on proper pushforwards

Let |$f:X\rightarrow Y$| be a partially proper morphism of germs, and suppose that we have sheaves of rings |$\mathcal{A}_X$| and |$\mathcal{A}_Y$| on X and Y, respectively, together with a morphism |$\mathcal{A}_Y\rightarrow f_*\mathcal{A}_X$| making f into a morphism of ringed spaces. The principal example for us will, of course, be the structure sheaves |$\mathcal{A}_X=\mathcal{O}_X$| and |$\mathcal{A}_Y=\mathcal{O}_Y$|⁠.  

4.9. Comparison with separated quotients

The next crucial result we need is a comparison between |$\mathbf{R}f_!$|⁠, as we have defined it here, and the classical notion of proper pushforwards for maps between locally compact topological spaces. To prepare for this, we note the following property of separated quotients:

Now, let |$f:X\rightarrow Y$| be a partially proper morphism of germs. Then we have a diagram

relating the separated quotients of X and Y. If Y is taut, then so is X by [10, Lemma 5.1.4 ii)], and hence Proposition 4.21 applies to both X and Y. In this situation, we may consider the usual functor |$[f]_!$| of sections whose support is topologically proper over |$[Y]$|⁠, together with its total derived functor |$\mathbf{R}[f]_!:\textbf{D}^+([X])\rightarrow \textbf{D}^+([Y])$|⁠.

5. KIEHL PARTIAL PROPERNESS AND COHOMOLOGICAL DIMENSION

In this section we will prove a result on the cohomological dimension of coherent sheaves for certain partially proper morphisms. The condition we require is in fact the original definition of partial properness given by Kiehl [11]. Any such morphism has to be locally of finite type, which excludes many examples of partially proper morphisms (in particular, Huber’s universal compactifications, constructed in [10], are generally not locally of finite type). For morphisms locally of finite type, partial properness in the sense of Kiehl coincides with partial properness in many cases of interest, although it is still open whether or not the two coincide in general.

5.1. Polydiscs and affine spaces over germs

To begin with, we recall the definitions of polydiscs and affine spaces over a germ Y. To begin with, suppose that |$Y=\mathrm{Spa}\left(R,R^+\right)$| is a Tate affinoid adic space, then we have the usual definition

of the closed unit polydisc over Y, using multi-index notation |$\boldsymbol{z}=(z_1,\ldots,z_d)$|⁠. If |$\varpi\in R$| is a quasi-uniformiser, and |$q\in {\mathbb Q}_{\geq 0}$|⁠, we write |$q=\frac{a}{b}$| in lowest terms and define

to be the ‘closed disc of radius |$\left\vert\varpi\right\vert^{q}$|’. Similarly, if q < 0, we set |$n=\lfloor q\rfloor$|⁠, write |$q-n=\frac{a}{b}$| in lowest terms and set

We then define

as well as analogous open discs

‘of radius |$\left\vert\varpi\right\vert^q$|’.

More generally, if Y is an adic space admitting an element |$\varpi\in \Gamma(Y,\mathcal{O}_Y)$|⁠, which is a quasi-uniformiser locally around every point |$y\in Y$|⁠, then we can define any of

by gluing. If Y is a germ admitting a similar global quasi-uniformiser |$\varpi\in\Gamma(Y,\mathcal{O}_Y)$|⁠, then, locally on some ambient adic space Y, we can define analogous spaces over Y by pulling back from those defined over Y, for example, |$\mathbb{D}^d_Y(0;\left\vert\varpi\right\vert^q)$| is defined by the Cartesian diagram

Finally, the definitions of |${\mathbb D}^d_Y(0;1), {\mathbb D}^d_Y(0;1^-)$| and |${\mathbb A}_Y^{d,\mathrm{an}}$| are independent of the choice of quasi-uniformiser, and hence the definition globalises to give |${\mathbb D}^d_Y(0;1), {\mathbb D}^d_Y(0;1^-)$| and |${\mathbb A}_Y^{d,\mathrm{an}}$| over an arbitrary germ Y. It is straightforward to check that if |$Y^{\prime}\rightarrow Y$| is any morphism of germs, then there is a natural Cartesian diagram

as well as the obvious analogues for |${\mathbb D}^d_Y(0;1^-)$| and |${\mathbb A}_Y^{d,\mathrm{an}}$|⁠.

in the category of (not necessarily analytic) adic spaces. Here |${\mathbb Z}$| and |${\mathbb Z}[\boldsymbol{z}]$| are given the discrete topology, and |${\mathbb Z}\unicode{x27E6} \boldsymbol{z} \unicode{x27E7}$| the z-adic topology. We will not use these constructions in this article.

To define Kiehl’s version of partial properness, we use the following result:

1. Locally on X and Y, there exist a quasi-uniformiser ϖ on Y, open covers |$\{V_i\}_{i\in I}$| and |$\{U_i\}_{i\in I}$| of X, integers |$N_i\geq 1$|⁠, closed immersions |$U_i\hookrightarrow {\mathbb D}_Y^{N_i}(0;1)$| over Y and integers |$m_i\geq 1$| such that |$V_i\subset U_i\cap {\mathbb D}_Y^{N_i}(0;\left\vert\varpi\right\vert^{\frac{1}{m_i}})$|⁠.

2. Locally on X and Y, there exist open covers |$\{V_i\}_{i\in I}$| and |$\{U_i\}_{i\in I}$| of X, integers |$N_i\geq 1$| and closed immersions |$U_i\hookrightarrow {\mathbb D}_Y^{N_i}(0;1)$| over Y such that |$V_i\subset U_i\cap {\mathbb D}_Y^{N_i}(0;1^-)$|⁠.

