2.1 Cohomology theories

Let K be an algebraically closed nonarchimedean field of arbitrary rank, $\Gamma $ be its value group and $\Gamma _{\infty }=\Gamma \cup \{\infty \}$ the associated tropical semi-group. In order to fully exploit Hrushovski-Loeser’s main theorem, part of the present article consists in developing in parallel a sheaf cohomology of definable (i.e., semi-linear) sets of $\Gamma _{\infty }$ . More generally, we lay out a sheaf cohomology for definable sets in o-minimal expansions of $\Gamma _{\infty }$ over the site of open definable sets, extending theorems from [Reference Edmundo, Jones and Peatfield22, Reference Edmundo and Prelli25, Reference Edmundo and Prelli26, Reference Edmundo and Prelli28] on sheaves on o-minimal expansions of $\Gamma $ , including the formalism of the six Grothendieck operations (see Remark 6.23). It is worth mentioning that the passage from $\Gamma $ to $\Gamma _{\infty }$ is more subtle than one might expect, as already noted by Hrushovski and Loeser (see the first paragraph in [Reference Hrushovski and Loeser35, Section 4.1]).

The following is the main theorem concerning our cohomology theories:

2.2 Applications

We obtain the following applications:

(I) In rank 1, Hrushovski and Loeser show finiteness of homotopy types in definable families (see [Reference Hrushovski and Loeser35, Theorems 14.3.1 and 14.4.4]). As they observed (see the beginning of [Reference Hrushovski and Loeser35, Section 14.3]), such a result is no longer true in higher ranks. However, using invariance of cohomology with definably compact support, we obtain a tameness result showing that given a definable family $Z_w$ of $\mathrm {v\!+\!g}$ -locally closed subsets of a variety, as w runs through W, there are finitely many isomorphisms types for the $\widehat {\mathrm {v\!+\!g}}$ -cohomology $H^*_c(\widehat {Z}_{w};{\mathbb Q})$ with definably compact supports. (See later Theorem 8.1.)

Results of A. Abbes and T. Saito [Reference Abbes and Saito1], later generalised by J. Poineau [Reference Poineau48], showed that for families parametrised by the norm of an analytic function, there is a finite partition of $\mathbb {R}$ into intervals such that on each piece, $\pi _0$ of the fibres are in canonical bijections. For definable families parametrised by ${\Gamma _{\infty } }$ , we get analogues of such results for $H_c^*$ and $\pi _0^{\mathrm {def}}$ (both independently of the rank). See Corollaries 8.2 and 8.3.

It is worth pointing out that assuming finiteness and invariance of cohomology without support, similar methods lead to bounds on the $\mathrm {v\!+\!g}$ -Betti numbers of a uniformly definable family of $\mathrm {v\!+\!g}$ -locally closed subsets of a variety. Such a result will be an analogue, in all ranks, of a result obtained by S. Basu and D. Patel ([Reference Basu and Patel4, Theorem 2]) in the case where the valued field has rank one, using the usual topological (singular) Betti numbers instead. In comparison with [Reference Basu and Patel4, Theorem 2], this approach will provide a much simpler proof by taking advantage of the fact that we can work in higher elementary extensions where we can endow the tropical semi-group ${\mathbf {\Gamma }_{\infty } }$ with the structure of a real closed field and therefore apply earlier results of Basu himself on bounds in o-minimal expansions of real closed fields ([Reference Basu3, Theorem 2.2]).

(II) We recover results of Berkovich [Reference Berkovich11] on the singular cohomology of $V^{\mathrm {an}}$ concerning finiteness and invariance (see later Theorems 7.12 and 7.13 and Corollaries 7.19 and 7.17).

(III) We extend Berkovich’s results in two different ways that we briefly explain. Let F be a nonarchimedean field of rank 1 (not necessarily complete) and V be an algebraic variety over F. Hrushovski and Loeser also introduced a topological space, called the model-theoretic Berkovich space associated to V, which they denoted by $B_{\mathbf F}(V)$ (see later Section 7 for its formal definition). When F is complete, $B_{\mathbf F}(V)$ is homeomorphic to the underlying topological space of $V^{\mathrm {an}}$ . Letting $F^{\max }$ be a maximally complete algebraically closed extension of F with value group $\mathbb {R}$ , they show the existence of a closed continuous surjection $\pi \colon \widehat {V}(F^{\max })\to B_{\mathbf F}(V)$ . Such a map allowed them to transfer results from $\widehat {V}(F^{\max })$ to $B_{\mathbf F}(V)$ and draw conclusions about the homotopy type of $V^{\mathrm {an}}$ for V quasi-projective. When $F=F^{\max }$ , the topological spaces $\widehat {V}(F)$ , $B_{\mathbf F}(V)$ and $V^{\mathrm {an}}$ are all homeomorphic, and we show that, in this particular case, our cohomology coincides with the singular cohomology of $V^{\mathrm {an}}$ (Theorems 7.16 and 7.18). Furthermore, the map $\pi $ allowed us to transfer results from the $\widehat {\mathrm {v\!+\!g}}$ -cohomology on $\widehat {V}(F^{\max })$ to the singular cohomology of $B_{\mathbf F}(V)$ and hence, when F is complete, to the singular cohomology of $V^{\mathrm {an}}$ . This is mainly how we recover Berkovich’s results. However, note that our results hold more generally for $B_{\mathbf F}(V)$ , a space that is well-defined without assuming F to be complete.

A second way in which we extend Berkovich’s results is that our theorems actually hold for semi-algebraic subsets of $B_{\mathbf F}(V)$ (respectively, for semi-algebraic subsets of $\widehat {V}(K)$ , for any nontrivially valued algebraically closed field extension of F). Here, a semi-algebraic subset should be understood in the sense of A. Ducros [Reference Ducros20] (see later Fact 7.1). Using that $\widehat {V}(K)$ has finitely many definably connected components (Lemma 5.21), we also recover the analogue result due to Ducros [Reference Ducros20] for $V^{\mathrm {an}}$ . In a similar vein, but using completely different methods, F. Martin [Reference Martin43] extended results of Berkovich concerning the étale cohomology of analytic spaces to the larger category of their semi-algebraic subsets. We would like to explore in subsequent research if the techniques here employed can also be used to develop an analogue of the étale cohomology over the spaces $\widehat {V}(K)$ and $B_{\mathbf F}(V)$ .

2.3 Layout of the article

Throughout, we will use a formalism of sheaves developed by M. Edmundo and L. Prelli ([Reference Edmundo and Prelli27]), which builds on a modification of M. Kashiwara and P. Schapira’s ([Reference Kashiwara and Schapira40]) notion of ${\mathcal {T}}$ -topology (a Grothendieck topology). The semi-algebraic site, the sub-analytic site, the o-minimal site and the $\widehat {\mathrm {v\!+\!g}}$ -site mentioned above will be examples of such ${\mathcal {T}}$ -topologies. The needed background and properties of such a formalism are presented in Section 3.

The novelty here is the introduction of the notions of ${\mathcal {T}}$ -normality and families of ${\mathcal {T}}$ -normal supports on spaces equipped with a ${\mathcal {T}}$ -topology. When we pass to the ${\mathcal {T}}$ -spectrum of such Grothendieck topologies, the corresponding categories of sheaves are isomorphic, and we get normal spectral spaces (respectively, families of normal and constructible supports on such spectral spaces). The latter notion will play the role that paracompactifying families of supports plays in sheaf theory in topological spaces. We study the notion of $\Phi $ -soft sheaves when $\Phi $ is a normal and constructible family of supports and set up the tools required to obtain the Base change formula, the Vietoris-Begle theorem and, in Section 6, Mayer-Vietoris sequences and bounds on cohomological $\Phi $ -dimension. These results were already known in the o-minimal case ([Reference Edmundo and Prelli25]) and the semi-algebraic case ([Reference Delfs17]), where ${\mathcal {T}}$ -normality and families of ${\mathcal {T}}$ -normal supports correspond to definably normal definable spaces and families of definably normal supports (respectively, regular and paracompact (locally) semi-algebraic spaces and paracompactifying (locally) semi-algebraic families of supports). Here we present a unifying approach to this theory.

In Section 4, we define the o-minimal site on a definable set in an o-minimal expansion ${\mathbf {\Gamma }_{\infty } }$ of ${\Gamma _{\infty } }$ and recall the notions of definable connectedness and definable compactness. Moreover, in order to be able to apply the tools of Section 3, we study the notion of definable normality in ${\mathbf {\Gamma }_{\infty } }$ . Unlike in o-minimal expansions ${\mathbf {\Gamma }}$ of ${\Gamma }$ , in ${\mathbf {\Gamma }_{\infty } }$ there are open definable sets that are not definably normal. The main result of Section 4, which ensures that we can later apply the tools of Section 3, is Theorem 4.28, showing that every definably locally closed set in ${\mathbf {\Gamma }_{\infty } }$ is the union of finitely many relatively open definable subsets that are definably normal. The proof of this result is rather long and is based in ideas from [Reference Edmundo and Prelli28].

