The Six Grothendieck Operations on O-Minimal Sheaves

THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES

MARIO´ J. EDMUNDO AND LUCA PRELLI

Abstract. In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coeﬃcient formula; (iii) derived base change formula; (iv) K¨unnethformula; (v) local and global Verdier duality.

1. Introduction The study of o-minimal structures ([16]) is the analytic part of model theory which deals with theories of ordered, hence topological, structures satisfying certain tameness properties. It generalizes piecewise linear geometry ([16, Chapter 1, §7]), semi-algebraic geometry ([4]) and globally sub-analytic geometry ([36], also called finitely sub-analytic in [15]) and it is claimed to be the formalization of Grothendieck’s notion of tame topology (topologie modérée).See [16] and [18]. The most striking successes of this model-theoretic point of view of sub-analytic geometry include, on the one hand, an understanding of the behavior at infinity of certain important classes of sub-analytic sets as in Wilkie’s ([55]) as pointed out by Bierstone and Milman [3], and on the other hand, the recent, somehow surprising, first unconditional proof of the André-Oortconjecture for mixed Shimura varieties expressible as products of curves by Pila [45] following previous work also using o-minimality by Pila and Zanier ([46]), Pila and Wilkie ([47]) and Peterzil and Starchenko ([43]). The goal of this paper is to contribute further to the claim that o-minimality does indeed realize Grothendieck’s notion of topologie modéréeby developing the formalism of the Grothendieck six operations on o-minimal sheaves, extending Delfs results for semi-algebraic sheaves ([12]) as well as Kashiwara and Schapira ([38]) and Prelli’s ([52]) results for sub-analytic sheaves restricted to globally sub-analytic spaces (as we deal, for now, only with definable spaces and not locally definable spaces - the later more general case will be dealt with in a sequel to this paper). In the semi-algebraic case nearly all of our results are completely new. Indeed, in [12, Section 8], Delfs constructs the semi-algebraic proper direct image functor and only proves two basic results about this functor (base change and commutativity

Date: July 21, 2014. 2010 Mathematics Subject Classification. 03C64; 55N30. The first author was supported by Funda¸cãopara a Ciênciae a Tecnologia, Financiamento Base 2008 - ISFL/1/209. The second author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitàe le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and was supported by Marie Curie grant PIEF-GA-2010-272021. This work is part of the FCT project PTDC/MAT/101740/2008. Keywords and phrases: O-minimal structures, proper direct image. 1 2 MARIO´ J. EDMUNDO AND LUCA PRELLI with small inductive limits) and then conjectures in [12, Remark 8.11 ii)] that: “It seems that the results of this section suffice to prove a semi-algebraic analogue of Verdier duality (by the same proof as in the theory of locally compact spaces c.f [54]).” In the globally sub-analytic case we introduce a new globally sub-analytic proper direct image functor which, unlike Kashiwara and Schapira ([38]) and Prelli’s ([52]) sub-analytic proper direct image functor, generalizes to arbitrary o-minimal structures including: (i) arbitrary real closed fields; (ii) the non-standard models of sub- analytic geometry ([17]); (iii) the non-standard o-minimal structure which does not came from a standard one as in [40, 33]. See Remark 3.15 for further details on this. Moreover, unlike [38, 52], our definition is compatible with the restriction to open subsets (see Remark 3.16).

The Grothendieck formalism developed here allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coefficient formula; (iii) derived base change formula; (iv) Künnethformula; (v) local and the global Verdier duality. It also sets up the framework for: (a) defining new o-minimal homology o theories H (X, ) := H∗(a ◦ a. ) extending o-minimal singular homology or ∗ Q X.o X Q BM ∗ .o the o-minimal Borel-Moore homology H∗ (X, Q) := H (aX∗ ◦ aX Q); (b) the full development of o-minimal geometry including the theory of characteristic classes, Hirzebruch-Riemann-Roch formula and Atiyah-Singer theorem in the non-standard o-minimal context. We expect that the theory developed here will have interesting applications even in algebraic analysis. With our definition it is possible to treat the global sub- analytic sites in which are defined sheaves of functions with growth conditions up to infinity (e.g. tempered and Whitney C∞ or holomorphic functions). This kind of objects are very important for applications as in [9] and in [10] (in which globally sub-analytic sheaves are hidden under the notion of ind-sheaves on a bordered space). Some quicker applications have already been obtained in the theory of definable groups in arbitrary o-minimal structures. See [22]. Using the notion of orientability defined here and the Künnethformula, we show that if G is a definably connected, definably compact definable group, then G is orientable and the o-minimal sheaf ∗ cohomology H (G; kG) of G with coefficients in a field k is a connected, bounded, Hopf algebra over k of finite type. Furthermore, using a consequence the Alexander duality proved here we develop degree theory which, when combined with a new o-minimal fundamental group in arbitrary o-minimal structure, gives the compu- tation of the subgroup of m-torsion points of G when G is abelian - extending the main result of [23] which was proved in o-minimal expansions of ordered fields using the o-minimal singular (co)homology. This result is enough to settle Pillay’s conjecture for definably compact definable groups ([50] and [34])in arbitrary o-minimal structures. Proofs of Pillay’s conjecture were recently also obtained in o-minimal expansions of ordered groups ([27] and [42]). However, in view of ([28]), it seems that those new proofs do not generalize to arbitrary o-minimal structures. Pillay’s conjecture is a non-standard analogue of Hilbert’s 5o problem for locally compact topological groups, roughly it says that after taking the quotient by a “small subgroup” (a smallest type-definable subgroup of bounded index) the quotient when equipped with the the so called logic topology is a compact real Lie group of the THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 3 same dimension.

The structure of the paper is the following. In Section 2 we introduce the setting where we will work on, listing all the preliminary results that will be needed throughout the rest of the paper. In Section 3 we deﬁne the proper direct image operation on o-minimal sheaves and prove its fundamental properties. In Section 4 we obtain the local and the global Verdier duality and its consequences. Finally we note that, for the readers convenience, at the beginning of each section, after we introduce the set up, we include an extended summary in which we compare the ideas involved in the proofs of the section with those present in the references cited in the bibliography.

2. Preliminaries

In this paper we work in an arbitrary o-minimal structure M = (M, <, (c)c∈C, (f)f∈F , (R)R∈R) with definable Skolem functions. We refer the reader to [16] for basic o-minimality. We let Def be the category whose objects are definable spaces and whose morphisms are continuous definable maps between definable spaces - here and below “definable” always means definable in M with parameters. (See [16, Chapter 10, §1]). If M is a real closed field (R, <, 0, 1, +, ·), then Def is the category whose objects are semi-algebraic spaces over R and whose morphisms are continuous semi-algebraic maps between such semi-algebraic space ([16, Chapter 2] and [12, Chapter I, Example 1.1]); if M is Ran = (R, <, 0, 1, +, ·, (f)f∈an) - the field of real numbers expanded by restricted analytic functions, then Def is the category whose objects are globally sub-analytic spaces together with continuous maps with globally sub-analytic graphs between such spaces ([14]); if M is an ordered vector space (V, <, 0, +, (d)d∈D) over an ordered division ring D, then Def is the category whose objects are the piecewise linear spaces in this vector space together with continuous piecewise linear maps between such spaces ([16, Chapter 1, §7]). Objects of Def are equipped with a topology determined by the order topology on (M, <). However, if (M, <) is non-archimedean then infinite definable spaces are totally disconnected and not locally compact, so one studies definable spaces equipped with the o-minimal site and replaces topological notions (connected, normal, compact, proper) by their definable analogues (definably connected, definably normal, definably compact, definably proper). The o-minimal site ([20]) generalizes both the semi-algebraic site ([12]) and the sub-analytic site ([38], [52]). Given an object X of Def the o-minimal site Xdef on X is the category Op(Xdef ) whose objects are open (in the topology of X mentioned above) definable subsets of X, the morphisms are the inclusions and the admissible covers Cov(U) of U ∈ Op(Xdef ) are covers by open definable subsets of X with finite sub-covers. As shown in [20, Proposition 3.2], if A is a commutative ring, then the category

Mod(AXdef ) of sheaves of A-modules on X (relative to the o-minimal site) is isomorphic to the category Mod(A ) of sheaves of A-modules on a certain spectral Xe topological space Xe, the o-minimal spectrum of X, associated to X. The o-minimal spectrum Xe of a definable space X is the set of ultra-filters of definable subsets of X (also know in model theory as types on X) equipped with the topology generated by the subsets Ue with U ∈ Op(Xdef ). If f : X → Y is a morphism in Def, then 4 MARIO´ J. EDMUNDO AND LUCA PRELLI one has a corresponding continuous map fe : Xe → Ye : α 7→ fe(α) where f(α) is the ultrafilter in Ye determined by the collection {A : f −1(A) ∈ α}. (See [20, Definitions 2.2 and 2.18] or [6] and [48] where these notions were first introduced). If M is a real closed field (R, <, 0, 1, +, ·), and V ⊆ Rn is an affine real algebraic variety over R, then Ve is homeomorphic to Sper R[V ], the real spectrum of the coordinate ring R[V ] of V ([4, Chapter 7, Section 7.2] or [12, Chapter I, Example 1.1]); and the isomorphism Mod(A ) ' Mod(A ) from [20] corresponds in this Vdef Ve case to [12, Chapter 1, Proposition 1.4]. Below we denote by Defg the corresponding category of o-minimal spectra of definable spaces and continuous definable maps and

Def → Defg the functor just deﬁned. Due to the isomorphism Mod(A ) ' Mod(A ) for Xdef Xe every object X of Def we will work in this paper in Def.g

2.1. Extended summary. In this section we will prove some preliminary results that will be crucial for the rest of the paper. These preliminary results are neces- sary since the category Defg in which we will be working is rather diﬀerent from the category Top of topological spaces.

An object of Defg is a T0, quasi-compact and a spectral topological space, i.e., it has a basis of quasi-compact open subsets, closed under taking finite intersections and each irreducible closed subset is the closure of a unique point. Such objects are not T1 (unless they are finite), namely not every point is closed, so they are not Hausdorff (unless they are finite). In particular, the usual (topological) definition of compact topological space cannot be used in Def.g Similarly the usual (topological) definition of proper maps is of no use in Defg. Since Defg has cartesian squares (which one should point out are not cartesian squares in the category of topological spaces Top), in this section we introduce a category theory definition of these notions in Defg just like in semi-algebraic geometry ([13, Section 9]) (and also in algebraic geometry [32, Chapter II, Section 4] or [31, Chapter II, Section 5.4]) and point out all the properties needed later.

A fiber fe−1(α) of a morphism fe : Xe → Ye in Defg is not in general an object of Def,g but following [1, Lemma 3.1] in the affine case, such fiber is homeomorphic to an object (f^S)−1(a) of Def(^S) where a is a realization of the type α and S is the prime model of the first-order theory of M over {a} ∪ M. Here Def(^S) is the same as Defg but defined in the o-minimal structure S. This phenomena is the model theoretic analogue of what happens in real-algebraic geometry ([12, Proposition 2.4], [53, Chapter II, 3.2]) (and also in algebraic geometry [32, Chapter II, Section 3]). Indeed, if f : X → Y is a morphism of real schemes over a real closed field R and α ∈ Y , then f −1(α) with the underlying topology is homeomorphic to the real scheme X ×Y Sper k(α) over the residue ordered field k(α) of α (recall that α is a prime cone). By [48] the real closure of k(α) is isomorphic to the prime model over R and a realization of α. THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 5

Due to the homeomorphism fe−1(α) ' (f^S)−1(a), when α ∈ Ye is a closed point, we will be able to use the theory from [24, Section 3] of normal and constructible families of supports on the object (f^S)−1(a) also on the ﬁber fe−1(α) after we show that working in a full subcategory Ae of Defg the family c of complete supports on (f^S)−1(a) is normal and constructible. Normal and constructible families of supports are the o-minimal analogue of the semi-algebraic and paracompactifying families of supports in semi-algebraic geometry ([12, Chapter II, Section 1]) and our main results on this, Lemma 2.21 and Corollary 2.22, are similar to [12, Chapter II, Proposition 8.2 and Corollary 8.3].

The full subcategory Ae of Defg on which we must work is the image under Def → Defg of a subcategory A of Def such that: (A0) cartesian products of objects of A are objects of A and locally closed subsets of objects of A are objects of A; (A1) in every object of A every open definable subset is a finite union of open and definably normal definable subsets; (A2) every object of A has a definably normal definable completion in A.

Examples of categories A as above include: (i) regular, locally definably compact definable spaces in o-minimal expansions of real closed fields; (ii) Hausdorff locally definably compact definable spaces in o-minimal expansions of ordered groups with definably normal completions; (iii) locally closed definable subspaces of cartesian products of a given definably compact definable group in an arbitrary o-minimal structure. For details on this see Example 3.1 below. In the semi-algebraic case the semi-algebraic analogues of (A0), (A1) and (A2) are also extensively used and they hold, as in our example (i), for regular locally complete semi-algebraic spaces. In fact what is used there is a stronger version of (A1), namely that every open semi-algebraic subset of a regular semi-algebraic space is semi-algebraically normal. In topology we do not have the problem of fibers described above and what we need from (A0), (A1) and (A2) (Propositions 2.15 and 2.16) holds on Hausdorff locally compact topological spaces. Finally note that even in algebraic geometry similar constrains occur since when one has to define the proper direct image functor of a separated morphism of finite type of schemes one has to use Nagata’s theorem on the existence of proper extensions of such morphisms (i.e., existence of completions as in (A2)).

2.2. Morphisms proper in Defg. Here we will introduce a category theory deﬁni- tion of the notions proper and complete in Defg just like in semi-algebraic geometry ([13, Section 9]) (and also in algebraic geometry [32, Chapter II, Section 4] or [31, Chapter II, Section 5.4]) and point out the properties needed later.

If X is an object of Def,g then a subset Z ⊆ X is called constructible if Z is also an object of Defg.