3. Locally on X and Y, there exist an open cover |$\{U_i\}_{i\in I}$| of X, integers |$N_i\geq 1$| and closed immersions |$U_i\hookrightarrow {\mathbb D}_Y^{N_i}(0;1^-)$| over Y.

We now set |$V_{i,n}$| to be |$U_{i,n}\cap {\mathbb D}^{2N_i}_Y(0;\left\vert\varpi\right\vert^{\frac{1}{n-1}})$|⁠, and we claim that the |$V_{i,n}$| still cover X. But |$V_{i,n}$| is defined in |$U_{i,n}$| by the two equivalent conditions

Thus |$V_{i,n}=U_{i,n-1}$|⁠, and so the |$V_{i,n}$| cover X as required.

1. If f is partially proper in the sense of Kiehl, then it is partially proper.

2. If |$f:X\rightarrow Y$| is partially proper in the sense of Kiehl, and |$g:Z\rightarrow Y$| is any morphism, then |$X\times_YZ \rightarrow Z$| is partially proper in the sense of Kiehl.

3. If f and g are partially proper in the sense of Kiehl, then so is |$g\circ f$|⁠. If |$g\circ f$| and g are partially proper in the sense of Kiehl, then so is f.

4. For any Y, the maps |${\mathbb D}^d_Y(0;1^-)\rightarrow Y$| and |${\mathbb A}^d_Y\rightarrow Y$| are partially proper in the sense of Kiehl.

5. Any closed immersion is partially proper in the sense of Kiehl.

6. If X and Y are quasi-separated adic spaces locally of finite type over a discretely valued affinoid field, then any partially proper map |$f:X\rightarrow Y$| is partially proper in the sense of Kiehl [10, Remark 1.3.19].

7. Any map that is partially proper in the sense of Kiehl is locally of finite type.

The main result of this section is then the following:  

Note that Theorem 5.6 is stated only for adic spaces, not for more general germs. We will therefore be dealing with adic spaces until the end of Section 5.3 below.

5.2. Cohomology of coherent sheaves

Before embarking on the proof of Theorem 5.6, we will need a couple of preliminary results on the cohomology of coherent sheaves on certain kinds of adic spaces. The first is the analogue of Theorems A and B for suitable ‘quasi-Stein’ adic spaces.  

We will also need a slight generalisation of [2, Proposition 1.3.6], giving conditions for the structure sheaf to have vanishing higher direct images along the separation map.  

Thanks to [6, Chapter 0, Corollary 2.3.31], this implies that |$[U]$| contains an open neighbourhood of |$\mathrm{sep}(x)$| in |$[X]$|⁠. Also note that, by definition, any very good adic space is necessarily taut.  

5.3. Proof of Theorem 5.6

We now return to the proof of Theorem 5.6, and there are two immediate reductions that we can make. First of all, we can assume that the base Y is Tate affinoid, and secondly we can assume (by Corollary 4.17) that X admits a closed immersion into some open unit polydisc |${\mathbb D}^N_Y(0;1^-)$|⁠.

Moreover, using Proposition 5.3 we can assume that |$X={\mathbb D}^N_Y(0;1^-)\cap Z$| for some closed immersion |$Z \hookrightarrow {\mathbb D}^N_Y(0;1)$|⁠, and that |$\mathscr{F}$| extends to Z. This allows us to make one further reduction.

for some |$m\geq 0$| and some coherent sheaf |$\mathscr{F}_1$| extending to Z. We therefore deduce that |$\mathbf{R}^qf_!\mathscr{F}\overset{\cong}{\longrightarrow} \mathbf{R}^{q+1}f_!\mathscr{F}_1$| for all q > d. Repeating the argument, we find a coherent sheaf |$\mathscr{F}_m$|⁠, extending to Z, such that |$\mathbf{R}^qf_!\mathscr{F}\overset{\cong}{\longrightarrow} \mathbf{R}^{q+m}f_!\mathscr{F}_m$| for all q > d. For m large enough we have |$\mathbf{R}^{q+m}f_!\mathscr{F}_m=0$| by Proposition 4.15, and hence |$\mathbf{R}^qf_!\mathscr{F}=0$| as required.

Now, thanks to Corollary 4.13, we have, for any |$y\in Y$|⁠, an identification

If we let |$\mathrm{sep}_{X_{(y)}}:X_{(y)}\rightarrow [X_{(y)}]$| denote the separation map, then Corollary 4.24 gives

Now applying Corollary 4.25, it suffices to show that |$\mathbf{R}^q\mathrm{sep}_{X_{(y)}*}(\mathcal{O}_X|_{X_{(y)}})=0$| for q > 0. Let |$y\in U\subset Y$| be a Tate open affinoid neighbourhood of y, with preimage |$f^{-1}(U)\subset X$| and separation map |$\mathrm{sep}_{f^{-1}(U)}:f^{-1}(U)\rightarrow [f^{-1}(U)]$|⁠.

and each |$f^{-1}(U)\cap {\mathbb D}^N_U(0;\left\vert\varpi\right\vert^{\frac{1}{n+1}-})$| is an overconvergent open subset of |$f^{-1}(U)$|⁠. Thus, for every point |$x\in f^{-1}(U)$|⁠, there is some n such that

and |$f^{-1}(U)\cap {\mathbb D}^N_U(0;\left\vert\varpi\right\vert^{\frac{1}{n}})$| is a Tate affinoid, since U is.

Thus Proposition 5.10 tells us that |$\mathbf{R}^q\mathrm{sep}_{f^{-1}(U)*}(\mathcal{O}_X|_{f^{-1}(U)})=0$| for q > 0, and Theorem 5.6 reduces to the following result:

is an isomorphism.

5.4. The case of overconvergent germs

We do not know whether Theorem 5.6 holds if Y is replaced by an arbitrary germ. We do at least have the following special case:  

Proof.