The $\widehat {\mathrm {v\!+\!g}}$ -site on the stable completion $\widehat {V}(K)$ of an algebraic variety V over an algebraically closed nonarchimedean field K (and more generally on the stable completion $\widehat {X}$ of a definable subset $X\subseteq V\times \Gamma _{\infty }^m$ ) is defined and studied in Section 5. Here, for the reader’s convenience, we recall some background from [Reference Hrushovski and Loeser35] on stable completions and the notions of definable connectedness and definable compactness. In particular, we include a proof (missing in [Reference Hrushovski and Loeser35]) of the fact that the stable completion $\widehat {X}$ of a definable subset $X\subseteq V\times {\Gamma ^n_{\infty } }$ has finitely many definably connected components (Lemma 5.21). The main result of this section is Corollary 5.31 showing that if X is a $\mathrm {v\!+\!g}$ -locally closed subset, then X is the union of finitely many basic $\mathrm {v\!+\!g}$ -open subsets that are weakly $\mathrm {v\!+\!g}$ -normal. When we pass to $\widehat {X}$ equipped with the $\widehat {\mathrm {v\!+\!g}}$ -site, this gives the required conditions to apply the tools of Section 3.

Section 6 is devoted to showing the Eilenberg-Steinrod axioms, finiteness, invariance and vanishing results for the associated sheaf cohomologies in the algebraic and ${\mathbf {\Gamma }_{\infty } }$ cases. The main difficulty concerns the homotopy axiom that, as usual, is proved using the Vietoris-Begle theorem. However, extra work is needed to verify the assumptions of the version of this theorem presented in Section 3. The results concerning finiteness and invariance in ${\mathbf {\Gamma }_{\infty } }$ are obtained by adapting the methods used in [Reference Berarducci and Fornasiero7] and [Reference Edmundo and Prelli26]. After verifying that the strong deformation retraction given by the main theorem of [Reference Hrushovski and Loeser35] is a morphism of $\widehat {\mathrm {v\!+\!g}}$ -sites, we obtain finiteness and invariance for stable completions in the quasi-projective case. The general case is then obtained using Mayer-Vietoris sequences based on the tools of Section 3.

Finally, Sections 7 and 8 gather the above-mentioned applications: the former is devoted to the relation with Berkovich spaces and the latter to the tameness results on cohomological complexity in families.

3.1 ${\mathcal {T}}$ -sheaves

Here, we recall the definition of ${\mathcal {T}}$ -space given in [Reference Edmundo and Prelli27], adapting the construction of Kashiwara and Schapira [Reference Kashiwara and Schapira40], and we recall some of the results obtained in that paper for the category of sheaves on $X_{{\mathcal {T}}}$ . Those results generalise results from the case of sub-analytic sheaves, semi-algebraic sheaves or o-minimal sheaves. See Examples 3.7 below.

Let X and ${\mathcal {T}}\subseteq \mathrm {Op} (X)$ satisfying (i) and (ii) of Definition 3.2. One can endow the category ${\mathcal {T}}$ with a Grothendieck topology, called the ${\mathcal {T}}$ -topology, in the following way: a family $\{U_i\}_i$ in ${\mathcal {T}}$ is a covering of $U \in {\mathcal {T}}$ if it admits a finite subcover. We denote by $X_{{\mathcal {T}}}$ the associated site. For short, we write $A_{{\mathcal {T}}}$ instead of $A_{X_{{\mathcal {T}}}}$ for the constant sheaf on $X_{{\mathcal {T}}}$ with value A, and we let $\mathrm {Mod}(A_{{\mathcal {T}}})$ be the category of sheaves of $A_{{\mathcal {T}}}$ -modules on $X_{{\mathcal {T}}}$ . If $\rho \colon X \to X_{{\mathcal {T}}}$ is the natural morphism of sites, then we have functors

with $\rho ^{-1} \circ \rho _{*} \simeq {\mathrm {id}}$ (equivalently, the functor $\rho _*$ is fully faithful). See [Reference Edmundo and Prelli27, Proposition 2.1.6].

By [Reference Edmundo and Prelli27, Propositions 2.1.7 and 2.1.8], we have:

Recall that any Boolean algebra ${\mathcal {A}}$ has an associated topological space that we denote by $S({\mathcal {A}})$ , called its Stone space. The points in $S({\mathcal {A}})$ are the ultrafilters $\alpha $ on ${\mathcal {A}}$ . The topology on $S({\mathcal {A}})$ is generated by a basis of open and closed sets consisting of all sets of the form

$$\begin{align*}\widetilde{C} = \{\alpha \in S({\mathcal{A}}): C \in \alpha \}, \end{align*}$$

where $C\in {\mathcal {A}}$ . The space $S({\mathcal {A}})$ is a compactFootnote ¹ totally disconnected space. Moreover, for each $C \in {\mathcal {A}}$ , the subspace $\widetilde {C}$ is compact. See [Reference Johnstone37] for an introduction to this subject.

Let X and ${\mathcal {T}}\subseteq \mathrm {Op} (X)$ satisfying (i) and (ii) of Definition 3.2. Then as in [Reference Edmundo and Prelli27], we let

$$ \begin{align*} {\mathcal{T}} _{loc}=\{U\in \mathrm{Op}(X): U \cap W \in {\mathcal{T}}\,\,\, \textrm{for every} \,\,\, W \in {\mathcal{T}}\}, \end{align*} $$

and we make the following definitions:

• • A subset S of X is a ${\mathcal {T}}_{loc}$ -subset if and only if $S\cap V$ is a $ {\mathcal {T}}$ -subset for every $V\in {\mathcal {T}}$ .

• • A closed (respectively, open) ${\mathcal {T}}_{loc}$ -subset is a ${\mathcal {T}}_{loc}$ -subset that is closed (respectively, open) in X.

• • A ${\mathcal {T}}_{loc}$ -connected subset is a ${\mathcal {T}}_{loc}$ -subset that is not the disjoint union of two proper clopen ${\mathcal {T}}_{loc}$ -subsets (in the induced topology).

Clearly, $\varnothing , X \in {\mathcal {T}}_{loc}$ , ${\mathcal {T}}\subseteq {\mathcal {T}}_{loc}$ and ${\mathcal {T}}_{loc}$ is closed under finite intersections. Moreover, if $\{S_i\}_i$ is a family of ${\mathcal {T}}_{loc}$ -subsets such that $\{i:S_i\cap W\neq \emptyset \}$ is finite for every $W\in {\mathcal {T}}$ , then the union and intersection of the family $\{S_i\}_i$ are ${\mathcal {T}}_{loc}$ -subsets. The complement of a ${\mathcal {T}}_{loc}$ -subset is also a ${\mathcal {T}}_{loc}$ -subset. Therefore, the ${\mathcal {T}}_{loc}$ -subsets form a Boolean algebra.

With this topology, for $U \in {\mathcal {T}}$ , the set $\widetilde {U}$ is quasi-compact in $\widetilde {X}_{{\mathcal {T}}}$ since it is quasi-compact in $S({\mathcal {A}})$ . Hence:

Furthermore, by [Reference Edmundo and Prelli27, Proposition 2.6.3], we also have:

Fact 3.6. There is an equivalence of categories

$$ \begin{align*} \mathrm{Mod}(A_{{\mathcal{T}}}) \simeq \mathrm{Mod}(A_{\widetilde{X}_{{\mathcal{T}}}}). \\[-3.4pc] \end{align*} $$

□

Note that in the statement of [Reference Edmundo and Prelli27, Proposition 2.6.3], X is assumed to be a ${\mathcal {T}}$ -space, but this is not necessary: all that is used is Fact 3.5 and [Reference Edmundo and Prelli27, Corollary 1.2.11] for $\widetilde {V}$ with $V\in {\mathcal {T}}$ .

We now recall the main known examples of ${\mathcal {T}}$ -spaces.

3.2 ${\mathcal {T}}$ -normality and families of ${\mathcal {T}}$ -supports

Let ${\mathfrak T}$ be the category whose objects are pairs $(X,{\mathcal {T}})$ with X a topological space, ${\mathcal {T}} \subseteq \mathrm {Op} (X)$ is a family of open subsets of X satisfying conditions (i) and (ii) of Definition 3.2 and such that $X\in {\mathcal {T}}$ , and morphisms $f\colon (X, {\mathcal {T}})\to (Y, {\mathcal {T}}')$ are (continuous) maps $f\colon X\to Y$ such that $f^{-1}({\mathcal {T}}')\subseteq {\mathcal {T}}$ .