Let f : X → Y be a morphism in Defg. We say that: 6 MARIO´ J. EDMUNDO AND LUCA PRELLI

• f : X → Y is closed in Defg if for every closed constructible subset A of X, its image f(A) is a closed subset of Y. • f : X → Y is a closed (resp. open) immersion if f : X → f(X) is a homeomorphism and f(X) is a closed (resp. open) subset of Y .

Since Def → Defg is an isomorphism of categories and cartesian squares exist in Def we have:

Fact 2.1. In the category Defg the cartesian square of any two morphisms f : X → Z and g : Y → Z in Defg exists and is given by a commutative diagram

pY X ×Z Y / Y

pX g f X / Z where the morphisms pX and pY are known as projections. The Cartesian square satisﬁes the following universal property: for any other object Q of Defg and morphisms qX : Q → X and qY : Q → Y of Defg for which the following diagram commutes, Q qY u

# pY % X ×Z Y / Y qX

pX g f X / Z there exist a unique natural morphism u : Q → X ×Z Y (called mediating morphism) making the whole diagram commute. As with all universal constructions, the cartesian square is unique up to a deﬁnable homeomorphism.

The following is a very important remark that one should always have in mind:

Remark 2.2. The cartesian square in Defg of two morphisms f : X → Z and g : Y → Z in Defg is not the same as the cartesian square in Top of the two morphisms f : X → Z and g : Y → Z in Top. In particular, if pt denotes a ﬁxed one point object of Def,g then the cartesian product

pY X ×pt Y / Y

pX aY

aX X / pt in Def,g also denoted by X × Y , of two objects X and Y in Defg is not the same as usual the cartesian product (in Top) of the objects X and Y in Top. THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 7

Given a morphism f : X → Y in Def,g the corresponding diagonal morphism is the unique morphism ∆ : X → X ×Y X in Defg given by the universal property of cartesian squares: X idX ∆

$ pY & X ×Y X / X idX pX f f X / Y. We say that: • f : X → Y is separated in Defg if the corresponding diagonal morphism ∆ : X → X ×Y X is a closed immersion.

We say that an object Z in Defg is separated in Defg if the morphism Z → {pt} to a point is separated.

Let f : X → Y be a morphism in Defg. We say that: • f : X → Y is universally closed in Defg if for any morphism g : Y 0 → Y in Defg the morphism f 0 : X0 → Y 0 in Defg obtained from the cartesian square

f 0 X0 / Y 0

g0 g f X / Y

in Defg is closed in Def.g

Deﬁnition 2.3. We say that a morphism f : X → Y in Defg is proper in Defg if f : X → Y is separated and universally closed in Def.g

Deﬁnition 2.4. We say that an object Z of Defg is complete in Defg if the morphism Z → pt is proper in Defg.

Directly from the deﬁnitions (as in [31, Chapter II, Proposition 5.4.2 and Corol- lary 5.4.3], see also [13, Section 9]) one has the following. See [21] for a detailed proof in Def which transfers to Def:g

Proposition 2.5. In the category Defg the following hold: (1) Closed immersions are proper in Defg. (2) A composition of two morphisms proper in Defg is proper in Defg. (3) If f : X → Y is a morphism over Z proper in Defg and Z0 → Z is a base 0 0 extension, then the corresponding base extension morphism f : X ×Z Z → 0 Y ×Z Z is proper in Defg. 8 MARIO´ J. EDMUNDO AND LUCA PRELLI

(4) If f : X → Y and f 0 : X0 → Y 0 are morphisms over Z proper in Defg, then 0 0 0 the product morphism f × f : X ×Z X → Y ×Z Y is proper in Defg. (5) If f : X → Y and g : Y → Z are morphisms such that g ◦ f is proper in Defg, then: (i) f is proper in Def;g (ii) if g is separated in Defg and f is surjective, then g proper in Defg. (6) A morphism f : X → Y is proper in Defg if and only if Y can be covered −1 by ﬁnitely many open constructible subsets Vi such that f| : f (Vi) → Vi is proper in Defg.

From Proposition 2.5 we easily have:

Corollary 2.6. Let f : X → Y be a morphism in Defg and Z ⊆ X a complete object of Defg. Then the following hold: (1) Z is a closed (constructible) subset of X.

(2) f|Z : Z → Y is proper in Defg. (3) f(Z) ⊆ Y is (constructible) complete in Defg. (4) If f : X → Y is proper in Defg and C ⊆ Y is a complete object of Defg, then f −1(C) ⊆ X is (constructible) complete in Defg.

Below, if C is a subcategory of Def we denote by Ce its image under Def → Defg. The following is a standard consequence of Proposition 2.5. See [21] for a detailed proof in Def:

Corollary 2.7. Let C be a full a subcategory of the category of deﬁnable spaces Def whose set of objects is: • closed under taking locally closed deﬁnable subspaces of objects of C, • closed under taking cartesian products of objects of C, Then the following are equivalent: (1) Every object X of Ce is completable in Ce i.e., there exists an object X0 of Ce which is complete in Defg together with an open immersion i : X,→ X0 in Ce with i(X) dense in X0. Such i : X,→ X0 is called a completion of X in Ce . (2) Every morphism f : X → Y in Ce is completable in Ce i..e, there exists a commutative diagram

i X / X0

f f 0 j Y / Y 0

of morphisms in Ce such that: (i) i : X → X0 is a completion of X in Ce ; (ii) j is a completion of Y in Ce . THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 9

(3) Every morphism f : X → Y in Ce has a proper extension in Ce i.e., there exists a commutative diagram ι X / P

f f Y of morphisms in Ce such that ι is a open immersion with ι(X) dense in P and f a proper in Defg.

Finally we observe the following which was proved in Def in the paper [21]:

Proposition 2.8. Let S be an elementary extension of M. Let f : X → Y a morphism in Defg. Then the following are equivalent: (1) f is separated (resp. proper) in Defg. (2) f S is separated (resp. proper) in Def(g S).

We end the subsection by describing the closed subsets of objects of Defg and also showing that a morphism in Defg is closed if and only if it closed in Defg.

T Remark 2.9. Let X be an object of Defg and C = i∈I Ci ⊆ X with the Ci’s constructible subsets. The following hold: • C is quasi-compact subset (with the induced topology). • A closed subset B of C is a quasi-compact subset. Indeed, B = D ∩ C with D a closed subset of X. So D is quasi-compact and B is also quasi-compact. • A quasi-compact subset B of C is a quasi-compact subset.

Lemma 2.10. Let f : X → Y be a morphism in Defg. Then the following hold: T (1) If C = Ci ⊆ X with the Ci’s constructible subsets, then f(C) = T i∈I i∈I f(Ci). T −1 (2) If D = j∈J Dj ⊆ Y with the Dj’s constructible subsets, then f (D) = T −1 j∈J f (Dj). T (3) C ⊆ X is closed if and only if C = i∈I Ci with each Ci ⊆ X constructible and closed. T Proof. (1) Suppose that C = i∈I Ci with the Ci’s constructible. Then C is closed and hence compact in the Stone topology of X (the topology generated by the constructible subsets). By [20, Remark 2.19], f : X → Y is continuous with respect to the Stone topologies on X and Y , so f(C) is compact in the Stone topology of X. Since Y is Hausdorﬀ in the Stone topology, f(C) is closed (in the Stone topology). Therefore, \ f(C) = {E : f(C) ⊆ E and E is constructible}. If E is a constructible subset such that f(C) ⊆ E, then C ⊆ f −1(E) and by −1 compactness, there is Ci such that C ⊆ Ci ⊆ f (E) and hence f(C) ⊆ f(Ci) ⊆ E. T Therefore, f(C) = i∈I f(Ci). 10 MARIO´ J. EDMUNDO AND LUCA PRELLI

(2) Obvious. (3) Suppose that C is closed. Then its complement is open and so a union of open constructible subsets. Then by [20, Remark 2.3], C is the intersection of closed constructible subsets. The converse is clear.

Proposition 2.11. Let f : X → Y be a morphism in Defg. Then the following are equivalent: (1) f is closed. (2) f is closed in Defg. Proof. Assuming (1) clearly (2) is immediate. Assume (2). If D ⊆ X be a closed constructible subset, then f(D) is closed (and constructible by [20, Remark T 2.19]); if C ⊆ X is a general closed subset, we have C = Ci with the Ci’s closed i∈I T and constructible (Lemma 2.10 (3)) and by Lemma 2.10 (1), f(C) = i∈I f(Ci) with each f(Ci) closed and constructible and hence, f(C) is closed as required.

2.3. Fibers in Defg. A fiber f −1(α) of a morphism f : X → Y in Defg is not in general an object of Def,g but a model theoretic trick such fibers can each be seen as objects of Def(^S) for some elementary extension S of M. This follows from our assumption that M has definable Skolem functions.

Before we proceed we recall the basic model theoretic consequences of our assumption on M that will be required later.

By [51, Theorem 5.1]: Fact 2.12. Since M has definable Skolem functions, then for any parameter v in some model of the first-order theory of M, there is a prime model S of the first- order theory of M over M ∪ {v} such that, for every s ∈ S, there is a definable map f : A → M (definable in M), such that v ∈ A(S) and s = f S(v). By Fact 2.12 we have: Fact 2.13. Since M has definable Skolem functions, if S is a prime model of the first-order theory of M over M ∪ {v} where v is a parameter in some model of the first-order theory of M, then if u ∈ S and K is a prime model of the theory of M over M ∪ {u}, then we have either K = M or K = S.

Indeed, consider the model theoretic algebraic closure operator aclM(•) in S given by a ∈ aclM(C) with C ⊆ S if and only if there is c ∈ C and a definable map f : A → M (definable in M) such that c ∈ A(S) and f S(c) = a. By [51, Theorem 4.1], this model theoretic algebraic closure operator satisfies the exchange property: if a ∈ aclM(Cb)\aclM(C), then b ∈ aclM(Ca). Thus, by Fact 2.12, if u ∈ S and K is a prime model of the theory of M over M ∪{u}, then we have either K = M or K = S.

Let S be an elementary extension of M. For each object Xe of Defg we have a restriction map r : X](S) → Xe such that given an ultraﬁlter α ∈ X](S), r(α) is the ultraﬁlter in Xe determined by the collection {A : A(S) ∈ α}. This is a continuous surjective map which is neither open nor closed. THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 11

The following result was proved in [1, Lemma 3.1] in the affine case - for continuous definable maps between definable sets, under the assumption that M is an o-minimal expansion of an ordered group. However, the only thing needed from M is that it has definable Skolem functions. In fact all that is required is Fact 2.12. A special case of this result, when the map is a projection X × [a, b] → X, was proved before in [20, Claim 4.5].

Lemma 2.14. Let fe : Xe → Ye be a morphism in Defg, α ∈ Ye, a |= α a realization of α and S a prime model of the ﬁrst-order theory of M over {a} ∪ M. Then there is a homeomorphism −1 −1 r| : (f^S) (a) → fe (α) induced by the restriction r : X](S) → X.e Proof. Since (f −1(B))(S) = (f S)−1(B(S)) for every deﬁnable subset B ⊆ Y , we have a commutative diagram

feS −1 (f^S) (a) / X](S) / Y](S)

r| r r

fe fe−1(α) / Xe / Ye

−1 −1 and so r| : (f^S) (a) → fe (α) is well deﬁned. −1 −1 We now have to show that r| : (f^S) (a) → fe (α) is a continuous and open −1 bijection. Let Yi be a deﬁnable chart of Y such that α ∈ Yei and let Zi = f (Yi). −1 ^S −1 ^S −1 −1 Since ((f|Zi ) ) (a) = (f ) (a) and (^f|Zi ) (α) = fe (α), the map we are inter- ested on is the same as the restriction in the following commutative diagram

feS ^S −1 ^ ] ((f|Zi ) ) (a) / Zi(S) / Yi(S)

r| r r −1 fe fg|Zi (α) / Zfi / Yei.

^S −1 ^S −1 ^ Let Xj be a deﬁnable chart of X. Since ((f|Zi∩Xj ) ) (a) = ((f|Zi ) ) (a) ∩ Xj(S) −1 −1 and (f^|Zi∩Xj ) (α) = fg|Zi (α) ∩ Xfj, then the restriction in the following commutative diagram

feS ^S −1 ^ ] ((f|Zi∩Xj ) ) (a) / (Zi ∩ Xj)(S) / Yi(S)

r| r r −1 fe f^|Zi∩Xj (α) / Z^i ∩ Xj / Yei is a continuous and open bijection by [1, Lemma 3.1]. Therefore, since the Xfj’s −1 ^ ^S −1 ^S −1 (resp. Xj(S)) cover fg|Zi (α) (resp. ((f|Zi ) ) (a)), it follows that r| : (f ) (a) → fe−1(α) is a continuous and open surjection. 12 MARIO´ J. EDMUNDO AND LUCA PRELLI

^S −1 ^S −1 −1 Let β1, β2 ∈ (f ) (a) = ((f|Zi ) ) (a) be such that r|(β1) = r|(β2) ∈ fe (α) = −1

2.4. The main assumptions. Here we introduce the main assumptions under which we develop the Grothendieck formalism of the six operations on o-minimal sheaves. We also explain exactly which consequences of these assumptions are used later.

Below we will work in a full subcategory Ae of Defg which is the image under Def → Defg of a subcategory A of Def such that: (A0) cartesian products of objects of A are objects of A and locally closed subsets of objects of A are objects of A; (A1) in every object of A every open definable subset is a finite union of open and definably normal definable subsets; (A2) every object of A has a definably normal definable completion in A.

The only consequences of the assumptions (A0), (A1) and (A2) that will be crucial and will be used in the paper are the two proposition below. Later for some results we will have to add also an assumption (A3) which we will discuss in the appropriate place.