Choose an ambient adic space Y of Y. By localising on Y we may assume that it is Tate affinoid, with quasi-uniformiser |$\varpi\in \Gamma(\boldsymbol{Y},\mathcal{O}_{\boldsymbol{Y}})$|⁠. By Corollary 4.17 we may assume that X admits a closed immersion |$u:X\hookrightarrow {\mathbb D}^N_Y(0;1^-)$| for some n. We can therefore extend f to a diagram of pairs

such that:

• |$f\colon \boldsymbol{X}\rightarrow \boldsymbol{Y}$| is smooth, and |$X=f^{-1}(Y)$|⁠;

• |$\mathscr{F}$| extends to a coherent sheaf on X;

• |$u \colon\boldsymbol{X}\rightarrow {\mathbb D}^N_{\boldsymbol{Y}}(0;1^-)$| is a locally closed immersion.

Note that |$X=\pi^{-1}(Y)\cap \boldsymbol{X}$| as subspaces of |${\mathbb D}^N_{\boldsymbol{Y}}(0;1^-)$|⁠. Let |$\boldsymbol{U}\subset {\mathbb D}^N_{\boldsymbol{Y}}(0;1^-)$| be open subspace such that X is a closed analytic subspace of U.

Since X is a locally closed analytic subspace of |${\mathbb D}^N_{\boldsymbol{Y}}(0;1^-)$|⁠, it is closed under generalisations, and since Y is an overconvergent closed subset of Y, it follows that |$\pi^{-1}(Y)$| is an overconvergent closed subset of |${\mathbb D}^N_{\boldsymbol{Y}}(0;1^-)$|⁠. Hence |$X=\pi^{-1}(Y)\cap \boldsymbol{X}$| is closed under generalisations inside |${\mathbb D}^N_{\boldsymbol{Y}}(0;1^-)$|⁠, that is, it is an overconvergent closed subset of |${\mathbb D}^N_{\boldsymbol{Y}}(0;1^-)$|⁠. It therefore follows from [6, Chapter 0, Proposition 2.3.17] that each |$X\cap {\mathbb D}^N_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}})$| admits a basis of neighbourhoods in |${\mathbb D}^N_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}})$| consisting of overconvergent open subsets. In particular there exist overconvergent open subsets |$V_n\subset{\mathbb D}^N_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}})$| such that

$$ X\cap {\mathbb D}^N_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}}) \subset V_n\subset \boldsymbol{U}\cap {\mathbb D}^N_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}}) . $$

Since V_n is overconvergent, it is the preimage of an open subset of |$[{\mathbb D}^N_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}})]$| via the separation map. Thus arguing as in the proof of Proposition 5.10, we see that V_n can be covered by open subsets |$V_{n,i}$| admitting closed immersions into |${\mathbb D}^N_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}})\times_{\boldsymbol{Y}} {\mathbb D}^{M_{n,i}}_{\boldsymbol{Y}}(0;1^-)$| over |${\mathbb D}^N_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}})$|⁠. It follows that

$$ V_n^-:=V_n\cap {\mathbb D}^{N}_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}-}) $$

admits a covering by open subsets |$V^-_{n,i}:=V_{n,i}\cap {\mathbb D}^{N}_{\boldsymbol{Y}}(0;\left\vert\varpi\right\vert^{\frac{1}{n}-})$|⁠, each of which admits a closed immersion into |${\mathbb D}^{2N+M_{n,i}}_{\boldsymbol{Y}}(0;1^-)$| over Y. Thus the |$V^-_{n,i}$| are a collection of open subspaces of |${\mathbb D}^N_{\boldsymbol{Y}}(0;1^-)$|⁠, covering X, and there are closed immersions

$$ \boldsymbol{X}\cap V^-_{n,i}\rightarrow V^-_{n,i} \rightarrow {\mathbb D}_{\boldsymbol{Y}}^{2N+M_{n,i}}(0;1^-) $$

of adic spaces over Y.

Therefore, by localising on X, and once more appealing to Corollary 4.17, we may reduce to the case that f extends to a diagram of pairs

such that |$u:\boldsymbol{X}\rightarrow {\mathbb D}^N_{\boldsymbol{Y}}(0;1^-)$| is a closed immersion, and |$\mathscr{F}$| extends to X. In this case, since |$Y\subset \boldsymbol{Y}$| is overconvergent, and |$X=f^{-1}(Y)$| inside X by smoothness of f, we can combine Lemma 4.14 with Theorem 5.6 to conclude.

6. THE TRACE MAP

In this section, we construct a trace map for the class of smooth morphisms that are partially proper in the sense of Kiehl, and whose target is an overconvergent and finite-dimensional germ. This is a morphism

in the derived category of |$\mathcal{O}_Y$|-modules, satisfying the conditions outlined in the introduction. We closely follow the argument of [15], see also [3, 5].

6.1. The relative open unit polydisc

We first construct a trace map when |$X={\mathbb D}^d_Y(0;1^-)$| is the relative open unit polydisc over a Tate affinoid adic space |$Y=\mathrm{Spa}\left(R,R^+\right)$|⁠. Choose a quasi-uniformiser |$\varpi\in R^\times\cap R^{\circ\circ}$|⁠. Since |$X={\mathbb D}^d_Y(0;1^-)$| is partially proper over Y, the support of a section of some sheaf |$\mathscr{F}$| on X is proper over Y if and only if it is quasi-compact over Y, if and only if it is quasi-compact. The closure |$\overline{{\mathbb D}}_n$| of |${\mathbb D}_Y^d(0;\left\vert\varpi\right\vert^{\frac{1}{n}})$| inside |${\mathbb D}^d_Y(0;1^-)$| is quasi-compact, and moreover any quasi-compact subset of |${\mathbb D}^d_Y(0;1^-)$| has to be contained in |$\overline{{\mathbb D}}_n$| for some n. Thus, if we let |${\rm H}^q_Z(X,-)$| denote cohomology groups with support in a closed subset |$Z\subset X$|⁠, we find that

for any sheaf |$\mathscr{F}$| on |${\mathbb D}^d_Y(0;1^-)$|⁠.