Note that a morphism $f\colon (X,{\mathcal {T}})\to (Y,{\mathcal {T}}')$ of ${\mathfrak T}$ defines a morphism of sites

$$\begin{align*}f\colon X_{{\mathcal{T}}}\to Y_{{\mathcal{T}}'} \end{align*}$$

which extends to a homomorphism from the boolean algebra of ${\mathcal {T}}'$ -subsets of Y to the boolean algebra of ${\mathcal {T}}$ -subsets of X: namely, $f^{-1}(C)$ is a ${\mathcal {T}}$ -subset of X whenever C is a ${\mathcal {T}}'$ -subset of Y. It follows that we have an induced map

$$\begin{align*}\widetilde{f}\colon \widetilde{X}_{{\mathcal{T}}}\to \widetilde{Y}_{{\mathcal{T}}'} \end{align*}$$

between the corresponding spectral topological spaces, where $\widetilde {f}(\alpha )$ is the ultrafilter in $\widetilde {Y}_{{\mathcal {T}}'}$ given by the collection

$$\begin{align*}\{C: C\,\, \textrm{is a}\,\, {\mathcal{T}}'\textrm{-subset and}\,\, f^{-1}(C)\in \alpha \}. \end{align*}$$

The map $\widetilde {f}\colon \widetilde {X}_{{\mathcal {T}}}\to \widetilde {Y}_{{\mathcal {T}}'}$ is continuous since if $V\in {\mathcal {T}}',$ then clearly $\widetilde {f}^{-1}(\widetilde {V})=\widetilde {f^{-1}(V)}$ .

Later, we will also require the following generalisation of the above constructions:

Remark 3.8. Let $\zeta \colon X_{{\mathcal {T}}}\to Y_{{\mathcal {T}}'}$ be a morphism of sites that extends to a homomorphism from the boolean algebra of ${\mathcal {T}}'$ -subsets of Y to the boolean algebra of ${\mathcal {T}}$ -subsets of X. If $\zeta (Y)=X$ and $\zeta (A)\subseteq \zeta (B)$ whenever A and B are ${\mathcal {T}}'$ -subsets such that $A\subseteq B,$ then $\zeta $ induces a continuous map $\widetilde {\zeta }\colon \widetilde {X}_{{\mathcal {T}}}\to \widetilde {Y}_{{\mathcal {T}}'}$ , where $\widetilde {\zeta }(\alpha )$ is the ultrafilter of ${\mathcal {T}}'$ -subsets given by the collection

$$\begin{align*}\{C: C\,\, \textrm{is a}\,\, {\mathcal{T}}'\textrm{-subset and}\,\, \zeta (C)\in \alpha \}. \end{align*}$$

If in addition $\zeta \colon X_{{\mathcal {T}}}\to Y_{{\mathcal {T}}'}$ is an isomorphism of sites, then $\widetilde {\zeta }$ is a homeomorphism.

Also recall the following facts:

Let $\widetilde {{\mathfrak T}}$ be category with objects $\widetilde {X}_{{\mathcal {T}}}$ for $(X,{\mathcal {T}})$ an object of ${\mathfrak T}$ and with morphisms $\widetilde {f}\colon \widetilde {X}_{{\mathcal {T}}}\to \widetilde {Y}_{{\mathcal {T}}'}$ for $f\colon (X,{\mathcal {T}})\to (Y,{\mathcal {T}}')$ a morphism of ${\mathfrak T}$ . Below, we denote by

$$\begin{align*}{\mathfrak T} \to \widetilde{{\mathfrak T}} \end{align*}$$

the functor sending an object $(X,{\mathcal {T}})$ of ${\mathfrak T}$ to $\widetilde {X}_{{\mathcal {T}}}$ and sending a morphism $f\colon (X,{\mathcal {T}})\to (Y,{\mathcal {T}}')$ of ${\mathfrak T}$ to $\widetilde {f}\colon \widetilde {X}_{{\mathcal {T}}}\to \widetilde {Y}_{{\mathcal {T}}'}$ .

The goal of this subsection is to extend to $({\mathfrak T}, \widetilde {{\mathfrak T}})$ some results from [Reference Edmundo, Jones and Peatfield22] and [Reference Edmundo and Prelli25] proved for the pair $(\mathrm {Def}, \widetilde {\mathrm {Def}})$ , where $\mathrm {Def}$ is the subcategory of ${\mathfrak T}$ of Example 3.7 (3): that is, it is the category of definable spaces in some fixed arbitrary o-minimal structure ${\mathbf {\Gamma }} =({\Gamma }, <,\ldots )$ without end points, with morphisms being continuous definable maps between such definable spaces, and $\widetilde {\mathrm {Def}}$ is the image of $\mathrm {Def}$ under ${\mathfrak T} \to \widetilde {{\mathfrak T}}$ .

For $({\mathfrak T}, \widetilde {{\mathfrak T}}),$ just like in the case of $(\mathrm {Def}, \widetilde {\mathrm {Def}}),$ normality will play the role that paracompactness plays in sheaf theory on topological spaces [Reference Bredon14, Chapter I, Section 6 and Chapter II, Sections 9–13] or the role that regular and paracompact plays in sheaf theory on locally semi-algebraic spaces [Reference Delfs17, Chapter II].

So we introduce and study a notion of normality adapted to ${\mathfrak T}:$

We let the reader convince her/himself of the equivalence of (1)–(3).

As in the case of $(\mathrm {Def}, \widetilde {\mathrm {Def}})$ ([Reference Edmundo, Jones and Peatfield22, Theorem 2.13]), we have the following. Here we give a simpler proof.

Proof. By Remark 3.10 and quasi-compactness of $\widetilde {X}$ , it is immediate that if $\widetilde {X}$ is normal, then X is ${\mathcal {T}}$ -normal. It is a standard fact that a spectral topological space is normal if and only if any two distinct closed points can be separated by disjoint open subsets (see [Reference Coste and Roy15, Proposition 2]). So let $\alpha $ and $\beta $ be two distinct closed points in $\widetilde {X}$ . Since $\alpha $ and $\beta $ are closed points, we have $\{\alpha \}=\bigcap _{i\in I}C_i$ and $\{\beta \}=\bigcap _{j\in J}D_j$ , where $C_i$ s and $D_j$ s are closed constructible subsets of $\widetilde {X}$ . Since $\alpha $ and $\beta $ are distinct, we have

$$\begin{align*}\bigcap \{C_i\cap D_j: i\in I, j\in J\}=\emptyset .\end{align*}$$

By quasi-compactness of $\widetilde {X},$ there are $C=C_{i_1}\cap \ldots \cap C_{i_k}$ and $D=D_{j_1}\cap \ldots \cap D_{j_l}$ such that $C\cap D=\emptyset $ . Since C and D are disjoint, closed constructible subsets of $\widetilde {X},$ by Remark 3.10, we can use the ${\mathcal {T}}$ -normality of X to find disjoint, open constructible subsets of $\widetilde {X}$ separating C and D and hence $\alpha $ and $\beta $ .

As usual, normality implies the shrinking lemma, whose proof is standard; see, for example, [Reference Edmundo, Jones and Peatfield22, Proposition 2.17] or [Reference van den Dries53, Chapter 6, (3.6)]. In the ${\mathcal {T}}$ -spectra, due to the quasi-compactness of the base, we get a stronger result:

Later we will require the following weaker notion:

We continue with the ${\mathfrak T}$ versions of some definitions from [Reference Edmundo and Prelli25] that are needed in the sequel.

Let $(X,{\mathcal {T}})$ be an object of ${\mathfrak T}$ . If Z is a ${\mathcal {T}}$ -subset of X and $\Phi $ is a family of ${\mathcal {T}}$ -supports on $(X,{\mathcal {T}}),$ then we have families of ${\mathcal {T}}\cap Z$ -supports

$$\begin{align*}\Phi \cap Z=\{S\cap Z:S\in \Phi \} \end{align*}$$

and

$$\begin{align*}\Phi _{|Z}=\{S\in \Phi : S\subseteq Z\} \end{align*}$$

on $(Z,{\mathcal {T}} \cap Z)$ .

If $f\colon (X,{\mathcal {T}})\longrightarrow (Y,{\mathcal {T}}')$ is a morphism in ${\mathfrak T}$ and $\Psi $ is a family of ${\mathcal {T}}'$ -supports on $(Y,{\mathcal {T}}'),$ then we have a family of ${\mathcal {T}}$ -supports

$$\begin{align*}f^{-1}\Psi =\{S\subseteq X: S\,\,\textrm{is a closed}\,\, {\mathcal{T}}\textrm{-subset and}\,\,\exists B\in \Psi \,\,(S\subseteq f^{-1}(B)\} \end{align*}$$

on $(X,{\mathcal {T}})$ .

Remark 3.17 (Constructible family of supports).