Condition (A1) alone gives us the following:

Proposition 2.15. Let X be an object of Ae . Then the following hold: (1) In X every open subset is a finite union of open and normal constructible subsets. (2) (Shrinking lemma) If {Ui : i = 1, . . . , n} is a covering of an open, normal constructible subset W of X by open subsets, then there are constructible open subsets Vi and constructible closed subsets Ci of W (1 ≤ i ≤ n) with Vi ⊆ Ci ⊆ Ui and W = ∪{Vi : i = 1, . . . , n}. (3) If α ∈ X, then there is an open, normal constructible subset U of X such that α ∈ U and α is closed in U. Proof. (1) Follows by (A1) and [20, Theorem 2.13]. (2) Is well known in normal spectral spaces, see for example [20, Proposition 2.17]. (3) The specializations of a point α in an object Y of Defg form finite chains by [20, Lemma 2.11] and if the object is normal, by [20, Proposition 2.12] or [7, Proposition 2], α has a unique closed specialization ρ which must be the minimum of any chain of specialization of α. By (1) let U be an open, normal constructible open subset of X such that α ∈ U. Let ρ be the unique closed specialization of α in U. If α is closed in U we are done, otherwise, U \{ρ} is open (in U and so THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 13 also in X), and so by (1) there is an open, normal constructible open subset V of U \{ρ} (and so of X) such that α ∈ V. Since every specialization of α in V is a also a specialization of α in U and the maximum length of chains of specializations of α in V is smaller that the maximum length of chains of specializations of α in U, repeating the process finitely many times we get the result.

Conditions (A0) and (A2) alone gives us the following:

Proposition 2.16. Let f : X → Y be a morphism in Ae . Then the following hold: (1) f has a proper extension in Ae . (2) f is separated in Defg. (3) If Z ⊆ X is a closed constructible subset and f|Z : Z → Y is proper in Defg, then for every open constructible neighborhood W of Z in X there is a closed constructible neighborhood B of Z in W such that f|B : B → Y is proper in Defg. Proof. (1) Follows by (A0), (A2) and Corollary 2.7. (2) Observe that the completions of X and Y in Ae are normal if and only if they are o-minimal spectra of definably normal objects of A ([20, Theorem 2.13]); a definably normal object of Def is Hausdorff; a morphism in Def between Hausdorff definable spaces has Hausdorff fibers and so is separated in Def. Therefore by (A2), f is separated in Defg. (3) By (1) we have a commutative diagram

ι X / P

f f Y of morphisms in Defg such that ι is an open immersion and f is proper in Defg. Since f|Z = f ◦ ι|Z is proper in Def,g by Proposition 2.5 (5) (i), ι|Z is proper in Defg and hence it is closed in Defg. So ι(Z) is a closed constructible subset of P . On the other hand, ι(Z) and P \ ι(W ) are closed disjoint constructible subsets of P . Since P is normal, by the shrinking lemma, there is a closed constructible neighborhood C of ι(Z) in P with C ⊆ ι(W ). Let B = ι−1(C). Then B is a closed constructible neighborhood of Z in X such that B ⊆ W . Since f|B = f ◦ ι|B, f is proper in

Defg and ι|B is a closed immersion, by Proposition 2.5 (1) and (2), f|B : B → Y is proper in Defg as required.

2.5. On families of supports on fibers in Defg. In the paper [24] normal and constructible families of supports on objects of Defg played a fundamental role. Here we introduce the notion of a normal and constructible family of supports Φ on a fiber fe−1(α) of a morphism fe : Xe → Ye in Defg. We show that working in Ae will guaranty: (i) the normality and constructibility of the family Φ of proper sup- fe ports; (ii) the normality and constructibility of the family of c complete supports on a fiber fe−1(α) of a closed point; (iii) the compatibility of the family of complete 14 MARIO´ J. EDMUNDO AND LUCA PRELLI supports on a fiber f −1(α) of a closed point with the family of proper supports Φ . e fe

First recall the following deﬁnitions from [24, page 1267].

Deﬁnition 2.17. Let X be an object in Def.g A family Φ of closed subsets of X is a family of supports on X if: • every closed subset of a member of Φ is in Φ; • Φ is closed under ﬁnite unions. A family Φ of supports on X is said to be constructible if: • every member of Φ is contained in a member of Φ which is constructible. A family Φ of supports on X is said to be normal if: • every member of Φ is normal; • for each member S of Φ, if U is an open neighborhood of S in X, then there exists a closed constructible neighborhood of S in U which is a member of Φ.

By Propositions 2.16 (3) and Proposition 2.16 (3) applied to the morphism aX : X → pt to a point respectively we have:

Example 2.18. Let f : X → Y be a morphism in Ae and Z an object of Ae . Then:

(1) The family Φf = {A : A ⊆ X is closed, A ⊆ B for some closed constructible

subset B of X such that f|B : B → Y is proper in Defg} of closed subsets of X is a normal and constructible family of supports on X. (2) The family c = {A : A ⊆ X is closed, A ⊆ B for some closed constructible complete in Defg subset B of X} of complete supports on Z is a normal and constructible family of supports on Z.

Working with the previous definition in Def(^S) we can use the homeomorphism −1 −1 r| : (f^S) (a) → fe (α) of Lemma 2.14 to define the notion of a normal and constructible family of supports on the fibers fe−1(α):

Deﬁnition 2.19. Let fe : Xe → Ye be a morphism in Def,g α ∈ Ye, a |= α a realization of α and S a prime model of the ﬁrst-order theory of M over {a} ∪ M. • A family of supports Ψ on the quasi-compact space fe−1(α) is constructible −1 if its inverse image (r|) Ψ is a constructible family of supports on the object (f^S)−1(a) of Def(^S). • A family of supports Ψ on the quasi-compact space fe−1(α) is normal if its −1 −1 inverse image (r|) Ψ is a normal family of supports on the object (f^S) (a) of Def(^S). If there is no risk of confusion, and since r| is a homeomorphism, we also use Ψ to denote the inverse image of Ψ by r|.

We also say that a subset Z of fe−1(α) is constructible if its inverse image −1 −1 (r|) (Z) is a constructible subset of the object (f^S) (a) of Def(^S). Of course, THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 15

−1 −1 a subset Z of fe (α) is quasi-compact if its inverse image (r|) (Z) is a quasi- compact subset of the object (f^S)−1(a) of Def(^S).

We can now introduce one of the main deﬁnitions of the paper:

Deﬁnition 2.20. Let fe : Xe → Ye be a morphism in Def,g α ∈ Ye, a |= α a realization of α and S a prime model of the ﬁrst-order theory of M over {a} ∪ M. • The family of complete supports on the object (f^S)−1(a) of Def(^S), denoted c, is the constructible family of supports on (f^S)−1(a) of all closed subsets A of (f^S)−1(a) with A ⊆ Ze for some closed constructible complete in Def(^S) subset Ze of the object (f^S)−1(a) of Def(^S). • The family of complete supports on fe−1(α), denoted c, is the (constructible) −1 family of supports on fe (α) whose inverse image by r| is the constructible family of complete supports on (f^S)−1(a).

Throughout the paper we will also require that the family c of complete supports on a ﬁber fe−1(α) of a morphism fe : Xe → Ye in Defg with α ∈ Ye, is such that, if α is closed, then c = Φ ∩ f −1(α) and c is a normal and constructible family of fe e supports. Working in Ae is enough to guarantee this.

First we need the following consequence of conditions (A0) and (A2):

Lemma 2.21. Let fe : Xe → Ye be a morphism in Ae . Let α ∈ Ye be closed, a |= α a realization of α and S a prime model of the ﬁrst-order theory of M over {a} ∪ M. Then for every constructible complete in Def(^S) subset K of (f^S)−1(a) there exists a closed constructible subset B of Xe such that fe|B : B → Ye is proper in Defg and K ⊆ B(S). Proof. By Proposition 2.16 (1), we have a commutative diagram

ι Xe / Pe

eh f e Ye of morphisms in Defg such that ι is an open immersion and eh is proper in Defg. Since K is a constructible complete in Def(^S) subset of (f^S)−1(a), by Corollary 2.6 (3) in Def(^S), we have that ιS(K) is constructible complete in Def(^S) and −1 −1 −1 hence closed in (h^S) (a) ⊆ P](S). The homeomorphism r| : (h^S) (a) → eh (α) −1 (Lemma 2.14) implies that r|(ιS(K)) is closed in eh (α) which is closed in Pe (α is closed). Hence r|(ιS(K)) is closed in Pe and moreover r|(ιS(K)) ∩ (Pe \ ι(Xe)) = ∅. Since Pe is normal, by the shrinking lemma, we can ﬁnd disjoint constructible open neighborhoods Ue and Ve of r|(ιS(K)) and Pe\ι(Xe) respectively in Pe. Set Ce = Pe\Ve. Then r|(ιS(K)) ⊂ Ce ⊂ ι(Xe) with Ce constructible and closed in both ι(Xe) and P.e 16 MARIO´ J. EDMUNDO AND LUCA PRELLI

From the commutative diagram,

r| (f^S)−1(a) / fe−1(α)

Defg since eh is proper in Defg and ι|B : B → Ye is also proper Defg being a closed immersion in Defg. On the other hand, since r(B(S)) = B, we get K ⊆ B(S) as required.

By Lemma 2.21 we have the following consequence of conditions (A0) and (A2):

Corollary 2.22. Let fe : Xe → Ye be a morphism in Ae . Let α ∈ Ye be closed. Let c be the family of complete supports of fe−1(α). Then c = Φ ∩ f −1(α) = {A ∩ f −1(α): A ∈ Φ } fe e e fe and c is a normal and constructible family of supports on fe−1(α) Proof. Let a |= α be a realization of α and S a prime model of the ﬁrst-order theory of M over {a} ∪ M. Let L ∈ c. Then by deﬁnition of c, there is a constructible complete in Def(^S) −1 −1 subset K of (f^S) (a) such that (r|) (L) ⊆ K. By Lemma 2.21, there exists a closed constructible subset B of Xe such that fe|B : B → Ye is proper in Defg and −1 K ⊆ B(S). Then L = r((r|) (L)) ⊆ r(B(S)) = B for some closed constructible subset B of X such that f : B → Y is proper. This means L ∈ Φ ∩ f −1(α). e e|B e fe e Let L ∈ Φ ∩ f −1(α). Then L = A ∩ f −1(α) with A ⊆ X closed, A ⊆ B fe e e e for some closed constructible subset B of Xe such that fe|B : B → Ye is proper in Def. By Proposition 2.8, (gf S) : B( ) → Y]( ) is proper in Def(^). Thus g |B(S) S S S ^S −1 S −1 ^ ^S −1 B(S) ∩ ((f ) (a)) = ((f )|B(S)) (a) is a complete in Def(S) subset of (f ) (a) −1 −1 −1 −1 −1 (Corollary 2.6 (4)). Since (r|) (L) = r (A ∩ fe (α)) ⊆ r (B ∩ fe (α)) = B(S) ∩ (f S^)−1(a)), we get L ∈ c. By Example 2.18 (2), c is a normal and constructible family of supports fe−1(α).

3. Proper direct image In this section we will develop the theory of proper direct image in a full subcategory Ae of Defg which is the image under Def → Defg of a certain full subcategory A of Def. THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 17

Notation: For the rest of the paper we let A be a noetherian ring (in fact we can assume more generally that A is a commutative ring with ﬁnite weak global dimension, since this is all that we shall use about A - see Remark 3.34). If X is a topological space we denote by Mod(AX ) the category of sheaves of A-modules on X and we call its objects A-sheaves on X. Since Defg is a subcategory of Top, if X is an object of Defg then we have the classical operations −1 HomAX (•, •), • ⊗AX •, f∗, f , (•)Z , ΓZ (X; •), Γ(X; •) on A-sheaves on X, where Z ⊆ X is a locally closed subset. Below we may use freely these operations and refer the reader to [37, Chapter II, Sections 2.1 - 2.4] for the details on sheaves on topological spaces and on the properties of these basic operations. Later in this section we may also use the derived versions of many of the properties relating the above operations and we refer to reader to [37, Chapter II, Section 2.6] for details. The reader can also see the classical references [5], [29] and [35] for similar details on sheaves on topological spaces.

3.1. Extended summary. Our goal below is to develop the theory of proper direct image in a full subcategory Ae of Defg which is the image under Def → Defg of a subcategory A of Def such that: (A0) cartesian products of objects of A are objects of A and locally closed subsets of objects of A are objects of A; (A1) in every object of A every open definable subset is a finite union of open and definably normal definable subsets; (A2) every object of A has a definably normal definable completion in A.

For some of our results about the proper direct image (base change formula (Propo- sition 3.33), derived base change formula (Theorem 3.39), the K¨unnethformula (Theorem 3.40) and dual base change formula (Proposition 4.6)) we will also require that: (A3) if f : X → Y is a morphism in Def and if u ∈ Y , then for every elementary

extension S of M and every F ∈ Mod(AXdef ) we have an isomorphism ∗ −1 ∗ S −1 Hc (f (u); F|f −1(u)) ' Hc ((f ) (u); F (S)|(f S)−1(u))

−1 ∗ where Fe(S) = r Fe , r : X](S) → Xe is the restriction and Hc is the cohomology with definably compact supports ([24, Example 2-10 and Definition 2.12]). We have: Example 3.1. Categories A satisfying also (A3) include: (i) Regular, locally definably compact definable spaces in o-minimal expansions of real closed fields. ((A1) is by [16, Chapter 10, Theorem (1.8)] and [16, Chapter 6, Lemma (3.5)]; (A2) and (A3) by [25, Fact 4.7 and Corollary 4.8].) (ii) Hausdorff locally definably compact definable spaces in o-minimal expansions of ordered groups with definably normal completions. ((A1) by [16, 18 MARIO´ J. EDMUNDO AND LUCA PRELLI

Chapter 6, Lemma (3.5)]; (A2) by assumption and (A3) by [25, Theorem 4.5].) (iii) Locally closed definable subspaces of cartesian products of a given definably compact definable group in an arbitrary o-minimal structure. (See [22].)

In these examples, (A3) is obtained in [25] after extending an invariance result for closed and bounded deﬁnable sets in o-minimal expansions of ordered groups from [2].