Using Proposition 5.7, we can see that |${\rm H}^q({\mathbb D}^d_Y(0;1^-),\mathscr{F})=0$| for any coherent |$\mathcal{O}_{{\mathbb D}^d_Y(0;1^-)}$|-module |$\mathscr{F}$| and any q > 0. Thus we deduce isomorphisms

We can cover |${\mathbb D}^d_Y(0;1^-)\backslash \overline{{\mathbb D}}_n$| by the spaces

each of which admits a closed immersion into an open polydisc over Y. Again, Proposition 5.7 implies that coherent sheaves have vanishing higher cohomology groups on each |$U_{i,n}$|⁠. The same reasoning applies to all intersections |$\cap_{i\in I} U_{i,n}$|⁠, so we can compute the cohomology of |$\mathscr{F}$| on |${\mathbb D}^d_Y(0;1^-) \backslash \overline{{\mathbb D}}_n$| as the cohomology of the Čech complex

In the particular case when |$\mathscr{F}=\omega_{{\mathbb D}^d_Y(0;1^-)/Y}$|⁠, we can therefore give a complete description of the cohomology groups |${\rm H}^q_c({\mathbb D}^d_Y(0;1^-)/Y,\mathscr{F})$| as follows. Choose coordinates |$z_1,\ldots,z_d$| on Y and let |$R\langle z_1^{-1},\ldots,z_d^{-1}\rangle^\dagger$| denote the set of overconvergent series in |$z_1^{-1},\ldots,z_d^{-1}$|⁠, that is, series of the form

for which there exists |$n\geq 1$| such that |$r_{i_1,\ldots,i_d}^n\varpi^{i_1+\ldots+i_d}\rightarrow 0$| as |$(i_1,\ldots,i_d)\rightarrow -\infty$|⁠. Then

We can therefore define the trace map

as in [15, Section 2.4] or [3, Section 2.1]. We can then globalise this construction to define

whenever the base Y is an adic space. When Y is an overconvergent germ, we pullback to Y from its ambient adic space Y using Lemma 4.14. Also note that by Corollary 5.14 we may view the trace map as a morphism

in |$\textbf{D}^b(\mathcal{O}_Y)$|⁠. The verification of the following is straightforward:

1. The trace map

   $$ \mathrm{Tr}_{z_1,\ldots,z_d}: \mathbf{R}^df_!\omega_{{\mathbb D}^{d}_Y(0;1^-)/Y} \rightarrow \mathcal{O}_Y $$

   vanishes on the image of |$\mathbf{R}^df_!\Omega^{d-1}_{{\mathbb D}^{d}_Y(0;1^-)/Y}$|⁠, and hence induces a map

   $$ \mathrm{Tr}_{z_1,\ldots,z_d}: \mathbf{R}f_!\Omega^\bullet_{{\mathbb D}^{d}_Y(0;1^-)/Y}[2d] \rightarrow \mathcal{O}_Y. $$

   This map is an isomorphism.
2. The trace map is compatible with composition in the following sense: let |$(z_1,\ldots,z_d)$| be coordinates on |${\mathbb D}^{d}_Y(0;1^-)$|⁠, let |$1\leq e\leq d$|⁠, and let |$h:{\mathbb D}^{d}_Y(0;1^-) \rightarrow {\mathbb D}^{e}_Y(0;1^-)$| be the projection |$(z_1,\ldots,z_d)\mapsto (z_1,\ldots,z_e)$|⁠. Let |$f:{\mathbb D}^{d}_Y(0;1^-)\rightarrow Y $| and |$g: {\mathbb D}^{e}_Y(0;1^-)\rightarrow Y$| be the canonical identification

   $$ \omega_{{\mathbb D}^{d}_Y(0;1^-)/Y} = h^*\omega_{{\mathbb D}^{e}_Y(0;1^-)/Y} \otimes \omega_{{\mathbb D}^{d}_Y(0;1^-)/{\mathbb D}^{e}_Y(0;1^-)}, $$

   and the resulting identification

   $$ \mathbf{R}f_!\left(\omega_{{\mathbb D}^{d}_Y(0;1^-)/Y} \right)[d] = \mathbf{R}g_!\left(\omega_{{\mathbb D}^{e}_Y(0;1^-)/Y} \otimes \mathbf{R}h_!\omega_{{\mathbb D}^{d}_Y(0;1^-)/{\mathbb D}^{e}_Y(0;1^-)}[d-e] \right)[e], $$

   we have

   $$ \mathrm{Tr}_{z_1,\ldots,z_d} = \mathrm{Tr}_{z_1,\ldots,z_e} \circ \mathbf{R}g_!\left(\mathrm{id} \otimes \mathrm{Tr}_{z_{e+1},\ldots,z_d}\right). $$

We will see later on that |$\mathrm{Tr}_{z_1,\ldots,z_d}$| is independent of the choice of coordinates |$z_1,\ldots,z_d$|⁠; for now we record a special case of this.

for sections |$w_i:{\mathbb D}^e_Y(0;1^-) \rightarrow {\mathbb D}^d_Y(0;1^-)$| of the natural projection. Then |$\mathrm{Tr}_{z_1,\ldots,z_d}=\mathrm{Tr}_{z_1^{\prime},\ldots,z^{\prime}_d}$|⁠.

is entirely similar, and the analogues of Proposition 6.1 and Lemma 6.2 hold.