Note that a family of ${\mathcal {T}}$ -supports $\Phi $ on an object $(X,{\mathcal {T}})$ of ${\mathfrak T}$ determines a family of supports

$$\begin{align*}\widetilde{\Phi }=\{A\subseteq \widetilde{X}: A\,\,\textrm{ is closed and}\,\,\exists B\in \Phi \,\,(A\subseteq \widetilde{B})\} \end{align*}$$

on the topological space $\widetilde {X}=\widetilde {X_{{\mathcal {T}}}}$ . By Remark 3.10, it follows that

$$\begin{align*}\widetilde{\Phi \cap Z}=\widetilde{\Phi }\cap \widetilde{Z},\,\, \widetilde{\Phi _{|Z}}=\widetilde{\Phi }_{|\widetilde{Z}}\,\,\mathrm{and}\,\,\widetilde{f^{-1}\Psi }=\widetilde{f}^{-1}\widetilde{\Psi }. \end{align*}$$

We will say that the family of supports $\Psi $ on $\widetilde {X}$ is a constructible family of supports on $\widetilde {X}$ if $\Psi =\widetilde {\Phi }$ for some family of ${\mathcal {T}}$ -supports on $(X,{\mathcal {T}})$ .

By Proposition 3.12, it follows that:

Below, we say that $\Psi $ is a family of normal and constructible supports on the spectral topological space $\widetilde {X}$ if $\Psi =\widetilde {\Phi }$ for some family of ${\mathcal {T}}$ -normal supports on $(X,{\mathcal {T}})$ .

3.3 Sheaf cohomology with ${\mathcal {T}}$ -supports

Let $(X,{\mathcal {T}})$ be an object of ${\mathfrak T}$ . Due to the isomorphism

$$\begin{align*}\mathrm{Mod} (A_{X_{{\mathcal{T}}}})\simeq \mathrm{Mod} (A_{\widetilde{X}}), \end{align*}$$

in analogy to what happened in the case $(\mathrm {Def}, \widetilde {\mathrm {Def}})$ ([Reference Edmundo, Jones and Peatfield22] for o-minimal sheaf cohomology without supports and in [Reference Edmundo and Prelli25] in the presence of families of definable supports), in this subsection, we will develop sheaf cohomology on $X_{{\mathcal {T}}}$ via this tilde isomorphism.

Definition 3.21. Let $(X,{\mathcal {T}})$ be an object of ${\mathfrak T}$ , $\Phi $ a family of ${\mathcal {T}}$ -supports in $(X,{\mathcal {T}})$ and ${\mathcal {F}} \in \mathrm {Mod} (A_{X_{{\mathcal {T}}}})$ . We define the ${\mathfrak T}$ -sheaf cohomology groups with ${\mathcal {T}}$ -supports in $\Phi $ via the tilde isomorphism by

$$\begin{align*}H^*_{\Phi}(X;{\mathcal{F}})=H^*_{\widetilde{\Phi}}(\widetilde{X};\widetilde{{\mathcal{F}}}), \end{align*}$$

where $\widetilde {{\mathcal {F}}}$ is the image of ${\mathcal {F}}$ via the isomorphism $\mathrm {Mod} (A_{X_{{\mathcal {T}}}})\simeq \mathrm {Mod} (A_{\widetilde {X}})$ .

If $f\colon (X,{\mathcal {T}})\longrightarrow (Y,{\mathcal {T}}')$ is a morphism in ${\mathfrak T},$ we define the induced homomorphism

$$\begin{align*}f^*\colon H^*_{\Phi }(Y;{\mathcal{F}})\longrightarrow H^*_{f^{-1}\Phi }(X;f^{-1}{\mathcal{F}})\end{align*}$$

in cohomology to be the same as the homomorphism

$$\begin{align*}\widetilde{f}^*\colon H^*_{\widetilde{\Phi }}(\widetilde{Y};{\widetilde {\mathcal{F}}})\longrightarrow H^*_{\widetilde{f}^{-1}\widetilde{\Phi }}(\widetilde{X};\widetilde{f}^{-1}{\widetilde {\mathcal{F}}})\end{align*}$$

in cohomology induced by the continuous map $\widetilde {f}\colon \widetilde {X}\longrightarrow \widetilde {Y}$ of topological spaces.

Below, we shall use a few facts about sheaf cohomology in topological spaces that can be found, for example, in [Reference Bredon14, Chapter II, Sections 1–8]. We will also need the following ${\mathfrak T}$ versions of [Reference Bredon14, Chapter II, 9.5] and [Reference Bredon14, Chapter II, 9.21], respectively.

Lemma 3.22. Let X be an object of $\widetilde {{\mathfrak T}}$ . Let Z be a subspace of X and Y a quasi-compact subset of Z having a fundamental system of normal and constructible locally closed neighbourhoods in X. Then for every ${\mathcal {G}}\in {\mathrm {Mod}}(A_Z)$ , the canonical morphism

$$\begin{align*}\underset{Y\subseteq U}{\underrightarrow{\lim}}\Gamma(U\cap Z;{\mathcal{G}}) \longrightarrow \Gamma(Y;{\mathcal{G}}_{|Y})\,\,\,\end{align*}$$

where U ranges through the family of open constructible subsets of X containing Y, is an isomorphism.

Proof. Since Y is quasi-compact, the family of open neighbourhoods of Y in Z of the form $V\cap Z$ , where V is an open constructible subset of X, is a fundamental system of neighbourhoods of Y in Z. Hence, the morphism of the lemma is certainly injective.

To prove that it is surjective, consider a section $s\in \Gamma (Y;{\mathcal {G}} _{|Y})$ . There is a covering $\{U_j:j\in J\}$ of Y by open constructible subsets of X and sections $s_j\in \Gamma (U_j\cap Z; {\mathcal {G}}_{|U_j\cap Y})$ , $j\in J$ , such that $s_{j|U_j\cap Y}=s_{|U_j\cap Y}$ . Since Y is quasi-compact, we can assume that J is finite, so $\cup \{U_j:j\in J\}$ is an open constructible subset of X. Therefore, there are an open constructible subset $O'$ in X and a normal, constructible locally closed subset $X'$ in X such that $Y\subseteq O'\subseteq X'\subseteq \cup \{U_j:j\in J\}$ . For each $j\in J$ , let $U^{\prime }_j=U_j\cap X'$ . Since $X'$ is normal and constructible, by the shrinking lemma, there are open constructible subsets $\{V^{\prime }_j:j\in J\}$ of $X'$ and closed constructible subsets $\{K^{\prime }_j:j\in J\}$ of $X'$ such that $V^{\prime }_j\subseteq K^{\prime }_j\subseteq U^{\prime }_j$ for every $j\in J$ and $X'= \cup \{V^{\prime }_j:j\in J\}$ . Since $X'$ is a constructible locally closed subset of $X,$ it is a constructible closed subset of an open subset of $X,$ which by quasi-compactness we may assume is also constructible. Replacing X by that constructible open subset, we may assume that $X'$ is a constructible closed subset of X. For each $j\in J$ , let $V_j=V^{\prime }_j\cap O'$ and $K_j=K^{\prime }_j$ . Then for each $j\in J, \ V_j$ is an open constructible subset of X and $K_j$ is a closed constructible subset of X with $V_j\subseteq K_j\subseteq U_j$ and $Y\subseteq \cup \{V_j:j\in J\}$ .

For $x \in Z\cap O'$ , set $J(x)=\{j\in J:x\in K_j\}$ , and let

$$\begin{align*}W_x = (\bigcap_{x \in V_l}V_l \cap \bigcap_{j\in J(x)}U_j) \setminus \bigcup_{k \notin J (x)}K_k. \end{align*}$$

Then $W_x$ is a constructible neighbourhood of x in X such that if $y\in W_x$ , then $J(y)\subseteq J(x)$ . Indeed, suppose that $y\in W_x$ , and let $i\in J(y)$ . Then $y\in K_i$ and either $x\in K_i$ , in which case $i\in J(x)$ , or $x\not \in K_i$ , in which case $i\not \in J(x)$ , so $y\in W_x\cap K_i=\emptyset $ .