Convention: For the rest of the paper we will work in the category Defg (resp. Ae or Def(^S)) and omit the tilde on objects and morphisms. We will say “proper” (resp. “separated” or “complete”) instead of “proper in Def”g (resp. “separated in Def”g or “complete in Def”).g We will assume that all morphism that we consider in Defg are separated (this is the case for morphisms in Ae (Proposition 2.16 (2))). If X in an object of Defg (resp. Ae ), then Op(X) denotes the category of open subsets of X with inclusions and Opcons(X) is the full sub-category of constructible open subsets of X.

Since our definition of the o-minimal proper direct image functor f associated .o to a morphism f : X → Y in Defg is similar to the definition of the topological proper direct image functor f!, we only replace proper in Top by proper in Def,g we follow closely the proofs in topology ([37, Chapter II, Sections 2.5 and 2.6]) but we have to deal with: (i) the fact that the fibers f −1(α) are often not objects of Def;g (ii) the fact that cartesian squares in Defg are not cartesian squares in Top; (iii) the fact that objects of Defg are not locally compact topological spaces. The most technical result is the Fiber formula (Corollary 3.13) which is a consequence of the Relative fiber formula (Proposition 3.9) in whose proof we use heavily the consequences of our assumptions (A0), (A1) and (A2) discussed in the Section 2. Our methods are different from those in semi-algebraic case: the fiber formula there ([12, Chapter II, Corollary 8.8]) is obtained as a consequence of a Base change formula ([12, Chapter II, Theorem 8.7]) and no semi-algebraic analogue of the Rel- ative fiber formula is proved by Delfs.

We then proceed with the theory of f-soft sheaves which gives us the f -injective .o objects (Proposition 3.26) and later the bound on the cohomological dimension of f (Theorem 3.42). The f-soft sheaves are those whose restriction to ﬁbers f −1(α) .o are c-soft. We show our results about f-soft sheaves after we remark (Remark 3.19) that by our assumptions (A0), (A1) and (A2) we can always reduce to the case where α is a closed point in which case c is a normal and constructible family of supports on f −1(α) and the results follow, via (f^S)−1(a) ' f −1(α), from similar results already proved in [24, Section 3]. This theory has not been considered in the semi-algebraic case.

The Projection formula (Proposition 3.28) is obtained in a standard way but the Base change formula (Proposition 3.33) requires assumption (A3) and, after we prove a technical lemma (Lemma 3.32), we use (A3) more or less in a similar way the corresponding fact ([12, Chapter I, Theorem 6.10]) is used in the prove of the semi-algebraic Base change formula ([12, Chapter II, Theorem 8.7]). Using THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 19 the f-soft resolutions we obtain in a standard way the Derived projection formula (Theorem 3.35) and the Derived base change formula (Theorem 3.39) after we prove a technical lemma (Lemma 3.38) requiring assumption (A3). This part was also not considered in the semi-algebraic case. The complication with the Base change formula and the Derived base change formula and the need to use (A3) arises from the fact that in a cartesian square

f 0 X0 / Y 0

g0 g f X / Y. in Defg, if γ ∈ Y 0, then we may have (f 0^S)−1(a) ' f 0−1(γ) and (f^K)−1(b) ' f −1(g(γ)) with K = M and S 6= M. So after we identify (f 0S)−1(a) with (f S)−1(b) −1 −1 0S ∗ 0 ∗ 0S via g we need to know that Hc (fe (b); Fe 0−1 ) ' Hc ((]f ) (b); Fe( ) −1 ). |fe (b) S |(]f 0S) (b)

3.2. Proper direct image. Here we deﬁne the proper direct image functor and prove some of its basic properties.

Definition 3.2. Let f : X → Y be a morphism in Defg and let F ∈ Mod(AX ). The cons proper direct image is the subsheaf of f∗F defined by setting for U ∈ Op (Y ) Γ(U; f F ) = limΓ (f −1(U); F ), .o −→ Z Z where Z ranges through the family of closed constructible subsets of f −1(U) such that f|Z : Z → U is proper. From the definition we have: Remark 3.3. The functor f is clearly left exact; if f : X → Y is proper, then .o f = f ; if i : Z → Y is the inclusion of a locally closed subset, then i is the .o ∗ .o extension by zero functor (i.e. (i F ) ' F or 0 according to α ∈ Z or α 6∈ Z) and .o α α (•) = i ◦ i−1(•). Compare with [37, Chapter II, Proposition 2.5.4]. Z .o

Remark 3.4. If we consider the morphism aX : X → {pt} to a point in Def,g then we have (a F ) ' Γ(pt; a F ) = Γ (X; F ). X.o pt X.o c Proposition 3.5. Let f : X → Y and g : Y → Z be morphisms in Defg. Then (g ◦ f) ' g ◦ f . .o .o .o Proof. Indeed, if V ∈ Opcons(Z), then a section of Γ(V, g ◦ f F ) is repre- .o .o sented by s ∈ Γ(f −1(g−1(V )); F ) such that supp(s) ⊂ f −1(S) ∩ Z, where Z,S are closed constructible in f −1(g−1(V )) and g−1(V ) respectively and such that −1 −1 f|Z : Z → g (V ), g|S : S → V are proper. The set Z ∩ f (S) is closed constructible and the restriction (g ◦ f): Z ∩ f −1(S) → V is proper (Proposition 2.5 (1) and (2)). Conversely if V ∈ Opcons(Z), then a section of Γ(V, (g ◦f) F ) is repre- .o −1 −1 −1 −1 sented by s ∈ ΓZ (f (g (V )); F ) where Z is closed constructible in f (g (V )) −1 such that (g ◦ f)|Z : Z → V is proper. But then f|Z : Z → g (V ) is proper and 20 MARIO´ J. EDMUNDO AND LUCA PRELLI g|f(Z) : f(Z) → V is proper (Proposition 2.5 (5)).

In particular, we have:

Remark 3.6. For f : X → Y a morphism in Def,g since aX = aY ◦ f, we have Γ (Y ; f F ) ' Γ (X; F ). c .o c Proposition 3.7. Let f : X → Y be a morphism in Def. The functor f commutes g .o with ﬁltrant inductive limits.

Proof. Let (Fi)i∈I be a filtrant inductive system in Mod(AX ). Since open constructible sets are quasi-compact, then for any V ∈ Opcons(X) and any closed constructible set S of V the functors Γ(V ; •) and Γ(V \ S; •) commute with filtrant −→lim ([24, Remark 2.7]). Hence ΓS(V ; •) commutes with filtrant−→ lim. We have Γ(U; limf F ) ' limΓ(U; f F ) −→ .o i −→ .o i i i −1 ' −→limΓZ (f (U); Fi) i,Z −1 ' −→limΓZ (f (U);−→ limFi) Z i ' Γ(U; f limF ), .o−→ i i where Z ranges through the family of closed constructible subsets of f −1(U) such that f|Z : Z → U is proper.

The following lemma follows immediately from the deﬁnition of proper direct image and we leave the details to the reader

Lemma 3.8. Let f : X → Y be a morphism in Defg and W an open subset of Y . Consider the commutative diagram j f −1(W ) / X

f| f i W / Y.

If F ∈ Mod(A ), then (f F ) ' (f ) (F −1 ). X .o |W | .o |f (W )

Remember that, given a morphism f : X → Y in Def,g the family Φf is deﬁned by Φf = {A : A ⊆ X is closed, A ⊆ B for some closed constructible subset B of X such that f|B : B → Y is proper}.

Proposition 3.9 (Relative ﬁber formula). Let f : X → Y and g : Y → Z be morphisms in Ae , β ∈ Z and let F be a sheaf in Mod(AX ). Let K be a subset of g−1(β). If K ∈ c, then Γ(K; f F ) ' Γ (f −1(K); F ). .o c Proof. First we show that we may assume without loss of generality that β is a closed point in Z and K is a closed subset of Y . By Proposition 2.15 (3), there exists Z0 ∈ Opcons(Z) with β ∈ Z0 such that 0 0 −1 0 0 −1 0 0 β is closed in Z . Let Y = g (Z ) and X = f (Y ). Let also f = f|X0 and THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 21

0 0 g = g|Y 0 . By Lemma 3.8 we have (foF )|K = ((foF )|Y 0 )|K ) ' (f F|X0 )|K ), and since . . .o 0−1 −1 f (K) = f (K), we have also F|f −1(K) = F|f 0−1(K)). Thus after replacing Z (resp., Y and X) by Z0 (resp. Y 0 and X0) and f (resp., g) by f 0 (resp., g0), we may assume without loss of generality that β is a closed point of Z. Thus β ∈ Z is a closed point, and so g−1(β) is a closed subset of Y . Since K is closed in g−1(β), then K is closed in Y .

Let b |= β be a realization of β and S a prime model of the ﬁrst-order theory of M over {b} ∪ M. Then we have the following commutative diagram

r| (f S)−1((gS)−1(b)) / f −1(g−1(β))

(f|)S f|

r| (gS)−1(b) / g−1(β) given by Lemma 2.14 (where we have omitted the tildes and used the fact that (g ◦ f)S = (gS) ◦ (f S)). By Proposition 2.16 (1) and Corollary 2.7 (2), we a commutative diagram

i X X / X0

f f 0

iY Y / Y 0

g g0

iZ Z / Z0 of completions in Ae . So we can make a similar diagram

r| (f 0S)−1((g0S)−1(b)) / f 0−1(g0−1(β)) ; c 0 S 0 (f| ) f|

r| (g0S)−1(b) / g0−1(β) O O S (iX|) (iY |)S iY | iX|

r| (gS)−1(b) / g−1(β) O O

(f|)S f|

r| (f S)−1((gS)−1(b)) / f −1(g−1(β)) given by Lemma 2.14 applied also to f 0 and g0.

First we observe the following:

−1 Claim 3.10. Φf ∩ f (K) = c|f −1(K). −1 −1 −1 Let Z ∈ Φf . We have to show that Z ∩ f (K) ∈ c in f (g (β)). We may assume that Z is a closed constructible subset of X. Since f|Z : Z → Y is proper in Def, f S : Z( ) → Y ( ) is proper in Def( ) (Proposition 2.8). Since K ∈ g |Z(S) S S g S 22 MARIO´ J. EDMUNDO AND LUCA PRELLI c in g−1(β), by deﬁnition, there is a C ⊆ (gS)−1(b) constructible and complete −1 −1 in (gS) (b) such that K ⊆ r|(C). By Corollary 2.6 (4), (f S) (C) ∩ Z(S) is a −1 −1 −1 constructible and complete in (f S) ((gS) (b)). Therefore, r|((f S) (C)∩Z(S)) = −1 −1 −1 −1 Z ∩ f (r|(C)) is a constructible and complete in f (g (β)). Since Z ∩ f (K) −1 −1 −1 −1 is a closed subset of Z ∩ f (r|(C)) we have that Z ∩ f (K) ∈ c in f (g (β)). Conversely, let Z ∈ c|f −1(K), then there is C a constructible complete subset of f −1(g−1(β)) such that Z ⊆ C ⊆ f −1(K). It follows from Corollary 2.22 that C is contained in a closed constructible subset B of X such that (g ◦ f)|B is proper, −1 −1 hence f|B is proper (Proposition 2.5 (5)). So C ⊆ B ∩ f (K) ∈ Φf ∩ f (K).

We proceed with the proof of the Proposition. We have Γ(K; f F ) ' limΓ(U; f F ) ' limΓ (f −1(U); F ), .o −→ .o −→ Z U U,Z where U ranges through the family of open constructible neighborhoods of K and −1 Z is closed constructible in f (U) and such that f|Z : Z → U is proper. By Claim 3.10 we can conclude the proof after the following two steps. Claim 3.11. We have an isomorphism −1 −1 limΓ (f (U); F ) ' limΓ −1 (f (U); F ), −→ Z −→ Φf ∩f (U) U,Z U where U ranges through the family of open constructible neighborhoods of K and Z −1 is closed constructible in f (U) and such that f|Z : Z → U is proper. Let s ∈ Γ(f −1(U); F ) whose support is contained in Z closed constructible in −1 −1 f (U) such that f|Z : Z → U is proper. Since K ∈ c, then r (K) ∈ c in S −1 S 0S −1 (g ) (b) and its image by iY belongs to c in (g ) (b). The homeomorphism ∼ 0S −1 0−1 S −1 0−1 r| :(g ) (b) → g (β) implies that r(iY (r (K))) = iY (K) is closed in g (β) and hence in Y 0, a completion of Y in Ae . Since Y 0 is normal ((A2) and [20, Theorem 2.12]), there exists an open constructible neighborhood V of K such that V ⊂ U. −1 Then f|Z∩f −1(V ) : Z ∩ f (V ) → V is proper and composing with the inclusion −1 V,→ Y , the morphism f|Z∩f −1(V ) : Z ∩ f (V ) → Y is proper (Proposition 2.5 (1) −1 −1 and (2)). Hence Z ∩ f (V ) ∈ Φf . The support of the restriction of s to f (V ) −1 −1 −1 −1 is contained in Z ∩ f (V ) = Z ∩ f (V ) ∩ f (V ) ∈ Φf ∩ f (V ) and the result follows.

Claim 3.12. We have an isomorphism −1 −1 limΓ −1 (f (U); F ) ' Γ −1 (f (K); F ), −→ Φf ∩f (U) Φf ∩f (K) U where U ranges through the family of open constructible neighborhoods of K.