6.2. Duality for regular immersions

To extend the trace map from open polydiscs to more general morphisms, we will need a form of duality for regular closed immersions. Luckily, this follows quite quickly from the scheme-theoretic case.

induces a t-exact equivalence of triangulated categories

compatible with internal homs.

is t-exact. To see that it is an equivalence, we consider the left adjoint

Essential surjectivity now follows from the fact that |$\mathcal{O}_X$| is R-flat, and full faithfulness follows from the fact that the adjunction map

is an isomorphism for any |$\mathscr{F}\in \textbf{D}^+_{\mathrm{coh}}(\mathcal{O}_X)$|⁠. Compatibility with internal homs now follows from the fact that the left adjoint |$-\otimes^{\mathbf{L}}_R \mathcal{O}_X$| is monoidal.

Recall that on a locally ringed space |$(X,\mathcal{O}_X)$|⁠, a perfect complex of |$\mathcal{O}_X$|-modules is one that is locally quasi-isomorphic to a bounded complex of finite free |$\mathcal{O}_X$|-modules. Similarly, if A is a ring, then a perfect complex of A-modules is a complex quasi-isomorphic to a bounded complex of finite projective A-modules. (Thus being a perfect complex of A-modules is a priori stronger than being a perfect complex of |$\mathcal{O}_{\mathrm{Spec}\left(A\right)}$|-modules.) The categories of such objects are viewed as full subcategories of |$\textbf{D}(\mathcal{O}_X)$| and |$\textbf{D}(A)$|⁠, respectively.

in |$\textbf{D}(\mathcal{O}_Y)$|⁠, natural in |$\mathscr{F}$|⁠. This is compatible with composition, in the sense that if |$v:Y\rightarrow Z$| is a regular closed immersion of codimension d, and |$\mathfrak{n}_{Y/Z}$| (resp. |$\mathfrak{n}_{X/Z}$|⁠) the determinant of its normal bundle (resp. the normal bundle of X in Z), then, via the identification |$\mathfrak{n}_{X/Z}= \mathfrak{n}_{X/Y}\otimes_{\mathcal{O}_X} u^*\mathfrak{n}_{Y/Z}$|⁠, the diagram

commutes.

that this does indeed land in |$\textbf{D}^+_{\mathrm{coh}}(\mathcal{O}_X)$| can be checked locally on Y, whence it follows from Lemma 6.4 together with the corresponding result for schemes [8, Chapter III, Proposition 6.1]. Next, the canonical morphism

induces, for any |$\mathscr{F}\in \textbf{D}^+_{\mathrm{coh}}(\mathcal{O}_X)$|⁠, a map

which we claim induces an isomorphism

for any |$\mathscr{G}\in \textbf{D}^+_{\rm coh}(\mathcal{O}_Y)$|⁠. Since the map |$\mathscr{F}\rightarrow u^\flat u_*\mathscr{F}$| is defined globally, the fact that it defines such an adjunction can be checked locally, when again it follows from Lemma 6.4 together with the analogous result for schemes [8, Chapter III, Theorem 6.7]. Now uniqueness of adjoints gives rise to a canonical isomorphism |$(v\circ u)^\flat\cong u^\flat\circ v^\flat$| whenever |$X\overset{u}{\rightarrow}Y \overset{v}{\rightarrow} Z$| is a pair of regular closed immersions between adic spaces, and this isomorphism can, locally, be identified with that from [8, Chapter III, Proposition 6.2].

The first claim therefore reduces to constructing a natural isomorphism

in |$\textbf{D}^+_{\mathrm{coh}}(\mathcal{O}_X)$|⁠. Given this, the second claim then boils down to showing that if |$X\overset{u}{\rightarrow}Y \overset{v}{\rightarrow} Z$| is a pair of regular closed immersions between adic spaces, then the diagram

commutes. Since the first claim in particular implies that the relative dualising complex |$u^\flat\mathcal{O}_Y$| is concentrated in a single degree, they may be jointly checked locally on Z. Thus we may assume, in the first case, that X is cut out by a global regular sequence in Y, and in the second case, that moreover Y is also cut out by a global regular sequence in Z. Under these assumptions, the claims are a consequence of the analogous results for schemes, in particular the calculation of |$u^\flat$| for a regular closed immersion in [8, Corollary 7.3].

6.3. Closed subspaces of open polydiscs

We will apply the results of Section 6.2 to a closed immersion |$u:X\rightarrow {\mathbb D}^N_Y(0;1^-)$| of adic spaces, over a finite-dimensional adic space Y, such that the composite |$f:=\pi\circ u$|

of u with the natural projection π is smooth of relative dimension d. Since |$\omega_{X/Y}\cong \mathfrak{n}_{X/{\mathbb D}^N_Y(0;1^-)}\otimes_{\mathcal{O}_X} u^*\omega_{{\mathbb D}^N_Y(0;1^-)/Y}$|⁠, by taking |$\mathcal{F}=\mathcal{O}_X$|⁠, tensoring both sides with |$\omega_{{\mathbb D}^N_Y(0;1^-)/Y}$|⁠, and using the projection formula, we obtain an isomorphism

Hence applying |$\mathbf{R}^d\pi_!$| gives an isomorphism

Restricting along |$\mathcal{O}_{{\mathbb D}^N_Y(0;1^-)}\rightarrow u_*\mathcal{O}_X$| gives a map

and finally composing with |$\mathrm{Tr}_{z_1,\ldots,z_N}$| for a choice of coordinates on |${\mathbb D}^N_Y(0;1^-)$| gives a trace map

Via Theorem 5.6 we may view this as a map

1. The induced map |$\mathrm{Tr}_{X/Y}: \mathbf{R}f_!\omega_{X/Y}[d]\rightarrow \mathcal{O}_Y$| does not depend on the choice of embedding |$u:X\hookrightarrow {\mathbb D}^N_Y(0;1^-)$| over Y.