Observe that for all $i,j \in J(x)$ , we have that $W_x$ is an open subset of both $U_i$ and $U_j$ . Hence, for every $i,j \in J(x)$ , we have $s_{i|W_x\cap Y}= s_{|W_x\cap Y}=s_{j|W_x\cap Y}$ . So, for $y \in W_x \cap Y$ , we have $(s_i)_y=(s_j)_y$ for any $i,j \in J (x)$ . This implies that the set

$$\begin{align*}W=\{z\in (\bigcup_{j\in J}V_j)\cap Z: (s_i)_z=(s_j)_z\,\,\textrm{for any}\,\, i,j\in J(z)\}\end{align*}$$

contains Y (clearly $Y\subseteq \bigcup _{x\in Z} W_x\cap Y\subseteq (\bigcup _{j\in J}V_j)\cap Z$ ). On the other hand, if $z\in W$ , then for any $i,j\in J(z)$ , we have $(s_i)_z=(s_j)_z$ , so $s_i=s_j$ in an open neighbourhood of z. Since $J(z)$ is finite, z has an open neighbourhood in Z on which $s_i=s_j$ for any $i,j \in J(z)$ . Thus W is an open neighbourhood of Y in Z. Since Y is quasi-compact, we may assume that W is of the form $U\cap Z$ for some open constructible subset U of X. Since $s_{i_{|W \cap V_i \cap V_j}} =s_{j_{|W \cap V_i \cap V_j}},$ there exists $t\in \Gamma (W;{\mathcal {G}} )$ such that $t_ {|W \cap V_j}=s_{j|W \cap V_j}$ . This proves that the morphism is surjective.

A general form of Lemma 3.22 is:

Lemma 3.23. Assume that X is an object of $\widetilde {{\mathfrak T}}$ . Let Z be a subspace of X, $\Phi $ a normal and constructible family of supports on X and Y a subset of Z such that for every constructible $D\in \Phi ,$ , $D\cap Y$ is a quasi-compact subset of Z having a fundamental system of normal and constructible locally closed neighbourhoods in X. Then for every ${\mathcal {G}} \in \mathrm {Mod} (A_Z)$ , the canonical morphism

$$\begin{align*}\underset{Y\subseteq U}{\underrightarrow{\lim}}\Gamma_{\Phi \cap U \cap Z}(U\cap Z;{\mathcal{G}}) \longrightarrow \Gamma_{\Phi \cap Y}(Y;{\mathcal{G}}_{|Y}),\,\,\,\end{align*}$$

where U ranges through the family of open constructible subsets of X containing Y, is an isomorphism.

For the locally semi-algebraic analogues of Lemmas 3.22 and 3.23, see [Reference Delfs17, Chapter II, Lemma 3.1 and Proposition 3.2]. Note that in the locally semi-algebraic case, one only requires that Y has a paracompact and regular locally semi-algebraic neighbourhood in $X,$ instead of a fundamental system of such neighbourhoods, since under this assumption, the family of all open locally semi-algebraic neighbourhoods of Y in X is a fundamental system of neighbourhoods of Y in X ([Reference Delfs17, Chapter I, Lemma 5.5]). Moreover, these are paracompact and regular: that is, they are normal, and one can apply the shrinking lemma ([Reference Delfs17, Chapter I, Proposition 5.1]). The assumption for $\Phi $ implies that the same holds for each $D\cap Y$ .

The o-minimal versions are [Reference Edmundo and Prelli25, Lemmas 3.2 and 3.3]. Unfortunately, in the statement of these lemmas and also in [Reference Edmundo and Prelli25, Proposition 3.7], the assumption that Y (respectively, $D\cap Y$ ) has a fundamental system of normal and constructible locally closed neighbourhoods in X is missing. However, note that this does not affect the main results of that paper since in those results, $Y=X$ and each D satisfies the hypothesis as $\Phi $ is normal and constructible. The only results affected are the Vietoris-Begle theorem and the homotopy axiom ([Reference Edmundo and Prelli25, Theorems 2.13 and 2.14]) that we fix below.

Recall that a sheaf ${\mathcal {F}}$ on a topological space X with a family of supports $\Phi $ is $\Phi $ -soft if and only if the restriction $\Gamma (X;{\mathcal {F}})\longrightarrow \Gamma (S;{\mathcal {F}}_{|S})$ is surjective for every $S\in \Phi $ . If $\Phi $ consists of all closed subsets of X, then ${\mathcal {F}}$ is simply called soft.

The topological analogue of the next result is [Reference Bredon14, Chapter II, 9.3].

The locally semi-algebraic analogue of Proposition 3.24 is [Reference Delfs17, Chapter II, Propositions 4.1 and 4.2], and the o-minimal version is [Reference Edmundo and Prelli25, Proposition 3.4].

The following topological result is often useful. It is an immediate consequence of Proposition 3.24, and we omit the proof and refer the reader to [Reference Edmundo and Prelli25, Proposition 3.6] (compare also with [Reference Bredon14, Chapter II, Propositions 9.2 and 9.12 and Corollary 9.13]).

We now come to the main result here. Its topological analogue is [Reference Bredon14, Chapter II, 9.6, 9.9 and 9.10], and the locally semi-algebraic analogue is [Reference Delfs17, Chapter II, Propositions 4.8, 4.12 and Corollary 4.13]. The o-minimal version is [Reference Edmundo and Prelli25, Proposition 3.7] (as mentioned above, the assumption that $D\cap Y$ has a fundamental system of normal and constructible locally closed neighbourhoods in X is missing).

Proof. Point (1) follows since on any topological space $Y,$ the full additive subcategory of $\mathrm {Mod}(A_Y)$ of injective (and flabby) A-sheaves is co-generating (see, for example, [Reference Kashiwara and Schapira39, Proposition 2.4.3]) and injective A-sheaves are flabby, so $\Phi \cap Y$ -soft by Lemma 3.23. On the other hand, (3) follows as usual from (2) by a simple diagram chase using Proposition 3.24 (3).

We now prove (2). Let $s" \in \Gamma _{\Phi \cap Y}(Y,{\mathcal {F}}")$ . We have to find $s \in \Gamma _{\Phi \cap Y}(Y,{\mathcal {F}})$ such that $\beta (s)=s"$ . Since $\Phi $ is normal and constructible, there exist an open constructible subset U of X and a closed constructible subset C of X such that $\mathrm {supp}\,s" \subseteq U\subseteq C$ and $C \in \Phi $ .

Suppose there is $t\in \Gamma (C\cap Y; {\mathcal {F}})$ such that $\beta (t)=s^{\prime \prime }_{\,\,|C\cap Y}$ . Then $\beta (t_{|(C\setminus U)\cap Y})=0,$ so $t_{|(C\setminus U)\cap Y}=\alpha (s')$ for some $s'\in \Gamma ((C\setminus U)\cap Y;{\mathcal {F}}')$ . Since ${\mathcal {F}}'$ is $\Phi \cap Y$ -soft, $s'$ can be extended to a section $s'\in \Gamma (C\cap Y; {\mathcal {F}}')$ . Then $(t-\alpha (s'))_{|(C\setminus U)\cap Y}=0$ and so can be extended, by $0,$ to a section $s\in \Gamma (Y;{\mathcal {F}})$ with $\mathrm {supp} \,s\subseteq C\cap Y\in \Phi \cap Y$ and $\beta (s)=s"$ .

Now, considering the exact sequence

$$\begin{align*}0\to{\mathcal{F}}^{\prime}_{C \cap Y}\stackrel{\alpha}\to{\mathcal{F}}_{C \cap Y}\stackrel{\beta}\to{\mathcal{F}}^{\prime\prime}_{C\cap Y}\to 0, \end{align*}$$

since by Fact 3.25 (iii) ${\mathcal {F}}^{\prime }_{C \cap Y}$ is still $\Phi \cap Y$ -soft, replacing ${\mathcal {F}}',{\mathcal {F}},{\mathcal {F}}"$ with ${\mathcal {F}}^{\prime }_{C \cap Y}, \ {\mathcal {F}}_{C \cap Y}$ and ${\mathcal {F}}^{\prime \prime }_{C\cap Y}$ , respectively, we are reduced to proving that the sequence

$$\begin{align*}0\to\Gamma(C\cap Y;{\mathcal{F}}')\stackrel{\alpha}\to\Gamma(C\cap Y; {\mathcal{F}})\stackrel{\beta}\to\Gamma(C\cap Y;{\mathcal{F}}")\to 0 \end{align*}$$

is exact.

Let $s" \in \Gamma (C\cap Y;{\mathcal {F}}")$ . There is a covering $\{U_j:j\in J\}$ of $C\cap Y$ by open constructible subsets of X and sections $s_j\in \Gamma (U_j\cap (C\cap Y); {\mathcal {F}})$ , $j\in J$ , such that $\beta (s_{j})=s^{\prime \prime }_{\,\,|U_j\cap (C\cap Y)}$ . By the assumptions on $C\cap Y$ , we can assume that J is finite, so $\cup \{U_j:j\in J\}$ is an open constructible subset of $X,$ and furthermore, there are an open constructible subset $O'$ in X and a normal, constructible locally closed subset $X'$ in X such that $C\cap Y\subseteq O'\subseteq X'\subseteq \cup \{U_j:j\in J\}$ . For each $j\in J$ , let $U^{\prime }_j=U_j\cap X'$ . Since $X'$ is normal and constructible, by the shrinking lemma, there are open constructible subsets $\{V^{\prime }_j:j\in J\}$ of $X'$ and closed constructible subsets $\{K^{\prime }_j:j\in J\}$ of $X'$ such that $V^{\prime }_j\subseteq K^{\prime }_j\subseteq U^{\prime }_j$ for every $j\in J$ and $X'= \cup \{V^{\prime }_j:j\in J\}$ . Now we may replace X by a constructible open subset and assume that $X'$ is a constructible closed subset of X. For each $j\in J$ , let $V_j=V^{\prime }_j\cap O'$ and $K_j=K^{\prime }_j$ . Then for each $j\in J, \ V_j$ is an open constructible subset of X and $K_j$ is a closed constructible subset of X with $V_j\subseteq K_j\subseteq U_j$ and $C\cap Y\subseteq \cup \{V_j:j\in J\}$ .