−1 −1 Let s ∈ Γ(f (K); F ) and let Z ∈ Φf closed constructible in f (U) containing its support. The section s is represented by a section t ∈ Γ(W ; F ) for some open −1 constructible neighborhood W of f (K). Since Z ∈ Φf and K ∈ c, we have Z ∩ f −1(K) ∈ c in f −1(g−1(β)) (Claim 3.10). Therefore, r−1(Z ∩ f −1(K)) ∈ c S −1 S −1 S 0S −1 0S −1 in (f ) ((g ) (b)) and its image by iX belongs to c in (f ) ((g ) (b)). The ∼ 0S −1 0S −1 0−1 0−1 S −1 homeomorphism r| :(f ) ((g ) (b)) → f (g (β)) implies that r(iX (r (Z∩ THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 23

−1 −1 0−1 0−1 0 f (K)))) = iX (Z ∩ f (K)) is closed in f (g (β)) and hence in X , a completion of X in Ae . Since X0 is normal ((A2) and [20, Theorem 2.12]) there exists an open constructible neighborhood V of Z ∩ f −1(K) with V ⊂ W . We have that f 0 is proper, so f is proper (Proposition 2.5 (5) (i)) and V ∈ Φ . |V |V f Since f 0−1(K) = f −1(K), we have f 0−1(K) ∩ ((V ∩ supp(t)) \ V ) = ∅ and so K ∩ f 0((V ∩ supp(t)) \ V ) = ∅. Since K and f 0((V ∩ supp(t)) \ V ) are closed subsets of Y 0 (recall that f 0 is closed) and Y 0 is normal, by shrinking lemma (Proposition |V 2.15 (2)), there exists an open constructible subset U of Y with K ⊆ U such that f −1(U) ∩ V ∩ supp(t) ⊂ V . Let us deﬁne et ∈ Γ(f −1(U); F ) by

et|f −1(U)\(V ∩supp(t)) = 0,

et|f −1(U)∩V = t|f −1(U)∩V .

−1 By construction et|f −1(K) = s. Note that supp(et) is contained in f (U) ∩ V and V ∈ Φf .

Setting Y = Z and g = id we obtain

Corollary 3.13 (Fiber formula). Let f : X → Y be a morphism in Ae and let F be a sheaf in Mod(A ). Let α ∈ Y . Then (f F ) ' Γ (f −1(α); F ). X .o α c

We end the subsection comparing the o-minimal proper direct image functor f .o with Verdier’s ([54]) topological proper direct image functor f! and Kashiwara and Schapira ([38]) and Prelli’s ([52]) sub-analytic proper direct image functor f!!. To make the comparison more clear we will use the isomorphism Mod(A ) ' Xe

Mod(AXdef ) and the fact that f : X → Y is proper in Def if and only if fe : Xe → Ye is proper in Def to introduce f : Mod(A ) → Mod(A ) as f : Mod(A ) → g .o Xdef Ydef e.o Xe Mod(A ). Ye

We now explain the relationship between the o-minimal proper direct image functor and Verdier’s ([54]) proper direct image functor. Remark 3.14. Consider the category Def associated to an o-minimal expansion M = (R, <, (c)c∈C, (f)f∈F , (R)R∈R) of the ordered set of real numbers. For a continuous map f : X → Y between locally compact topological spaces Verdier’s proper direct image functor f! is given by: for G ∈ Mod(AX ) and U ∈ Op(Y ) we have −1 Γ(U; f!G) =−→ limΓZ (f (U); G), Z −1 where Z ranges through the family of closed subsets of f (U) such that f|Z : Z → U is proper (in Top). If X is an object of Def then X is also a topological space (with the usual topology generated by open deﬁnable subsets) and we have the natural morphism of sites ρ : X → Xdef induced by the inclusion Op(Xdef ) ⊂ Op(X). Since if f|Z : Z → Y is proper in Def then f|Z : Z → Y is proper in Top ([21]), we have Γ(U; f ◦ ρ F ) ⊆ Γ(U; ρ ◦ f F ) for U ∈ Op(X ). However, in general, we .o ∗ ∗ ! def have ρ ◦ f 6= f ◦ ρ . ∗ ! .o ∗ 24 MARIO´ J. EDMUNDO AND LUCA PRELLI

For example, suppose that X = R2, Y = R, f : R2 → R is the projection onto the ﬁrst coordinate and U = (0, +∞). Let φ : (0, +∞) → (0, +∞) be given 1 1 by φ(t) = t (resp. φ(t) = ln(t) ) if M is semi-bounded ([19]) (resp. polynomially bounded ([18])). Set S = {(x, y) ∈ (0, +∞) × R : y = φ(x))} and consider the sheaf −1 AS. Let s ∈ Γ(f (U); AS). Then f : supp(s) → U is proper (it is a homeomorphism into its image), so s ∈ Γ(U; f A ) = Γ(U; ρ ◦ f A ), but s∈ / Γ(U; f ◦ ρ A ) ! S ∗ ! S .o ∗ S −1 since there is no closed deﬁnable subset Z of f (U) such that f|Z : Z → U is proper in Def and S ⊆ Z.

Now we explain the relationship between the o-minimal proper direct image functor and Kashiwara and Schapira ([38]) and Prelli’s ([52]) sub-analytic proper direct image functor. Remark 3.15. Consider the category Def associated to the o-minimal structure Ran = (R, <, 0, 1, +, ·, (f)f∈an) - the ﬁeld of real numbers expanded by restricted globally analytic functions ([18]). As explained in [18], in this case, Def is the category of globally sub-analytic spaces with continuous maps with globally sub- analytic graphs. If X is a real analytic manifold, then we can equip X with its sub-analytic site, denoted Xsa, where the objects are open sub-analytic subsets Op(Xsa) and S ⊂ Op(Xsa) ∩ Op(U) is covering of U ∈ Op(Xsa) if for any compact subset K of X there is a ﬁnite subset S ⊆ S such that K ∩ S V = K ∩ U, see [38]. In this 0 V ∈S0 context we have ([52]) a sub-analytic proper direct image functor f!! given by: for

F ∈ Mod(AXsa ) and U ∈ Op(Ysa) we have −1 Γ(U; f!!F ) =−→ limΓK∩f −1(U)(f (U); F ), K where K ranges through the family of compact sub-analytic subsets of X. We recall that this is a direct construction of Kashiwara and Schapira ([38]) sub-analytic proper direct image functor originally constructed as a special case of a more general construction within the theory of ind-sheaves. If X is also globally sub-analytic (i.e. a definable subset in the o-minimal structure Ran) we have the natural morphism of sites ν : Xsa → Xdef induced by the inclusion Op(Xdef ) ⊂ Op(Xsa). Since a compact sub-analytic subset is a compact globally sub-analytic subset (hence a definably compact definable subset in the o-minimal structure Ran), we have Γ(U; ν ◦ f F ) ⊆ Γ(U; f ◦ ν F ) for U ∈ Op(X ). However, in general, we ∗ !! .o ∗ def have ν ◦ f 6= f ◦ ν . ∗ !! .o ∗ For example, suppose that X = R2, Y = R, f : R2 → R is the projection onto the first coordinate and U = (0, 1). Let φ : (0, 1) → (0, +∞) be given by 1 φ(t) = x(1−x) . Set S = {(x, y) ∈ (0, 1) × R : y = φ(x))} and consider the sheaf AS. −1 Let s ∈ Γ(f (U); AS). Then f| : supp(s) → U is definably proper (and so proper in Def by [21]), so s ∈ Γ(U; f ◦ ν A ), but s∈ / Γ(U; ν ◦ f A ) = Γ(U; f A ) since .o ∗ S ∗ !! S !! S there is no compact (globally) sub-analytic subset K of X such that supp(s) ⊆ K.

Remark 3.16. The o-minimal proper direct image functor f in comparison with .o the sub-analytic proper direct image functor f!! seems to be more natural since it commutes with restrictions in the sense of Lemma 3.8, as the classical Verdier THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 25 proper direct image functor f! does in topological spaces. The property of that lemma is not satisﬁed by the functor f!! since there are less compact subsets on f −1(W ) than there are intersections of f −1(W ) with compact subsets of X.

3.3. f-soft sheaves. Here we introduce the f-soft sheaves where f : X → Y is a morphism in Def.g We show that the full additive subcategory of f-soft sheaves is f -injective and is stable under lim and whose ﬂat objects are also stable under .o −→ • ⊗ F for all F ∈ Mod(AX ).

Consider a ﬁber f −1(α) of a morphism f : X → Y in Defg and let c be the family of complete supports on f −1(α). Recall that a sheaf F on f −1(α) is c-soft if and only if the restriction Γ(f −1(α); F ) → Γ(K; F ) is surjective for every K ∈ c.

Deﬁnition 3.17. Let f : X → Y be a morphism in Defg and let F be a sheaf in Mod(AX ). We say that F is f-soft if for any α ∈ Y the sheaf F|f −1(α) in Mod(Af −1(α)) is c-soft. By Remark 3.4 we have:

Remark 3.18. Let aX : X → {pt} be the morphism to a point in Defg and let F be a sheaf in Mod(AX ). Then F is aX -soft if and only if it is c-soft.

Remark 3.19. Let f : X → Y be a morphism in Ae . To prove a property of f-soft sheaves in Mod(AX ) we have to take an arbitrary α ∈ Y and prove the corresponding property for c-soft sheaves in Mod(Af −1(α)). We will use Proposition 2.15 (4) to be able to replace f : X → Y by a suitable morphism f 0 : X0 → Y 0 in Ae such that α ∈ Y 0 and is closed in Y 0. After that we take a |= α a realization of α, S a prime 0 −1 0−1 model of the ﬁrst-order theory of M over {a} ∪ M and r| :(f S) (a) → f (α) the homeomorphism of Lemma 2.14. It follows that (f 0S)−1(a) is an object of Def(^S) and c is a normal and constructible family of supports on (f 0S)−1(a) because f 0 : X0 → Y 0 is a morphism in Ae and by Corollary 2.22, c is a normal and constructible family of supports on f 0−1(α). Therefore, we will be able to transfer results for c-soft sheaves on o-minimal spectral spaces ([24, Section 3]) to c-soft sheaves on the ﬁber f −1(α) since all we need is that c is a family of normal and constructible supports.

Remark 3.20. In the paper [24] we assumed that A is a ﬁeld, but that was only used to ensure that • ⊗ G ' G ⊗ •, for G ∈ Mod(AX ), is exact. For this reason, our results here about • ⊗ G ' G ⊗ • will come with the ﬂatness assumption.

The ﬁrst application of the above method is:

Proposition 3.21. Let f : X → Y be a morphism in Ae . Then ﬁltrant inductive limits of f-soft sheaves in Mod(AX ) are f-soft.

Proof. Let (Fi)i∈I be a ﬁltrant inductive family of f-soft sheaves in Mod(AX ). We have to show that lim F is an f-soft sheaf in Mod(A ), i.e., for every α ∈ Y −→ i X the sheaf (lim F ) −1 = lim(F −1 ) in Mod(A −1 ) is c-soft. −→ i |f (α) −→ i|f (α) f (α) 26 MARIO´ J. EDMUNDO AND LUCA PRELLI

So ﬁx α ∈ Y and by Proposition 2.15 (4), let Y 0 ∈ Opcons(Y ) be such that α ∈ Y 0 0 0 −1 0 0 0−1 −1 and α is closed in Y . Let X = f (Y ) and f = f|Y 0 . Then f (α) = f (α) and so F −1 = F 0−1 . Hence the sheaf lim(F −1 ) in Mod(A −1 ) is c-soft if i|f (α) i|f (α) −→ i|f (α) f (α) and only if the sheaf lim(F 0−1 ) in Mod(A 0−1 ) is c-soft. −→ i|f (α) f (α) Let a |= α a realization of α, S a prime model of the ﬁrst-order theory of M 0 −1 0−1 over {a} ∪ M and r| :(f S) (a) → f (α) the homeomorphism of Lemma 2.14. Then the sheaf lim(F 0−1 ) in Mod(A 0−1 ) is c-soft if and only if the sheaf −→ i|f (α) f (α) −1 lim(r F 0 −1 ) in Mod(A 0 −1 ) is c-soft. −→ i|(f S) (a) (f S) (a) But (f 0S)−1(a) is an object of Def(^S) and c is a normal and constructible family of supports on (f 0S)−1(a) because f 0 : X0 → Y 0 is a morphism in Ae and by Corollary 2.22, c is a normal and constructible family of supports on f 0−1(α). Since the sheaves Fi|f −1(α) in Mod(Af −1(α)) are c-soft if and only if the sheaves −1 Fi|f 0−1(α) in Mod(Af 0−1(α)) are c-soft if and only if the sheaves r Fi|(f 0S)−1(a) in

Mod(A(f 0S)−1(a)) are c-soft, by [24, Corollary 3.5] in Def(^S), we conclude that the −1 sheaf lim(r F 0 −1 ) in Mod(A 0 −1 ) is c-soft as required. −→ i|(f S) (a) (f S) (a) Since restriction is exact and commutes with ⊗, by the method of Remark 3.19 in combination with [24, Proposition 3.13] and Remark 3.20 we have:

Proposition 3.22. Let f : X → Y be a morphism in Ae . If G ∈ Mod(AX ) is f-soft and ﬂat, then for every F ∈ Mod(AX ) we have that G ⊗ F is f-soft. By Relative ﬁber formula (Proposition 3.9) we have:

Proposition 3.23. Let f : X → Y and g : Y → Z be morphisms in Ae and let F be a sheaf in Mod(A ). If F is (g ◦ f)-soft, then f F is g-soft. X .o

Proof. Let β ∈ Z, we shall prove that the restriction (f F ) −1 is c-soft. Let .o |g (β) K ⊂ K0 be elements of the family of complete supports c on g−1(β). By Relative ﬁber formula (Proposition 3.9), there is a commutative diagram

ψ 0 Γ(K0; f F ) K Γ (f −1(K0); F ) .o / c

ψ Γ(K; f F ) K Γ (f −1(K); F ) .o / c induced by the restrictions, with ψK0 and ψK isomorphisms. Since F is (g ◦ f)-soft, the arrow on the right is surjective by [24, Proposition 3.4 (3)]. Therefore, the arrow on the left is also surjective. This implies that, if K0 is a constructible element −1 of the family of complete supports c on g (β), then (f F ) 0 is soft, which implies .o |K (f F ) −1 c-soft by [24, Proposition 3.4 (4)]. .o |g (β) A special case which follows by Remark 3.6 and [24, Proposition 3.4] is the following. Compare also with [37, Chapter II, Proposition 2.5.7 (ii)].