2. Suppose that |$g\colon Y \rightarrow Z$| is a smooth morphism of relative dimension e, factoring through a closed embedding into some relative open disc |${\mathbb D}^{M}_Z(0;1^-)$|⁠. Then, via the identification |$\omega_{X/Z} = \omega_{X/Y}\otimes f^*\omega_{Y/Z}$|⁠, the diagram

   

   commutes.
3. The trace map vanishes on the image of

   $$ \mathbf{R}^df_!\Omega^{d-1}_{X/Y} \rightarrow \mathbf{R}^df_!\omega_{X/Y}, $$

   and hence descends to a map

   $$ \mathrm{Tr}_{X/Y}: \mathbf{R}f_!\Omega^\bullet_{X/Y}[2d]\rightarrow \mathcal{O}_Y. $$

Proof.

For ease of notation, we will drop |$(0;1^-)$| from the notation for open polydiscs during the proof. Given Lemma 6.2, part (1) is proved verbatim as in the case of closed subspace of analytic affine space over a height one affinoid field treated in [15, Theorem 3.5] (which is, in turn, based on [5, Section 5]). For part (2), we can choose a commutative diagram

with all horizontal arrows closed immersions, all vertical arrows the natural projections and the upper right-hand square Cartesian. Writing things out painfully explicitly, we need to show that the diagram below is commutative.

Commutativity of the two left-hand triangles follows from Lemma 6.7 and Proposition 6.1, respectively, and commutativity of the upper hexagon is straightforward. It therefore suffices to prove commutativity of the central octagon |$(\star)$|⁠, which essentially expresses the fact that the maps |$\mathrm{Tr}_v$| and |$\mathrm{Tr}_{{\mathbb D}^N}$| arising from the Cartesian upper right-hand square of (6.1) commute.

To prove this commutativity, we may remove |$\mathbf{R}\rho_!$| from every term appearing in this octagon, as well as tensor everything in sight by (an appropriate pullback of) |$\omega_{{\mathbb D}^M_Z/Z}^{\otimes -1}$|⁠, and shift by −M. We will also simplify notation slightly and replace |${\mathbb D}^M_Z$| by Z, thus the upper right-hand square becomes

Finally, we can replace |$u_*\mathcal{O}_X$| by |$\mathcal{O}_{{\mathbb D}^N_Y}$|⁠, which has the effect of removing the right-hand map labelled ‘|$\mathrm{res}$|’, in the diagram, thus turning the octagon into a heptagon. The diagram that we need to show commutes is therefore the one below, where c is the codimension of Y in Z.

Now, by localising on Z, we can assume that Y is defined in Z by a regular sequence |$f_1,\ldots,f_c\in \Gamma(Z,\mathcal{O}_Z)$|⁠. Choosing coordinates |$z_1,\ldots,z_N$| on |${\mathbb D}^N$|⁠, we may use |$z_1,\ldots,z_N$| and |$f_1,\ldots,f_c$| to trivialise the canonical and normal sheaves appearing in the above diagram. We then consider the Koszul resolution

$$ \mathcal{K}^\bullet_{Y/Z}:= \left[ \mathcal{O}_Z \longrightarrow \ldots \longrightarrow \mathcal{O}_Z^{\oplus \binom{c}{2}}\longrightarrow \mathcal{O}_Z^{\oplus c} \overset{(f_1,\ldots,f_c)}{\longrightarrow}\mathcal{O}_Z \right] $$

of |$v_*\mathcal{O}_Y$|⁠, viewed as being concentrated in the interval |$[-c,0]$|⁠. Thus the natural map |$\mathcal{O}_Z\rightarrow v_*\mathcal{O}_Y$| corresponds to the inclusion

$$ \iota_0:\mathcal{O}_Z \rightarrow \mathcal{K}^\bullet_{Y/Z} $$

of the term in degree 0, and the trace morphism |$\mathrm{Tr}_v$| corresponds to the canonical isomorphism of complexes

$$ \mathrm{can}\colon \mathcal{K}^\bullet_{Y/Z}\overset{\cong}{\longrightarrow} \underline{\mathrm{Hom}}( \mathcal{K}^\bullet_{Y/Z},\mathcal{O}_Z)[c] $$

(see [8, Section 7]). Of course, |$\pi^*\mathcal{K}^\bullet_{Y/Z}$| is a resolution of |$v_*\mathcal{O}_{{\mathbb D}^N_Y}$| as a |$\mathcal{O}_{{\mathbb D}^N_Z}$|-module, with similar descriptions of |$\mathrm{Tr}_v$| and |$\mathcal{O}_{{\mathbb D}^N_Z}\rightarrow v_*\mathcal{O}_{{\mathbb D}^N_Y}$|⁠. The diagram that we are required to show the commutativity of then becomes the following:

The claim now follows from the explicit description of |$\mathbf{R}\pi_!\mathcal{O}_{{\mathbb D}^N}$| and the construction of the trace map |$\mathrm{Tr}_{z_1,\ldots,z_N}$| in Section 6.1.

For part (3), the question is local on Y, and on X by Corollary 4.17. Hence we may assume that there exists a closed immersion |$X\hookrightarrow {\mathbb D}^N_Y(0;1^-)$| over Y. We may also assume that Y is Tate affinoid, with quasi-uniformiser |$\varpi\in\Gamma(Y,\mathcal{O}_Y)$|⁠.