We now proceed by induction on $\#J$ . Suppose that $J=I\cup \{j\}$ , and let $V_I=\cup \{V_i:i\in I\}, \ K_I=\cup \{K_i:i\in I\}$ and $U_I=\cup \{U_i:i\in I\}$ . By induction hypothesis, let $s_I\in \Gamma (K_I\cap (C\cap Y); {\mathcal {F}})$ be such that $\beta (s_{I})=s^{\prime \prime }_{\,\,|K_I\cap (C\cap Y)}$ . Since $\beta ((s_I-s_j)_{|K_I\cap K_j})=0,$ there is $s'\in \Gamma (K_I \cap K_j\cap (C\cap Y);{\mathcal {F}}')$ such that $\alpha (s')=(s_I-s_j)_{|K_I\cap K_j}$ that extends to $s' \in \Gamma (C\cap Y;{\mathcal {F}}')$ since ${\mathcal {F}}'$ is $\Phi \cap Y$ -soft. Replacing $s_I$ with $s_I-\alpha (s^{\prime }_{\,|K_I\cap (C\cap Y)})$ , we may suppose that $s_I=s_j$ on $K_I \cap K_j\cap (C\cap Y)$ . Then since $(K_I\cup K_j)\cap (C\cap Y)= C\cap Y,$ there exists $s \in \Gamma (C\cap Y;{\mathcal {F}})$ such that $s_{| K_I\cap (C\cap Y)}=s_I$ and $s_{|K_j\cap (C\cap Y)}=s_j$ . Thus the induction proceeds.

We now include several corollaries that will be useful later.

Corollary 3.27. Assume that X is an object in $\widetilde {{\mathfrak T}}$ . Suppose either that $\Phi $ is a normal and constructible family of supports on X and Z is a (constructible) locally closed subset of X or that $\Phi $ is any family of supports on X and Z is a closed subset of X. If ${\mathcal {F}}\in \mathrm {Mod} (A_Z)$ , then

$$\begin{align*}H_{\Phi }^*(X;i_{Z!}{\mathcal{F}})=H_{\Phi |Z}^*(Z; {\mathcal{F}}). \end{align*}$$

It also follows that if $X, \ Z, Y$ and $\Phi $ are as in Proposition 3.26, then $H^q_{\Phi \cap Y}(Y; {\mathcal {G}} )$ is the qth cohomology of the cochain complex

$$\begin{align*}0\rightarrow \Gamma _{\Phi \cap Y}(Y; {\mathcal{I}} ^0)\rightarrow \Gamma _{\Phi \cap Y} (Y;{\mathcal{I}}^1)\rightarrow \ldots \end{align*}$$

where

$$\begin{align*}0\rightarrow {\mathcal{G}} \rightarrow {\mathcal{I}} ^0_{|Y}\rightarrow {\mathcal{I}}^1_{|Y}\rightarrow \ldots \end{align*}$$

is a resolution of ${\mathcal {G}} $ by $\Phi \cap Y$ -soft sheaves in $\mathrm {Mod} (A_Y)$ .

Moreover, if ${\mathcal {G}}$ is any flabby sheaf in $\mathrm {Mod} (A_X)$ , then the restriction $\Gamma _{\Phi }(X; {\mathcal {G}})\to \Gamma _{\Phi \cap Y}(Y;{\mathcal {G}} _{|Y})$ is surjective (by Lemma 3.23) and $H^p_{\Phi \cap Y}(Y;{\mathcal {G}}_{|Y})=0$ for all $p>0$ (since ${\mathcal {G}} _{|Y}$ is $\Phi \cap Y$ -soft by Lemma 3.22). This means Y is $\Phi $ -taut in X (see [Reference Bredon14, Chapter II, Definition 10.5]).

Since on any topological space X an open subset is $\Psi $ -taut in X for any family of supports $\Psi $ on $X,$ it follows from [Reference Bredon14, Chapter II, Theorem 10.6] that:

Corollary 3.28. Assume that X is an object in $\widetilde {{\mathfrak T}}$ . Let Z be a subspace of $X, \ \Phi $ a normal and constructible family of supports on X and Y a subspace of Z such that for every constructible $D\in \Phi , \ D\cap Y$ is a quasi-compact subset of Z having a fundamental system of normal and constructible locally closed neighbourhoods in X. Then for every ${\mathcal {G}}\in \mathrm {Mod} (A_Z)$ , the canonical homomorphism

$$\begin{align*}\underset{Y\subseteq U}{\underrightarrow{\lim}}H^q_{\Phi \cap U\cap Z}(U\cap Z;{\mathcal{G}}) \longrightarrow H^q_{\Phi \cap Y}(Y;{\mathcal{G}}_{|Y}),\,\,\, \end{align*}$$

where U ranges through the family of open constructible subsets of X containing Y, is an isomorphism for every $q\geq 0$ .

For the locally semi-algebraic analogue of the above, compare with [Reference Delfs17, Chapter II, Theorem 5.2].

Since fibres of morphisms in $\widetilde {{\mathfrak T}}$ are quasi-compact (Remark 3.10), applying Corollary 3.28, we obtain the following result (compare with [Reference Delfs17, Chapter II, Theorem 7.1]).

Theorem 3.29 (Base change formula).

Let $f\colon X\to Y$ be a morphism in $\widetilde {{\mathfrak T}}$ . Assume that f maps constructible closed subsets of X to closed subsets of Y and that every $\alpha \in Y$ has an open constructible neighbourhood W in Y such that $\alpha $ is closed in W and $f^{-1}(W)$ is a normal and constructible subset of X. Let ${\mathcal {F}} \in \mathrm {Mod} (A_X)$ . Then for every $\alpha \in Y,$ the canonical homomorphism

$$\begin{align*}(R^qf_*{\mathcal{F}})_{\alpha } \longrightarrow H^q(f^{-1}(\alpha );{\mathcal{F}}_{|f^{-1}(\alpha )})\,\,\, \end{align*}$$

is an isomorphism for every $q\geq 0$ .

Proof. Fix $\alpha \in Y$ . First, notice that

$$\begin{align*}\{f^{-1}(V): V\subseteq W\,\,\textrm{an open constructible neighbourhood of}\,\,\alpha \} \end{align*}$$

is a fundamental system of open neighbourhoods of $f^{-1}(\alpha )$ . Indeed, let U be an open neighbourhood of $f^{-1}(\alpha )$ . Since $f^{-1}(\alpha )$ is quasi-compact, we may assume that U is an open constructible neighbourhood of $f^{-1}(\alpha )$ . Then by assumption, $f(X\setminus U)$ is a closed subset of Y not containing $\alpha $ . Let V be an open constructible neighbourhood of $\alpha $ in W contained in $Y\setminus f(X\setminus U)$ . Then $f^{-1}(V)\subseteq U$ .

Also notice that $R^qf_*{\mathcal {F}}$ is the sheaf associated to the presheaf sending V to $H^q(f^{-1}(V);{\mathcal {F}})$ . So

$$\begin{align*}(R^qf_*{\mathcal{F}})_{\alpha }=\underset{\alpha \subseteq V\subseteq W}{\underrightarrow{\lim}}H^q(f^{-1}(V);{\mathcal{F}})=\underset{f^{-1}(\alpha )\subseteq U}{\underrightarrow{\lim}}H^q(U;{\mathcal{F}}),\end{align*}$$

where V (respectively, U) ranges through the family of open constructible subsets of Y (respectively, X).

Now, by assumption, $\alpha \in W$ is closed in W and $f^{-1}(W)$ is an open normal and constructible subset of X containing $f^{-1}(\alpha )$ . So the family of all closed subsets of $f^{-1}(W)$ is a normal and constructible family of supports on $f^{-1}(W)$ (Example 3.20); and since $f^{-1}(\alpha )$ is closed in $f^{-1}(W),$ for every constructible closed subset $D\subseteq f^{-1}(W)$ , $D\cap f^{-1}(\alpha )$ is a quasi-compact subset of $f^{-1}(W)$ having a fundamental system of normal and constructible locally closed neighbourhoods in $f^{-1}(W)$ .