Remark 3.24. Let f : X → Y be a morphism in Defg and let F be a sheaf in Mod(AX ). Suppose that the family c of supports on X (resp. on Y ) is such that every C ∈ c has a neighborhood D in X (resp. in Y ) such that D ∈ c. If F is c-soft, then f F is c-soft. .o THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 27

Lemma 3.25. Let f : X → Y be a morphism in Defg with X an open subspace of a normal space in Defg and let F be a sheaf in Mod(AX ). If F is injective and hence ﬂabby, then F is f-soft. In particular, the full additive subcategory of Mod(AX ) of f-soft sheaves is cogenerating. Proof. Let α ∈ Y and let K be an element of the family c of complete supports on f −1(α). Then K is quasi-compact in X (Remark 2.9). Since X is an open subspace of a normal space in Defg, by [24, Lemma 3.2], the canonical morphism

−→lim Γ(U; F ) → Γ(K;(F|f −1(α))|K ) K⊆U where U ranges through the family of open constructible subsets of X, is an isomorphism. Since F is ﬂabby Γ(X; F ) → Γ(U; F ) is surjective. The exactness of ﬁltrant lim−→ implies that Γ(X; F ) → Γ(K; F|f −1(α))|K ) is surjective. This morphism −1 factors through Γ(f (α); F|f −1(α)) and the result follows.

Proposition 3.26. Let f : X → Y be a morphism in Ae . Then the full additive subcategory of Mod(A ) of f-soft sheaves is f -injective, i.e.: X .o 0 (1) For every F ∈ Mod(AX ) there exists an f-soft F ∈ Mod(AX ) and an exact sequence 0 → F → F 0. 0 00 0 (2) If 0 → F → F → F → 0 is an exact sequence in Mod(AX ) and F is f-soft, then 0 → f F 0 → f F → f F 00 → 0 is an exact sequence. .o .o .o 0 00 0 (3) If 0 → F → F → F → 0 is an exact sequence in Mod(AX ) and F and F are f-soft, then F 00 is f-soft.

Proof. (1) By Lemma 3.25 the full additive subcategory of Mod(AX ) of f-soft sheaves is cogenerating. 0 00 0 (2) Let 0 → F → F → F → 0 be an exact sequence in Mod(AX ) with F is f- 0 00 soft. Then for every α ∈ Y , the sequence 0 → F|f −1(α) → F|f −1(α) → F|f −1(α) → 0 0 is exact and F|f −1(α) is a c-soft sheaf in Mod(Af −1(α)). By the method of Remark −1 0 3.19 in combination with [24, Proposition 3.7 (2)], 0 → Γc(f (α); F|f −1(α)) → −1 −1 00 Γc(f (α); F|f −1(α)) → Γc(f (α); F|f −1(α)) → 0 is an exact sequence for every α ∈ Y . By the Fiber formula (Corollary 3.13), the sequence 0 → f F 0 → f F → .o .o f F 00 → 0 is exact. .o 0 00 (3) Let 0 → F → F → F → 0 be an exact sequence in Mod(AX ) with 0 0 F and F both f-soft. Then for every α ∈ Y , the sequence 0 → F|f −1(α) → 00 0 F|f −1(α) → F|f −1(α) → 0 is exact and F|f −1(α) and F|f −1(α) are c-soft sheaves in Mod(Af −1(α)). By the method of Remark 3.19 in combination with [24, Proposi- 00 00 tion 3.7 (3)], F|f −1(α) is c-soft. Since α was arbitrary, F is f-soft.

3.4. Projection and base change formulas. Here we prove the projection and base change formulas.

For our next proposition we need the following topological lemma which is an adaptation of [37, Lemma 2.5.12]: 28 MARIO´ J. EDMUNDO AND LUCA PRELLI

Lemma 3.27. Let X be a quasi-compact topological space with a basis of open quasi-compact neighborhoods. Let F ∈ Mod(AX ), M a ﬂat A-module and Φ a family of supports on X. Then there is a natural isomorphism.

ΓΦ(X; F ) ⊗ M ' ΓΦ(X; F ⊗ MX ).

In particular, if F is Φ-soft, then F ⊗ MX is Φ-soft.

Proof. We may assume that X ∈ Φ. If X = ∪jUj is a ﬁnite cover of X by open quasi-compact neighborhoods, then

λ µ 0 → Γ(X; F ) → ⊕jΓ(Uj; F ) → ⊕j,iΓ(Uj ∩ Ui; F ) is exact. Applying the exact functor • ⊗ M, recall M is ﬂat, we obtain the commutative diagram with exact rows:

λ µ 0 / Γ(X; F ) ⊗ M / ⊕j Γ(Uj ; F ) ⊗ M / ⊕j,iΓ(Uj ∩ Ui; F ) ⊗ M

ϕ ψ ϑ

0 λ0 µ 0 / ΓΦ(X; F ⊗ MX ) / ⊕j ΓΦ(Uj ; F ⊗ MX ) / ⊕j,iΓΦ(Uj ∩ Ui; F ⊗ MX )

Observe also that for every x ∈ X we have

−→lim(Γ(U; F ) ⊗ M) ' −→limΓ(U; F ⊗ MX ), U U where U ranges through the family of open quasi-compact neighborhoods of x. In fact, both sides of this auxiliary isomorphism are isomorphic to Fx ⊗M. Thus, if s ∈ Γ(X; F ) ⊗ M is such that ϕ(s) = 0, then we can find, by the auxiliary isomorphism and quasi-compactness of X, a finite covering X = ∪jUj by open quasi-compact neighborhoods such that λ(s) = 0. Therefore, s = 0 and ϕ is injective. If we apply the same argument to Uj and Uj ∩ Ui instead of X we see that ψ and ϑ are also injective. To show that ϕ is surjective, take t ∈ Γ(X; F ⊗ MX ). By the auxiliary isomorphism above, there exists a finite covering X = ∪jUj by open quasi-compact neighborhoods such that λ0(t) is in the image of ψ. But by injectivity of ϑ it follows that t is in the image of ϕ.

By the Fiber formula (Corollary 3.13) and Lemma 3.27 we have:

Proposition 3.28 (Projection formula). Let f : X → Y be a morphism in Ae , F ∈ Mod(AX ) and G ∈ Mod(AY ). If G is ﬂat, then the natural morphism

f F ⊗ G → f (F ⊗ f −1G) .o .o is an isomorphism.

Proof. The morphism f −1 ◦f → f −1 ◦f → id induces the morphism f −1(f F ⊗ .o ∗ .o G) ' f −1 ◦ f F ⊗ f −1G → F ⊗ f −1G and we obtain the morphism f F ⊗ G → .o .o f (F ⊗ f −1G) by adjunction. .o THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 29

To prove that it is an isomorphism, let α ∈ Y . Then

−1 −1 −1 (f (F ⊗ f G)) ' Γ (f (α); (F ⊗ f G) −1 ) .o α c |f (α) −1 −1 ' Γc(f (α); F|f −1(α) ⊗ (f G)|f −1(α)) −1 ' Γc(f (α); F|f −1(α) ⊗ Gα) −1 ' Γc(f (α); F|f −1(α)) ⊗ Gα ' (f F ) ⊗ G .o α α ' (f F ⊗ G) , .o α by the Fiber formula (Corollary 3.13), Lemma 3.27 and using also the fact that −1 (f G)|f −1(α) ' Gα.

We now proceed to the proof of the base change formula. By Lemma 2.14, we have:

Remark 3.29. Let f : X → Y be a morphism in Defg and α ∈ Y . Let a |= α a realization of α and S a prime model of the theory of M over {a} ∪ M. Then we have a commutative diagram

S −1 f (f S) (a) / X(S) / Y (S)

Lemma 3.30. Consider a cartesian square in Defg

f 0 X0 / Y 0

g0 g f X / Y.

Let γ ∈ Y 0, v |= γ a realization of γ and S a prime model of the ﬁrst-order theory of M over {v} ∪ M. Let u = gS(v) (so u |= g(γ) is a realization of g(γ)). Then we have a commutative diagram

0S 0 −1 0 f 0 (f S) (v) / X (S) / Y (S)

0S 0 g| g S gS

S −1 f (f S) (u) / X(S) / Y (S) in Def(^S) with g0S :(f 0S)−1(v) → (f S)−1(u) an homeomorphism. 30 MARIO´ J. EDMUNDO AND LUCA PRELLI

Proof. Indeed, after removing the tilde, we have a similar commutative diagram of continuous S-definable maps between S-definable spaces with the restriction to the S-definable fibers an S-definable homeomorphism.

By Lemmas 2.14 and 3.30 we have:

Lemma 3.31. Consider a cartesian square in Defg

f 0 X0 / Y 0

g0 g f X / Y. Let γ ∈ Y 0, v |= γ a realization of γ and S a prime model of the ﬁrst-order theory of M over {v} ∪ M. Let u = gS(v) (so u |= g(γ) is a realization of g(γ)) and let K is a prime model of the ﬁrst-order theory of M over {u} ∪ M. If K = S, then we have a homeomorphism 0 0−1 −1 g| : f (γ) → f (g(γ)) which induces an isomorphism

0−1 0−1 −1 Γc(f (γ); (g F )|f 0−1(γ)) ' Γc(f (g(γ)); F|f −1(g(γ))) for every F ∈ Mod(AX ). Proof. We have the following commutative diagram

0 −1 0 r 0 _ 0−1 (f S) (v) / X (S) / X o ? f (γ)

0S 0 0 0 g| g S g g| −1 r _ −1 (f S) (u) / X(S) / Xo ? f (g(γ)).

0S 0S −1 S −1 Since g| :(f ) (v) → (f ) (u) is a homeomorphism by Lemma 3.30 and the −1 0 −1 0−1 −1 −1 −1 restrictions (r|) :(f S) (v) → f (γ) and (r|) :(f S) (u) → f (g(γ)) are also homeomorphisms (Lemma 2.14 and K = S) the result follows.

From now on until the end of the subsection we need to assume also (A3).

Lemma 3.32. Consider a cartesian square in Defg

f 0 X0 / Y 0

g0 g f X / Y. Suppose that f : X → Y satisﬁes (A3). If γ ∈ Y 0, then there exists an isomorphism

0−1 0−1 −1 Γc(f (γ); (g F )|f 0−1(γ)) ' Γc(f (g(γ)); F|f −1(g(γ))) for every F ∈ Mod(AX ). THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 31

Proof. Let v |= γ be a realization of γ and S a prime model of the ﬁrst-order theory of M over {v} ∪ M. Set u = gS(v) and note that u |= g(γ) is a realization of g(γ). Let K is a prime model of the ﬁrst-order theory of M over {u} ∪ M. Thus since u ∈ S, by Fact 2.13, we have either K = M and M 6= S or K = S. So we proceed with the proof by considering the two cases.

Case K = M and M 6= S: We have u = g(γ). Then we have −1 −1 Γc(f (g(γ)); F|f −1(g(γ))) = Γc(f (u); F|f −1(u)) S −1 ' Γc((f ) (u); F (S)|(f S)−1(u)) 0S −1 0S −1 ' Γc((f ) (v); (g ) F (S)|(f 0S)−1(v)) 0−1 0−1 ' Γc(f (γ); (g F )|f 0−1(γ)) where the ﬁrst isomorphism follows by (A3), the second follows from Lemma 3.30 and the third follows from Lemma 2.14 together with Remark 3.29. Case K = S: Then by Lemma 3.31 we have −1 0−1 0−1 Γc(f (g(γ)); F|f −1(g(γ))) ' Γc(f (γ); (g F )|f 0−1(γ)).

We are now ready to prove the base change formula:

Proposition 3.33 (Base change formula). Consider a cartesian square in Defg

f 0 X0 / Y 0

g0 g f X / Y

0 0 0 and let F ∈ Mod(AX ). Suppose that f : X → Y and f : X → Y are in Ae and that f : X → Y satisﬁes (A3). Then

−1 ∼ 0 0−1 g ◦ foF → f ◦ g F. . .o −1 0 0−1 Proof. Let us construct the morphism g ◦ fo → f ◦ g . We shall construct . .o ﬁrst the morphism 0 0 fo ◦ g → g∗ ◦ f . . ∗ .o cons 0 Let U ∈ Op (Y ) and G ∈ Mod(A 0 ). A section t ∈ Γ(U; f ◦ g G) is deﬁned X .o ∗ by a section s ∈ Γ((f ◦ g0)−1(U); G) such that supp(s) ⊂ g0−1(Z) for a closed −1 constructible subset Z of f (U) such that f|Z : Z → U is proper. Since

0 f| g0−1(Z) / g−1(U)

0 g g| |

f| Z / U is a cartesian square in Def,g (f ◦ g0)−1(U) = (g ◦ f 0)−1(U) and by Proposition 2.8 0 0−1 −1 (5), the restriction f|g0−1(Z) : g (Z) → g (U) is proper. Therefore, s ∈ Γ((g ◦ 32 MARIO´ J. EDMUNDO AND LUCA PRELLI f 0)−1(U); G) = Γ(f 0−1(g−1(U)); G) such that supp(s) ⊂ g0−1(Z) for a closed con- 0−1 0−1 −1 0 0−1 0−1 structible subset g (Z) of f (g (U)) such that f|g0−1(Z) : g (Z) → g (U) 0 0 0 is proper. Then s deﬁnes a section of Γ(U; g∗ ◦ f G) and we obtain fo ◦ g∗ → g∗ ◦ f . .o . .o −1 0 0−1 0 0−1 To construct g ◦ fo → f ◦ g consider the morphism fo → fo ◦ g∗ ◦ g → . .o . . 0 0−1 0 0 g∗ ◦ f ◦ g , where the second arrow is induced by fo ◦ g∗ → g∗ ◦ f . We deﬁne .o . .o −1 0 0−1 g ◦ fo → f ◦ g by adjunction. . .o −1 0 0−1 0 To prove that g ◦ foF → f ◦ g F is an isomorphism, let us take γ ∈ Y . . .o Then by the Fiber formula (Corollary 3.13) (g−1 ◦ f F ) ' (f F ) .o γ .o g(γ) −1 ' Γc(f (g(γ)); F|f −1(g(γ))) and 0 0−1 0−1 0−1 (f ◦ g F )γ ' Γc(f (γ); (g F )|f 0−1(γ)). .o Therefore, we have to show that −1 0−1 0−1 Γc(f (g(γ)); F|f −1(g(γ))) ' Γc(f (γ); (g F )|f 0−1(γ)). But this is proved in Lemma 3.32.