Since X is smooth over Y, the module of differentials |$\Omega^1_{X/Y}$| is locally free. Since Y is affinoid, it follows from Proposition 5.7 that we may choose, for any |$x\in X$|⁠, functions |$t_1,\ldots,t_d\in \Gamma(X,\mathcal{O}_X)$| such that |${\rm d}t_1,\ldots,{\rm d}t_d$| form a basis of |$\Omega^1_{X/Y,x} \otimes_{\mathcal{O}_{X,x}}k(x)$|⁠, and thus (by Nakayama’s lemma) a basis of |$\Omega^1_{X/Y,x}$|⁠. The locus of points xʹ where |${\rm d}t_1,\ldots,{\rm d}t_d$| are not a basis of |$\Omega^1_{X/Y,x^{\prime}}$| is then a closed analytic subspace of X.

We claim that the complement of any such subspace locally admits a closed immersion into an open unit polydisc over Y. To see this, let us set |$X_n:=X\cap {\mathbb D}^N_Y(0;\left\vert\varpi\right\vert^{\frac{1}{n}})$| for each |$n\geq 0$|⁠. Then, for any |$f\in \Gamma(X_n,\mathcal{O}_X)$|⁠, the Zariski open subset |$D(f):=\{x\in X_n\mid f(x)\neq 0\}$| of X_n admits a closed immersion into |${\mathbb A}^{2,\mathrm{an}}_{X_n}$| via

$$ \left(f,\frac{1}{f}\right):D(f)\rightarrow {\mathbb A}^{2,\mathrm{an}}_{X_n}. $$

Hence, for any |$m\in {\mathbb Z}_{\leq 0}$|⁠, |$D(f)\cap {\mathbb D}^N_Y(0;\left\vert\varpi\right\vert^{\frac{1}{n}-})\cap {\mathbb D}^N_{X_n}(0;\left\vert\varpi\right\vert^{m-})$| admits a closed immersion into a unit polydisc over Y, which proves the claim.

We may therefore reduce to the case where |${\rm d}t_1,\ldots,{\rm d}t_d$| are a basis of |$\Omega^1_{X/Y}$| on the whole of X. It then follows from [10, Proposition 1.6.9 iii)] that the morphism |$\boldsymbol{t}:=(t_1,\ldots,t_d): X $|  |$ \rightarrow {\mathbb A}^{d,\mathrm{an}}_{Y}$| is étale. Further localising on X, we may assume that the image of this morphism lands inside |${\mathbb D}^d_Y(0;1^-)$|⁠. But now, by applying part (2) to the composition

$$ X \overset{\boldsymbol{t}}{\rightarrow} {\mathbb D}^d_Y(0;1^-)\overset{\pi}{\rightarrow} Y, $$

we deduce that the diagram

commutes. Since the diagram

also commutes, the claim therefore reduces to the case |$X={\mathbb D}^{d}_{Y}(0;1^-)$| that we have already handled.

6.4. Smooth morphisms of adic spaces

As a penultimate case, we construct the trace morphism when Y is a finite-dimensional adic space, and |$f:X\rightarrow Y$| is a smooth morphism of relative dimension d, which is moreover partially proper in the sense of Kiehl. Then, locally on Y, there exists a cover of X by opens U_i admitting closed embeddings |$U_i\hookrightarrow {\mathbb D}^{N_{i}}_Y(0;1^-)$| over Y. Moreover, each |$U_i\cap U_j$| admits a closed embedding into |${\mathbb D}^{N_{i}+N_j}_Y(0;1^-)$| over Y. Using the spectral sequence from Corollary 4.17, together with Proposition 6.9, the trace maps

factor uniquely through a map

which can be checked not to depend on the choice of the U_i using Proposition 6.9. Since |$\mathrm{Tr}_{X/Y}$| does not depend on the choice of open cover of X, it glues over an open cover of Y, and by Theorem 5.6 this can be viewed as a map

in |$\textbf{D}(\mathcal{O}_Y)$|⁠. Moreover, |$\mathrm{Tr}_{X/Y}$| vanishes on the image of

since the same is true locally on X and Y, and thus induces a map

in |$\textbf{D}(\mathcal{O}_Y)$|⁠.

6.5. The general case

Finally, we consider again a smooth morphism |$f:X\rightarrow Y$| of relative dimension d, partially proper in the sense of Kiehl, but now with the base Y allowed to be any overconvergent, finite-dimensional germ. Then arguing exactly as in the proof of Corollary 5.14, we see that, locally on X, we can extend f to a diagram of pairs

such that |$f:\boldsymbol{X}\rightarrow \boldsymbol{Y}$| is smooth, |$u:\boldsymbol{X}\rightarrow {\mathbb D}^{N}_{\boldsymbol{Y}}(0;1^-)$| is a closed immersion over Y, and |$X=\pi^{-1}(Y)\cap \boldsymbol{X}$|⁠. Using Corollary 5.14, together with Lemma 4.14, we can therefore carry through all the arguments of Sections 6.3 and 6.4 to construct a trace morphism

which again can be viewed as a map

1. |$\mathrm{Tr}_{X/Y}$| vanishes on the image of |$\mathbf{R}^df_!\Omega^{d-1}_{X/Y}$| and descends to a map

   $$ \mathrm{Tr}_{X/Y}: \mathbf{R}f_!\Omega^\bullet_{X/Y}[2d] \rightarrow \mathcal{O}_Y. $$
2. If |$g:Y\rightarrow Z$| is smooth morphism of relative dimension e, partially proper in the sense of Kiehl, with Z overconvergent, then the diagram

   

   commutes.