Therefore the result follows from Corollary 3.28 applied to $X=Z=f^{-1}(W)$ , $\Phi $ the family of all closed subsets of $f^{-1}(W)$ and $Y=f^{-1}(\alpha )$ .

Using classical arguments, the Base change formula implies the following form of the Vietoris-Begle theorem:

Theorem 3.30 (Vietoris-Begle theorem).

Let $f\colon X\longrightarrow Y$ be a surjective morphism in ${\mathfrak T}$ . Assume that f maps constructible closed subsets of X onto closed subsets of Y and that every $\alpha \in Y$ has an open constructible neighbourhood W in Y such that $\alpha $ is closed in W and $f^{-1}(W)$ is a normal and constructible subset of X. Let ${\mathcal {F}}\in \mathrm {Mod} (A_Y),$ and suppose that $f^{-1}({\alpha })$ is connected and $H^q(f^{-1}({\alpha }); f^{-1}{\mathcal {F}}_{|f^{-1}({\alpha })}) \ =0$ for $q>0$ and all $\alpha \in Y$ . Then for any constructible family of supports $\Psi $ on Y, the induced map

$$\begin{align*}f^*\colon H^*_{\Psi }(Y;{\mathcal{F}})\longrightarrow H^*_{f^{-1}\Psi }(X;f^{-1}{\mathcal{F}})\end{align*}$$

is an isomorphism.

Proof. The homomorphism $f^*\colon H^*_{\Psi }(Y;{\mathcal {F}})\longrightarrow H^*_{f^{-1}\Psi }(X;f^{-1}{\mathcal {F}})$ is the composition $\epsilon \circ \eta $ , where

$$ \begin{align*} \epsilon \colon H^*_{\Psi }(Y; f_*(f^{-1}{\mathcal{F}}))\to H^*_{f^{-1}\Psi}(X,f^{-1}{\mathcal{F}}) \end{align*} $$

is the canonical edge homomorphism $E_2^{*,0}\to E^*$ in the Leray spectral sequence

$$\begin{align*}H^p_{\Psi}(Y;R^qf_*(f^{-1}{\mathcal{F}}))\Rightarrow H^{p+q}_{f^{-1}\Psi }(X; f^{-1}{\mathcal{F}}) \end{align*}$$

of $f^{-1}{\mathcal {F}}$ with respect to f and $\eta \colon H^*_{\Psi }(Y;{\mathcal {F}})\to H^*_{f^{-1}\Psi }(X;f_*(f^{-1}{\mathcal {F}}))$ is induced by the canonical adjunction homomorphism

$$\begin{align*}{\mathcal{F}} \to f_*f^{-1}{\mathcal{F}}. \end{align*}$$

By Theorem 3.29 and the hypothesis, $R^qf_*(f^{-1}{\mathcal {F}})=0$ for all $q>0$ , so the Leray sequence splits and $\epsilon $ is an isomorphism. On the other hand, by Theorem 3.29 and since $f^{-1}(\alpha )$ is connected, we have

$$ \begin{align*} (f_*f^{-1}{\mathcal{F}})_{\alpha } &= (R^0f_*f^{-1}{\mathcal{F}})_{\alpha }\\ &\simeq H^0(f^{-1}(\alpha ); (f^{-1}{\mathcal{F}})_{|f^{-1}(\alpha )})\\ &\simeq H^0(f^{-1}(\alpha ); {\mathcal{F}}_{\alpha })\\ &\simeq {\mathcal{F}} _{\alpha}. \end{align*} $$

Hence adjunction homomorphism ${\mathcal {F}} \to f_*f^{-1}{\mathcal {F}}$ is an isomorphism and $\eta $ is also an isomorphism.

We end this section with the following, which follows quickly from previous results exactly the same way as its topological analogue ([Reference Bredon14, Chapter II, 16.1]):

Proof. (1) $\Rightarrow $ (2) follows from Proposition 3.26 and Fact 3.25 (iii). (2) $\Rightarrow $ (3) is trivial. (3) is equivalent to (4) by Corollary 3.27. For (4) $\Rightarrow $ (1), consider a constructible closed set C in $\Phi $ and the exact sequence $0\longrightarrow {\mathcal F}_{X\setminus C}\longrightarrow {\mathcal F}\longrightarrow {\mathcal F}_C\longrightarrow 0$ . The associated long exact cohomology sequence

$$ \begin{align*}\dots \rightarrow \Gamma _{\Phi }(X;{\mathcal F})\rightarrow \Gamma _{\Phi |C}(X;{\mathcal F}_{|C})\rightarrow H_{\Phi |X\setminus C}^1(X\setminus C;{\mathcal F}_{|X\setminus C})\rightarrow \dots \end{align*} $$

shows that $\Gamma _{\Phi }(X;{\mathcal F})\longrightarrow \Gamma _{\Phi |C}(X;{\mathcal F}_{|C})$ is surjective. Hence ${\mathcal F}$ is $\Phi $ -soft by Proposition 3.24 (5).

4.1 The o-minimal site on ${\mathbf {\Gamma }_{\infty } }$ -definable sets

Here we introduce the natural first-order logic structure ${\mathbf {\Gamma }_{\infty } }$ induced by ${\mathbf {\Gamma }}$ on ${\Gamma _{\infty } }$ and show that ${\mathbf {\Gamma }_{\infty } }$ -definable sets are equipped with a site making them ${\mathcal {T}}$ -spaces.

For $x=(x_1, \ldots , x_m)\in {\Gamma ^m_{\infty } },$ the support of x, denoted $\mathrm {supp}\, x,$ is defined by

$$\begin{align*}\mathrm{supp} \, x=\{i\in \{1, \ldots, m\}: x_i\neq \infty \}. \end{align*}$$

For $L\subseteq \{1,\ldots , m\}$ , let

$$\begin{align*}({\Gamma^m_{\infty} } )_{L}=\{x\in {\Gamma^m_{\infty} }: \mathrm{supp} \, x= L\}. \end{align*}$$

Then ${\Gamma ^m_{\infty } }=\bigsqcup _{L\subseteq \{1,\ldots , m\}}({\Gamma ^m_{\infty } } )_{L}$ , $({\Gamma ^m_{\infty } } )_{\{1,\ldots , m\}}={\Gamma ^m}$ and, if $\tau _{L}\colon {\Gamma ^m_{\infty } }\to {\Gamma ^{|L|}_{\infty } }$ is the projection onto the $|L|$ coordinates in $L,$ then the restriction $\tau _{L|}\colon ({\Gamma ^m_{\infty } })_{L}\to {\Gamma ^{|L|}}$ is a bijection and $\tau _{\{1,\ldots ,m\} |}\colon {\Gamma ^m} \to {\Gamma ^m}$ is the identity.

If $\pi \colon {\Gamma ^m_{\infty } } \to {\Gamma ^k_{\infty } }$ is the projection onto the first k coordinates, $\pi '\colon {\Gamma ^m_{\infty } } \to \Gamma ^{m-k}_{\infty }$ is the projection onto the last $m-k$ coordinates, and for $L\subseteq \{1,\ldots , m\}$ , $\pi (L)=\{1,\ldots , k\}\cap L$ and $\pi '(L)=-k+\{k+1,\ldots , m\}\cap L,$ then

• (*)

  $$\begin{align*}\mathrm{supp} \,x=\mathrm{supp} \, \pi (x)\sqcup (k+\mathrm{supp} \, \pi'(x))\end{align*}$$

• and moreover,

  
  where $\pi ^L_k$ is a projection onto the first $\# \pi (L)$ coordinates and
  
  are commutative diagrams.

If, for $X\subseteq {\Gamma ^m_{\infty } }$ and for $L\subseteq \{1,\ldots , m\}$ , we set

$$\begin{align*}X_{L}=X\cap ({\Gamma^m_{\infty} })_L,\end{align*}$$

then $X=\bigsqcup _{L\subseteq \{1,\ldots , m\}}X_{L}$ and $X_{\{1,\ldots , m\}}=X\cap \Gamma ^m$ . Furthermore, the restriction $\tau _{L|}\colon X_{L}\to \tau _{L}(X_{L})$ is a bijection, $\tau _L(X_L)\subseteq {\Gamma } ^{|L|}$ and $\tau _{\{1,\ldots ,m\} |}\colon X\cap {\Gamma ^m}\to X\cap {\Gamma ^m}$ is the identity.