3.5. Derived proper direct image. Here we derive the proper direct image and prove the derived projection and base change formulas. As corollaries we obtain the universal coeﬃcients formula and the K¨unnethformula.

Remark 3.34. Since A is a commutative ring with finite weak global dimension, b + by the observations on page 110 in [37], if F ∈ D (AX ) (resp. F ∈ D (AX )), then F is quasi-isomorphic to a bounded complex (resp. a complex bounded form below) of flat A-sheaves. Therefore, we may define the left derived functor ∗ ∗ ∗ • ⊗ • :D (AX ) × D (AX ) → D (AX ) with ∗ = −, +, b.

Let f : X → Y be a morphism in Ae . We are going to consider the right derived functor of proper direct image Rf :D+(A ) → D+(A ). .o X Y + 0 If F ∈ D (AX ) and F is a complex of f-soft sheaves quasi-isomorphic to F (which exists by Proposition 3.26), then Rf F ' f F 0. .o .o Furthermore, if g : Y → Z is another morphism in Ae , then by Proposition 3.23, R(g ◦ f) ' Rg ◦ Rf . .o .o .o Note also that by Theorem 3.42, the functor Rf induces a functor: .o Rf :Db(A ) → Db(A ). .o X Y Deriving the projection formula (Proposition 3.28) we have: THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 33

Theorem 3.35 (Derived projection formula). Let f : X → Y be a morphism in + + Ae . Let F ∈ D (AX ) and G ∈ D (AY ). Then there is a natural isomorphism Rf F ⊗ G ' Rf (F ⊗ f −1G). .o .o Proof. First note that, if G is flat, then by Lemma 3.27, • ⊗ f −1G sends f- soft sheaves to f-soft sheaves. (Indeed, for every α ∈ Y , the restriction (•)|f −1(α) −1 commutes with ⊗ and (f G)|f −1(α) = Gα which is also flat.) + + 0 Now let F ∈ D (AX ) and G ∈ D (AY ). Let F be a complex of f-soft sheaves quasi-isomorphic to F (Proposition 3.26). By Remark 3.34, there exists a complex G0 of flat sheaves quasi-isomorphic to G. Then F 0 ⊗ f −1G0 is a complex of f-soft sheaves quasi-isomorphic to F ⊗ f −1G. Therefore, by Proposition 3.26, Rf F ⊗ G ' f F 0 ⊗ G0 ' f (F 0 ⊗ f −1G0) ' Rf (F ⊗ f −1G), .o .o .o .o where the second isomorphism follows from Proposition 3.28.

Corollary 3.36 (Universal coeﬃcients formula). Let X be an object of Defg. Let M be a A-module. Suppose that c is a normal and constructible family of supports on X. Then there is an isomorphism

RΓc(X; MX ) ' RΓc(X; AX ) ⊗ M. Proof. In the above proof if we use [24, Proposition 3.4] (resp. Lemma 3.27 ) instead of Proposition 3.26 (resp. Proposition 3.28), we obtain a natural isomorphism −1 RΓc(X; F )⊗M ' RΓc(F ⊗MX ). Since MX = aX Mpt ⊗AX , the result follows.

Corollary 3.37. Let X be an object of Defg. Let M be a A-module. Suppose that c is a normal and constructible family of supports on X. Then there is an exact sequence for each p ∈ Z p p p+1 0 → Hc (X; AX ) ⊗ M → Hc (X; MX ) → Tor1(Hc (X; AX ),M) → 0.

Recall that F is acyclic with respect to a left exact functor ϕ if RkϕF = 0 if k 6= 0. In such a situation F is also ϕ-injective.

In order to prove the derived base change formula, we need the following lemma in which we assume (A3): Lemma 3.38. Consider a cartesian square

f 0 X0 / Y 0

g0 g f X / Y in Defg. Suppose that f : X → Y satisﬁes (A3). Then g0−1(•) sends f-soft sheaves to f 0-acyclic sheaves. .o 34 MARIO´ J. EDMUNDO AND LUCA PRELLI

−1 Proof. Let F ∈ Mod(AX ) be f-soft. Then F|f −1(α) is Γc(f (α); •)-acyclic for 0−1 0−1 every α ∈ Y . We must show that the restriction (g F )|f 0−1(γ) is Γc(f (γ); •)- acyclic for every γ ∈ Y 0. So take γ ∈ Y 0. Let v |= γ be a realization of γ and S a prime model of the ﬁrst-order theory of M over {v} ∪ M. Set u = gS(v) and note that u |= g(γ) is a realization of g(γ). Let K is a prime model of the ﬁrst-order theory of M over {u} ∪ M. Thus since u ∈ S, by Fact 2.13, we have either K = M and M 6= S or K = S. So we proceed with the proof by considering the two cases.

Case K = M and M 6= S: We have u = g(γ) and F|f −1(g(γ)) = F|f −1(u). Since −1 S −1 F|f −1(u) is Γc(f (u); •)-acyclic, by (A3), F (S)|(f S)−1(u) is Γc((f ) (u); •)-acyclic. Since g0S :(f 0S)−1(v) → (f S)−1(u) is a homeomorphism (Lemma 3.30) and on 0S −1 0S −1 the other hand, ((g ) F (S))|(f 0S)−1(v) = (g| ) (F (S)|(f S)−1(u)), we conclude that 0S −1 0S −1 −1 0−1 0S −1 ((g ) F (S))|(f 0S)−1(v) is Γc((f ) (v); •)-acyclic. As (r|) : f (γ) → (f ) (v) 0−1 is a homeomorphism (Lemma 2.14), by Remark 3.29, we have that (g F )|f 0−1(γ) 0−1 is Γc(f (γ); •)-acyclic.

Let f : X → Y be a morphism in Ae . The full additive subcategory of Mod(AX ) of f -acyclic sheaves is f -injective. Therefore, if F ∈ D+(A ) and F 0 is a complex .o .o X of f -acyclic sheaves quasi-isomorphic to F , then .o

Rf F ' f F 0. .o .o

Deriving the base change formula (Proposition 3.33) we have:

Theorem 3.39 (Derived base change formula). Consider a cartesian square

f 0 X0 / Y 0

g0 g f X / Y in Defg. Suppose that f : X → Y and f 0 : X0 → Y 0 are in Ae and that f : + X → Y satisﬁes (A3). Then there is an isomorphism in D (AY 0 ), functorial in + F ∈ D (AX ):

−1 0 0−1 g ◦ RfoF ' Rf ◦ g F. . .o

Proof. Let F ∈ D+(A ) and let F 0 be a complex of f -soft sheaves quasi- X .o 0−1 0 isomorphic to F . By Lemma 3.38, g sends fo-soft sheaves to f -acyclic sheaves . .o and the f 0-acyclic sheaves are f 0-injective. So g0−1F 0 is a complex of f 0-acyclic .o .o .o sheaves quasi-isomorphic to g0−1F . Therefore, by Proposition 3.26 and Proposition THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 35

3.33 we have g−1 ◦ Rf F ' g−1 ◦ f F 0 .o .o ' f 0 ◦ g0−1F 0 .o ' Rf 0 ◦ g0−1F. .o

Combining the the derived projection and base change formulas we obtain: Theorem 3.40 (K¨unnethformula). Consider a cartesian square

f 0 X0 / Y 0 δ g0 g f ! X / Y in Defg where δ = f ◦ g0 = g ◦ f 0. Suppose that f : X → Y and f 0 : X0 → Y 0 are in Ae and that f : X → Y satisﬁes (A3). There is a natural isomorphism Rδ (g0−1F ⊗ f 0−1G) ' Rf F ⊗ Rg G .o .o .o + + for F ∈ D (AX ) and G ∈ D (AY 0 ). Proof. Using the derived projection formula and the derived base change formula we deduce

0 0−1 0−1 0 0−1 −1 Rf (g F ⊗ f G) ' (Rf ◦ g F ) ⊗ G ' (g ◦ RfoF ) ⊗ G. .o .o . Using the derived projection formula once again we ﬁnd

0 0−1 0−1 −1 Rgo ◦ Rf (g F ⊗ f G) ' Rgo((g ◦ RfoF ) ⊗ G) ' RfoF ⊗ RgoG . .o . . . . 0 and the result follows since Rδo ' Rgo ◦ Rf . . . .o

Corollary 3.41. Consider the cartesian square

pY X × Y / Y

aX×Y pX aY

aX # X / pt in Ae . Suppose that aX : X → pt satisﬁes (A3). Then for any k ∈ Z there is a natural isomorphism

k M q p Hc (X × Y ; AX×Y ) ' (Hc (X; AX ) ⊗ Hc (Y ; AY )). p+q=k

−1 −1 Proof. We have AX×Y ' pX AX ' pY AY . By Theorem 3.40 we obtain Ra A ' Ra A ⊗ Ra A . X×Y .o X×Y X.o X Y .o Y 36 MARIO´ J. EDMUNDO AND LUCA PRELLI

3.6. A bound for the cohomology of proper direct image. Here we ﬁnding a bound for the cohomology of the proper direct image.

The cohomological dimension of f is the smallest n such that Rkf F = 0 for all .o .o k > n and all sheaves F in Mod(AX ).

Theorem 3.42. Let f : X → Y be a morphism in Ae . Then the cohomological dimension of f is bounded by dim X. .o

Proof. Let F ∈ Mod(AX ) and let α ∈ Y . Taking a f-soft resolution of F −1 (Proposition 3.26) one checks easily that (Rf F ) ' RΓ (f (α); F −1 ) (the .o α c |f (α) Fiber formula - Corollary 3.13). By the method of Remark 3.19 in combination with [24, Theorem 3.12], we see k k k −1 that H (Rf F ) ' (R f F ) ' R Γ (f (α); F −1 ) = 0 if k > dim X. Since α .o α .o α c |f (α) was arbitrary the result follows.

4. Poincare-Verdier´ Duality In this section we prove the local and the global Verdier duality, we introduce the A-orientation sheaf and prove the Poincar´eand the Alexander duality.

The proofs here follow in a standard way from the results already obtained in the previous section. Compare with the topological case in [37, Chapter III, Sections 3.1 and 3.3].

4.1. Poincar´e-Verdier duality. Here we show that the derived proper direct image functor Rf extends to D(A ) and both Rf and its extension have a right .o X .o adjoint f .o. We then deduce the basic properties of the right adjoint f .o and the local and the global Verdier duality.

Let f : X → Y be a morphism in Ae . Let J be the full additive subcategory of Mod(A ) of f-soft sheaves. As a consequence of the results we proved for f and X .o for f -acyclic sheaves we have the following properties: .o  J is cogenerating;  fo has ﬁnite cohomological dimension;  . J is fo-injective;  . J is stable under small ⊕;  f commutes with small ⊕. .o Therefore, by [39, Proposition 14.3.4] the functor Rf :D+(A ) → D+(A ) extends .o X Y to a functor Rf : D(A ) → D(A ) .o X Y such that:

(i) for every F ∈ D(AX ) we have Rf F ' f F 0 .o .o THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 37

where F 0 is a complex of f -acyclic sheaves quasi-isomorphic to F ; .o (ii) Rf commutes with small ⊕. .o

Theorem 4.1. Let f : X → Y be a morphism in A. Then the functor Rf : e .o D(AX ) → D(AY ) admits a right adjoint o f . : D(AY ) → D(AX ). The functor f .o will thus satisfy an isomorphism Hom (Rf F ; G) ' Hom (F ; f .oG) D(AY ) .o D(AX ) functorial in F ∈ D(AX ) and G ∈ D(AY ). Moreover, the restriction o + + f . :D (AY ) → D (AX ) is well deﬁned and it is the right adjoint to the restriction Rf :D+(A ) → D+(A ). .o X Y o Proof. The existence of right adjoint f . : D(AY ) → D(AX ) is a consequence of the Brown representability theorem (see [39, Corollary 14.3.7] for details). + o + For the second part we have to show that if G ∈ D (AY ), then f .G ∈ D (AX ). ≥0 We may assume that G ∈ D (AY ). Let N0 be the dimension of X. Then the cohomological dimension of f is bounded by N (Theorem 3.42). Set a = −N − 1 .o 0 0 for short. If F ∈ D≤a(A ), then Rf F ∈ D≤−1(A ) and X .o Y 0 = Hom (Rf F,G) ' Hom (F, f .oG). D(AY ) .o D(AX ) ≤a .o Hence for each F ∈ D (AX ) we have HomD(AX )(F, f G) = 0. In particular, if F = τ ≤af .oG, then ≤a o o ≤a o ≤a o . . ≤a . . HomD(AX )(τ f G, f G) ' HomD (AX )(τ f G, τ f G) = 0 ≤a o o + and so τ f .G = 0. This implies, by deﬁnition, that f .G ∈ D (AX ).