6.6. Duality morphism

We can now construct the duality morphism. Let |$f:X\rightarrow Y$| be a partially proper morphism of germs. If |$\mathscr{I},\mathscr{J}$| are |$\mathcal{O}_X$|-modules then the natural map

induces

and sheafifying this gives a pairing

By taking resolutions, we deduce that if |$\mathscr{E}$| and |$\mathscr{F}$| are bounded complexes of |$\mathcal{O}_X$|-modules, and both X and Y are finite-dimensional, then there is a natural pairing

in |$\textbf{D}^-(\mathcal{O}_Y)$|⁠. In particular, if |$\mathscr{G}$| is a third bounded complex, then any pairing

induces a corresponding pairing

in cohomology. If we now assume, moreover, that:

• f is smooth of relative dimension d, and partially proper in the sense of Kiehl;

• Y is overconvergent;

• |$\mathscr{E}$| is a perfect complex on X,

then, setting |$\mathcal{E}^\vee:=\mathbf{R}\underline{\mathrm{Hom}}(\mathcal{E},\mathcal{O}_X)$|⁠, we have a natural evaluation pairing

Together with the trace map |$\mathrm{Tr}_{X/Y}$| this induces a pairing

There is, of course, a similar pairing

7. A COUNTEREXAMPLE

In this section we give a counterexample showing that the formalism of |$\mathbf{R}f_!$| cannot be extended beyond the partially proper case in any reasonable way. Our example also shows that the analogue of Lemma 4.14 fails in general if Z is replaced by a non-maximal point of Y. The example is based upon a suggestion of B. Le Stum.  

Proof.

Let |$\kappa=(k,k^\circ)$| be a height one affinoid field, |$X={\mathbb D}^1_\kappa(0;1)=\mathrm{Spa}\left(k\langle z \rangle,k^\circ\langle z \rangle\right)$| the (ordinary) closed unit disc over κ. Let |$j_U:U \rightarrow X$| denote the inclusion of the open subspace defined by |$\{x\in X \mid v_x(z-1)=1\}$|⁠, |$f:X\rightarrow X$| the (finite, thus proper) morphism defined by |$z\mapsto z^2$|⁠, and |$V:=f^{-1}(U)\subset X$|⁠. Thus V is the intersection of U with the open subspace defined by |$\{x \in X \mid v_x(z+1)=1\}$|⁠, we let |$j_V\colon V\rightarrow X$| denote the inclusion. We therefore have a Cartesian square

To prove the theorem, it suffices to show that |$\mathbf{R}f_!\circ j_{V!}\ncong j_{U!}\circ \mathbf{R}f_!$|⁠.

To see this, we take |$\mathscr{F}$| to be the constant sheaf |$\underline{{\mathbb Z}}$| on V and compute the stalks of both sides at the Type V apex point ξ₁ of the open disc |$\{x\in X \mid v_{[x]}(z-1)\lt1\}$|⁠. Clearly we have that |$\left(j_{U!}\mathbf{R}f_!\underline{{\mathbb Z}}\right)_{\xi_1}=0$|⁠, and we shall show that |$\left(\mathbf{R}f_!j_{V!}\underline{{\mathbb Z}}\right)_{\xi_1}=\left(\mathbf{R}f_*j_{V!}\underline{{\mathbb Z}}\right)_{\xi_1}\neq 0$|⁠.

Indeed, we may base change to the set |$G(\xi_1)$| of generalisations of ξ₁, which is a two-point space |$\{\xi_1,\xi\}$| consisting of ξ₁ together with the Gauss point ξ. The fibre |$X_{(\xi_1)}=f^{-1}(G(\xi_1))$| is a three-point space |$\{\xi,\xi_1,\xi_{-1}\}$| consisting of |$\xi,\xi_1$| and the apex point ξ₋₁ of the open disc |$\{x\in X \mid v_{[x]}(z+1)\lt1\}$|⁠, and the induced map |$f:X_{(\xi_1)}\rightarrow G(\xi_1)$| sends |$\xi_{\pm 1}$| to ξ₁ and ξ to ξ. We then have

$$ \left(\mathbf{R}f_*j_{V!}\underline{{\mathbb Z}}\right)_{\xi_1} = \mathbf{R}\Gamma(X_{(\xi_1)},(j_{V!}\underline{{\mathbb Z}})|_{X_{(\xi_1)}}), $$

and we can identify the restriction of |$j_{V!}\underline{{\mathbb Z}}$| to |$X_{(\xi_1)}$| as the extension by zero of the constant sheaf |$\underline{{\mathbb Z}}$| along the open immersion |$\{\xi \}\rightarrow X_{(\xi_1)}$|⁠. In particular, we have the exact sequence

$$ 0 \rightarrow \left(j_{V!}\underline{{\mathbb Z}}\right)|_{X_{(\xi_1)}} \rightarrow \underline{{\mathbb Z}}\rightarrow i_{1*}\underline{{\mathbb Z}} \oplus i_{-1*}\underline{{\mathbb Z}} \rightarrow 0 $$

where |$i_{\pm1}$| is the inclusion of the closed point |$\xi_{\pm1}$| inside |$X_{(\xi_1)}$|⁠. Taking cohomology then gives the exact triangle

$$ \left(\mathbf{R}f_*j_{V!}\underline{{\mathbb Z}}\right)_{\xi_1}\rightarrow {\mathbb Z} \rightarrow {\mathbb Z}\oplus {\mathbb Z} \overset{+1}{\rightarrow} $$

where the second map is the diagonal morphism. Thus

$$ \left(\mathbf{R}f_*j_{V!}\underline{{\mathbb Z}}\right)_{\xi_1}\cong {\mathbb Z}[-1] $$

is non-zero as claimed.

Acknowledgements

We would like to thank Yoichi Mieda for reading the manuscript carefully and giving us several helpful comments, and Jérôme Poineau for answering some of our questions. We also thank Bernard Le Stum for pointing out the failure of the proper base change theorem in general. Additionally, we would also like to thank the anonymous referee for their thorough reading of the manuscript and for their helpful suggestions to improve it.

Funding

T.A. was supported by JSPS KAKENHI Grant Numbers 16H05993, 18H03667, 20H01790.

References