For each m, let

$$\begin{align*}\mathfrak{G}_m=\{X\subseteq {\Gamma^m_{\infty} }: \tau _{L}(X_{L})\subseteq {\Gamma} ^{|L|}\,\,\textrm{is } {\mathbf{\Gamma}}\textrm{-definable}\,\,\textrm{for every}\,L\subseteq \{1, \ldots ,m\}\}. \end{align*}$$

Recall that an o-minimal structure (possibly with end points) has definable Skolem functions if, given a definable family $\{Y_t\}_{t\in T}$ , there is a definable function $f\colon T\to \bigcup _{t\in T} Y_t$ such that $f(t)\in Y_t$ for each $t\in T$ . By the (observations before the) proof of [Reference van den Dries53, Chapter 6, (1.2)] (see also Comment (1.3) there), the o-minimal structure has definable Skolem functions if and only if every nonempty definable set X has a definable element $e(X)\in X$ .

The following Proposition is left to the reader.

Results in o-minimality usually are stated and proved for o-minimal structures without end points. Nearly all of those results can be checked to hold with exactly the same proof when there is an end point. Nevertheless, for convenience, we will now introduce a new structure without end points that contains as a substructure a copy of ${\mathbf {\Gamma }_{\infty } }$ .

Let ${\Sigma }=\{0\}\times {\Gamma _{\infty } } \cup \{1\}\times {\Gamma }$ be equipped with the natural order extending $<$ on ${\Gamma }$ and on ${\Gamma _{\infty } }$ such that $(0,x) < (1,y)$ for all $x\in {\Gamma _{\infty } }$ and all $y\in {\Gamma }$ .

Set ${\Sigma } _0=\{0\}\times {\Gamma _{\infty } }$ , ${\Sigma } _1=\{1\}\times {\Gamma }$ and, for $\alpha \in 2^m,$ which below we identify with a sequence of 0s and 1s of length $m$ , let

$$\begin{align*}{\Sigma} _{\alpha }=\Pi _{i=1}^m{\Sigma} _{\alpha (i)}. \end{align*}$$

Then ${\Sigma ^m}=\bigsqcup _{\alpha \in 2^m}{\Sigma } _{\alpha }$ , ${\Sigma } _{\overline {0}}={\Sigma } _0^m$ and ${\Sigma } _{\overline {1}}={\Sigma } _1^m$ . If

$$\begin{align*}\sigma _m\colon {\Sigma ^m} \to {\Gamma^m_{\infty} } \colon ((\alpha (1),x_1), \ldots, (\alpha (m),x_m))\mapsto (x_1,\ldots ,x_m) \end{align*}$$

is the natural projection, then the restriction $\sigma _{m|}\colon {\Sigma } _{\alpha }\to {\Gamma ^m_{\infty } }$ is injective, $\sigma _{m|}\colon {\Sigma } _{\overline {0}}\to {\Gamma ^m_{\infty } }$ is a bijection with inverse the natural inclusion $\iota _m\colon {\Gamma ^m_{\infty } } \to {\Sigma } _0^m\colon (x_1,\ldots , x_m)\mapsto ((0,x_1), \ldots , (0,x_m))$ , and $\sigma _{m|}\colon {\Sigma } _{\overline {1}}\to {\Gamma ^m}$ is a bijection with inverse the natural inclusion $\iota _m\colon {\Gamma ^m} \to {\Sigma } _1^m\colon (x_1,\ldots , x_m)\mapsto ((1,x_1), \ldots , (1,x_m))$

If $\pi \colon {\Sigma ^m} \to {\Sigma ^k}$ is the projection onto the first k pairs of coordinates, $\pi '\colon {\Sigma ^m} \to {\Sigma ^{m-k}}$ is the projection onto the last $m-k$ pairs of coordinates, for $\alpha \in 2^m, \ \pi (\alpha )\in 2^k$ is given by $\pi (\alpha )=\alpha _{|\{1,\ldots , k\}}$ and $\pi '(\alpha )\in 2^{m-k}$ is given by $\pi '(\alpha )(s)=\alpha (s+k),$ then:

• (**)

  $$\begin{align*}\alpha =\pi (\alpha )* \pi'(\alpha ), \end{align*}$$
  where $*$ is a concatenation of sequences, and moreover,
  
  and
  
  are commutative diagrams.

If, for $X\subseteq {\Sigma ^m}$ and for $\alpha \in 2^m$ , we set

$$\begin{align*}X_{\alpha }=X\cap {\Sigma} _{\alpha }, \end{align*}$$

then $X=\bigsqcup _{\alpha \in 2^m}X_{\alpha }$ . Furthermore, the restriction $\sigma _{m|}\colon X_{\alpha }\to \sigma _{m}(X_{\alpha })$ is a bijection, $\sigma _m(X_{\alpha })\subseteq {\Gamma ^m_{\infty } }$ .

For each m, let

$$\begin{align*}\mathfrak{S}_m=\{X\subseteq {\Sigma ^m}: \sigma _{m}(X_{\alpha })\subseteq {\Gamma^m_{\infty} }\,\,\textrm{is } {\mathbf{\Gamma}_{\infty} }\textrm{-definable}\,\,\textrm{for every}\,\sigma \in 2^m\}. \end{align*}$$

Similarly to Proposition 4.1, we have:

The following remarks are obvious from the constructions above:

We now make a couple of observations comparing ${\mathbf {\Gamma }}$ , ${\mathbf {\Gamma }_{\infty } } $ and ${\mathbf {\Sigma }}$ .

Recall the notion of cell in ${\Gamma ^n_{\infty } }$ defined inductively as follows:

• • A cell in ${\Gamma _{\infty } }$ is either a singleton $\{a\}$ or $\{\infty \}$ or an open interval of the form $(-\infty , a)$ or $(a,b)$ for $a,b\in {\Gamma }$ with $a<b$ .

• • A cell in $\Gamma ^{n+1}_{\infty }$ is a set of the form

  $$\begin{align*}\Gamma(h)=\{(x,y)\in \Gamma ^{n+1}_{\infty }: x\in X\,\,\textrm{and}\,\,y=h(x)\}, \end{align*}$$
  or
  $$\begin{align*}(f,g)_X=\{(x,y)\in \Gamma ^{n+1}_{\infty }: x\in X\,\,\textrm{and}\,\,f(x)<y<g(x)\}, \end{align*}$$
  or
  $$\begin{align*}(-\infty , g)_X=\{(x,y)\in \Gamma ^{n+1}_{\infty }: x\in X\,\,\textrm{and}\,\,y<g(x)\}, \end{align*}$$
  for some ${\mathbf {\Gamma }_{\infty } }$ -definable and continuous maps $f, g,h\colon X\to {\Gamma _{\infty } }$ with $f<g$ , where X is a cell in ${\Gamma ^n_{\infty } }$ .

In either case, X is called the domain of the defined cell.

In an arbitrary o-minimal structure, a definable set X is definably connected if and only if the only clopen definable subsets of X are $\emptyset $ and X. We say that a ${\mathbf {\Gamma }_{\infty } }$ -definable subset of ${\Gamma ^m_{\infty } }$ is ${\mathbf {\Gamma }_{\infty } }$ -definably locally closed if and only if it is the intersection of an open ${\mathbf {\Gamma }_{\infty } }$ -definable subset of ${\Gamma ^m_{\infty } }$ and a closed ${\mathbf {\Gamma }_{\infty } }$ -definable subset of ${\Gamma ^m_{\infty } }$ or equivalently if and only if it is open in its closure in ${\Gamma ^m_{\infty } }$ . A similar definition applies in ${\mathbf {\Gamma }}$ .

Just like in ${\mathbf {\Gamma }},$ if $({\Gamma },<)$ is nonarchimedean as a linear order, then infinite ${\mathbf {\Gamma }_{\infty } }$ -definable spaces with the topology generated by open definable subsets are in general totally disconnected and not locally compact. So, as in Example 3.7 (3), to develop cohomology, one studies ${\mathbf {\Gamma }_{\infty } }$ -definable spaces X equipped with the o-minimal site:

Proposition 4.9. Let ${\mathbf {\Gamma }}= ({\Gamma },<, \ldots )$ be an arbitrary o-minimal structure without end points. If X is a ${\mathbf {\Gamma }_{\infty } }$ -definable subset of ${\Gamma ^m_{\infty } }$ , let

$$\begin{align*}{\mathcal{T}}=\{U \in \mathrm{Op}(X): U\,\, \text{is } {\mathbf{\Gamma}_{\infty} }\text{-definable}\}. \end{align*}$$

Then X is a ${\mathcal {T}}$ -space, $X_{{\mathcal {T}}}=X_{\mathrm {def}}$ , and $\widetilde {X}_{{\mathcal {T}}}$ is the o-minimal spectrum $\widetilde {X}$ of X: that is, its points are types over ${\Gamma _{\infty } }$ concentrated on X. Furthermore, there is an equivalence of categories

$$\begin{align*}\mathrm{Mod}(A_{X_{\mathrm{def}}}) \simeq \mathrm{Mod}(A_{\widetilde{X}}). \end{align*}$$