The bound of the f-soft dimension of Mod(AX ) implies the following result:

Proposition 4.2. Let f : X → Y be a morphism in Ae . Let F ∈ Mod(AX ), G ∈ Mod(AY ). Let N0 be the dimension of X. Then Hom(F,H−N0 f .oG) ' Hom(RN0 f F,G). .o Proof. Let I• be a complex of injective objects quasi-isomorphic to f .oG. As in the proof of Theorem 4.1 we have f .oG ' τ ≥−N0 I• and hence we have the exact sequence 0 → H−N0 f .oG → I−N0 → I−N0+1 that implies the isomorphism

−N0 o o . + . Hom(F,H f G) ' HomD (AX )(F, f G[−N0]). On the other hand the complex Rf F [N ] is concentrated in negative degree and .o 0 hence

0 N0 Hom + (Rf F [N ],G) ' Hom(R f F [N ],G) ' Hom(R f F,G). D (AY ) .o 0 .o 0 .o Then the result follows from Theorem 4.1. 38 MARIO´ J. EDMUNDO AND LUCA PRELLI

If a morphism f : X → Y in Ae is a homeomorphism onto a locally closed subset of Y , then f is (isomorphic to) the extension by zero functor. What about f .o? .o

Proposition 4.3. Let f : X → Y be a morphism in Ae . If f : X → Y is a homeomorphism onto a locally closed subset of Y , then o −1 −1 f .G ' f ◦ RHom(Af(X),G) ' f ◦ RΓf(X)(G) + for every G ∈ D (AY ). In particular: • if f : X → Y is a closed immersion, then id ' f .o ◦ Rf ; .o • if f : X → Y is an open immersion, then f .o ' f −1.

+ + Proof. Let G ∈ D (AY ) and F ∈ D (AX ). Then

Hom + (f F,G) ' Hom + (f F,RHom(A ,G)) D (AY ) .o D (AY ) .o f(X) −1 −1 ' Hom + (f ◦ f F, f ◦ RHom(A ,G)) D (AX ) .o f(X) −1 + ' HomD (AX )(F, f ◦ RHom(Af(X),G)).

o −1 Hence f .G ' f ◦ RHom(Af(X),G). Suppose now that f : X → Y is a closed immersion. Then f is proper, Rf ' Rf ∗ .o −1 and f ◦ Rf∗ ' id. On the other hand, we have the isomorphisms o −1 f . ◦ Rf∗F ' f ◦ RHom(Af(X), Rf∗F ) −1 ' f ◦ Rf∗RHom(Af(X),F ) −1 ' f ◦ Rf∗F.

Hence, id ' f .o ◦ Rf . .o −1 Suppose now that f : X → Y is an open immersion. Then f ◦ Rf∗ ' id and −1 o −1 RΓf(X) ' Rf∗ ◦ f . Hence, f . ' f .

We now prove several useful properties of the dual f .o of the derived proper direct image functor Rf . .o

Proposition 4.4. Let f : X → Y and g : Y → Z be morphisms in Ae . Then (g ◦ f).o ' f .o ◦ g.o. Proof. This follows from R(g ◦ f) ' Rg ◦ Rf and the adjunction in Theorem .o .o .o 4.1.

Proposition 4.5 (Dual projection formula). Let f : X → Y be a morphism in Ae . b + Let F ∈ D (AY ) and G ∈ D (AY ). Then we have a natural isomorphism

f .o ◦ RHom(F,G) ' RHom(f −1F, f .oG). Proof. This follows from the derived projection formula (Theorem 3.35), the adjunction in Theorem 4.1 and the adjunction

+ + HomD (AZ )(F ⊗ H,G) ' HomD (AZ )(H,RHom(F,G)). THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 39

Proposition 4.6 (Dual base change formula). Consider a cartesian square

f 0 X0 / Y 0

g0 g f X / Y in Defg. Suppose that f : X → Y and f 0 : X0 → Y 0 are in Ae and that f : + X → Y satisﬁes (A3). Then there is an isomorphism in D (AX ), functorial in + F ∈ D (AY 0 ): .o 0 0o. f ◦ Rg∗F ' Rg∗ ◦ f F. Proof. This follows from the derived base change formula (Theorem 3.39), the adjunction in Theorem 4.1 and the adjunction −1 + + HomD (AZ )(H, Rh∗G) ' HomD (AX )(h H,G) for every morphism h : X → Z in Ae .

Theorem 4.7 (Local and global Verdier duality). Let f : X → Y be a morphism b + in Ae . Then for F ∈ D (AX ) and G ∈ D (AY ), we have the local Verdier duality Rf ◦ RHom(F, f .oG) ' RHom(Rf F,G) ∗ .o and the global Verdier duality RHom(F, f .oG) ' Hom(Rf F,G). .o Proof. We obtain the morphism Rf ◦ RHom(F, f .oG) → RHom(Rf F,G) ∗ .o by composing the canonical morphism Rf ◦ RHom(F, f .oG) → RHom(Rf F, Rf ◦ f .oG) ∗ .o .o with the morphism Rf ◦ f .oG → G obtained by adjunction (Theorem 4.1). .o Let V ∈ Op(Y ). Then we have j o o H (RΓ(V ; Rf ◦ RHom(F, f .G))) ' Hom + (F −1 , f .G[j] −1 ) ∗ D (Af−1(V )) |f (V ) |f (V )

' Hom + (Rf F ,G[j] ) D (AV ) .o |V |V ' Hj(RΓ(V ; RHom(Rf F,G))) .o completing the proof of the ﬁrst isomorphism. The second isomorphism is obtained from the ﬁrst one by applying the functor RΓ(Y ; •).

4.2. Orientation and duality. Here we introduce the A-orientation sheaf and prove the Poincar´eand the Alexander duality theorems.

Remark 4.8. Let X be an object of Defg such that c is a normal and constructible family of supports on X. Then as the reader can easily verify all our previous results for the proper direct image f of a morphism f : X → Y in A that do not require .o e (A3) hold also for the proper direct image a of the morphism a : X → pt. X.o X Indeed, (A0), (A1) and (A2) were used only to show: 40 MARIO´ J. EDMUNDO AND LUCA PRELLI

−1 (i) if α ∈ Y is closed, then c = Φf ∩f (α) and c is a normal and constructible family of supports on f −1(α); (ii) ﬁber formula; (iii) the theory of f-soft sheaves.

For the morphism aX : X → pt we have that (i) holds if we assume that c is a normal and constructible family of supports on X, (ii) is Remark 3.4 and (iii) is the theory of c-soft sheaves developed already in [24, Section 3].

Let X be an object of Defg such that c is a normal and constructible family of supports on X. Then the functor Ra :D+(A ) → D+(Mod(A)) admits a right X.o X adjoint .o aX : D(Mod(A)) → D(AX ) (Theorem 4.1) and we have the dual projection formula (Proposition 4.5) and local and the global Verdier duality (Theorem 4.7) for a . In particular, we obtain the X.o form of the global Verdier duality proved already in [24] (assuming A is a ﬁeld), .o where aX AX is the dualizing complex:

Theorem 4.9 (Absolute Poincar´eduality). Let X be an object of Defg such that c is a normal and constructible family of supports on X. Then we have a natural isomorphism .o RHom(F, aX A) ' RHom(RΓc(X; F ),A) b as F varies through D (AX ).

Let X be an object of Defg such that c is a normal and constructible family of supports on X. We say that X has an A-orientation sheaf if for every U ∈ cons Op (X) there exists an admissible (ﬁnite) cover {U1,...,U`} of U such that for each i we have  A if p = dimX  p  (1) Hc (Ui; AX ) =   0 if p 6= dimX.

−dimX .o If X has an A-orientation sheaf, we set OrX := H aX A and call it the A-orientation sheaf on X. For A = Z we simply say orientation sheaf on X.

Theorem 4.10. Let X be an object of Defg of dimension n such that c is a normal and constructible family of supports on X. Suppose that X has an A-orientation sheaf OrX . Then OrX is the A-sheaf on X with sections n Γ(U; OrX ) ' Hom(Hc (U; AX ),A).

Moreover OrX is locally constant and there is a quasi-isomorphism

.o OrX [n] ' aX A. THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 41

Proof. Setting f = aX ,F = AU , G = A in Proposition 4.2 we obtain n n Γ(U; OrX ) ' Hom(Hc (X; AU ),A) ' Hom(Hc (U; AX ),A), where the second isomorphism follows from [24, Corollary 3.9]. .o On the other hand, (1), implies RΓ(Ui; aX A) ' RHom(RΓc(Ui; AX ),A) ' .o −n .o A[−n], i.e. aX A is concentrated in degree −n and the sheaf H aX A is locally isomorphic to AX .

In particular we recover [24, Theorem 4.11]: Remark 4.11 (Poincar´eduality in cohomology). When A is a ﬁeld, setting (•)∨ = Hom(•,A) we obtain: p n−p ∨ H (X; OrX ) ' Hc (X; AX ) . Using the pure homological algebra result [35, Proposition VI.4.6] we also have:

Corollary 4.12. Let X be an object of Defg of dimension n such that c is a normal and constructible family of supports on X. Suppose that X has a Z-orientation sheaf OrX . Then there is a short exact sequence of abelian groups: 1 k+1 n−k k 0 → Ext (Hc (X; ZX ), Z) → H (X; OrX ) → Hom(Hc (X, ZX ), Z) → 0. n−k k k+1 In particular H (X; OrX ) ' Hom(Hc (X, ZX ), Z) when Hc (X, ZX ) has no torsion.

Proof. By the pure homological algebra result [35, Proposition VI.4.6] we have 1 k+1 • −k • k • 0 → Ext (H C , Z) → H RHom(C , Z) → Hom(H C , Z) → 0 • • for any bounded complex C of abelian groups. Applying this to C = RΓc(X; ZX ) .o and using RHom(RΓc(X; ZX ), Z) ' RHom(ZX , aX Z) ' RΓ(X; OrX [n]) (by The- orems 4.9 and 4.10) the result follows.

Corollary 4.13. Let X be an object of Defg of dimension n such that c is a normal and constructible family of supports on X. Suppose that X has a Z-orientation sheaf OrX . Then there exists an isomorphism n 0 l H (X; OrX ) ' Hom(Hc (X; ZX ), Z) ' Z where l is the number of complete connected components of X. 0 l Proof. By Corollary 4.12 (with k = 0) and since Hc (X; ZX ) = Z where l is the number of complete connected components of X, the result follows once we show 1 that Hc (X; ZX ) is torsion free. But this is [5, Chapter I, Exercise 11 and Chapter II, Exercise 28].

By an A-orientation we understand an isomorphism AX 'OrX . We shall say that X is A-orientable if an A-orientation exists and A-unorientable in the opposite case. For A = Z we simply say orientation, orientable or unorintable.

From [24, Theorem 3.12] (dim X is a bound on the cohomological c-dimension of X) and Corollary 4.12, arguing as in [24, Proposition 4.13] we have: 42 MARIO´ J. EDMUNDO AND LUCA PRELLI

Proposition 4.14. Let X be an object of Defg of dimension n such that c is a normal and constructible family of supports on X. Suppose that X has an orientation sheaf OrX . Then n (1) Hc (X; ZX ) ' Z if X is orientable. n (2) Hc (X; ZX ) ' 0 if X is unorientable.

If Z a closed constructible subset of X, then setting F = AZ , G = A in Theorem 4.9 we obtain:

Theorem 4.15 (Alexander duality). Let X be an object of Defg of dimension n such that c is a normal and constructible family of supports on X. Suppose that X is A-orientable. If Z a closed constructible subset of X, then there exists a quasi-isomorphism

RΓZ (X; AX ) ' RHom(RΓc(Z; AX ),A)[n]. In particular we recover [24, Theorem 4.14]: Remark 4.16 (Alexander duality in cohomology). When A is a ﬁeld, setting (•)∨ = Hom(•,A) we obtain:

p n−p ∨ HZ (X; AX ) ' Hc (Z; AX ) . Using the pure homological algebra result [35, Proposition VI.4.6] we also have:

Corollary 4.17. Let X be an object of Defg of dimension n such that c is a normal and constructible family of supports on X. Suppose that X is Z-orientatable. Then there is a short exact sequence of abelian groups:

1 k+1 n−k k 0 → Ext (Hc (Z; ZX ), Z) → HZ (X; ZX ) → Hom(Hc (Z, ZX ), Z) → 0. n−k k k+1 In particular HZ (X; ZX ) ' Hom(Hc (Z; ZX ), Z) when Hc (Z; ZX ) has no torsion.

Proof. By the pure homological algebra result [35, Proposition VI.4.6] we have

1 k+1 • −k • k • 0 → Ext (H C , Z) → H RHom(C , Z) → Hom(H C , Z) → 0 • • for any bounded complex C of abelian groups.Applying this to C = RΓc(Z; ZX ) and using RHom(RΓc(Z; ZX ), Z)[n] ' RΓZ (X; ZX ) (by Theorem 4.15) the result follows.

Corollary 4.18. Let X be an object of Defg of dimension n such that c is a normal and constructible family of supports on X. Suppose that X is Z-orientatable. Then there exists an isomorphism

n 0 l HZ (X; ZX ) ' Hom(Hc (Z; ZX ), Z) ' Z induced by the given orientation, where l is the number of complete connected components of Z. THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 43

0 l Proof. By Corollary 4.17 (with k = 0) and since Hc (Z; ZX ) = Z where l is the number of complete connected components of Z, the result follows once we show 1 that Hc (Z; ZX ) is torsion free. But this is [5, Chapter I, Exercise 11 and Chapter II, Exercise 28].

References [1] A. Berarducci Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup J. Symb. Logic 74 (3) (2009) 891-900. [2] A. Berarducci and A. Fornasiero O-minimal cohomology: finiteness and invariance results J. Math. Logic 9 (2) (2009) 167–182. [3] E. Bierstone and P. Milman Sub-analytic geometry in Model Theory, Algebra and Geometry, D. Haskell, A. Pillay and C. Steinhorn (eds), MSRI Publications 39 (2000) 151–172. [4] J. Bochnak, M. Coste and M-F. Roy Real Algebraic Geometry Ergebnisse der Math. (3) 36, Springer-Verlag, Berlin (1998). [5] G. Bredon Sheaf theory Second Edition, Graduate Texts in Math. 170, Springer- Verlag, New York, (1997). [6] M. Coste An introduction to o-minimal geometry Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa (2000). [7] M. Carral and M. Coste Normal spectral spaces and their dimensions J. Pure Appl. Algebra 30 (3) (1983) 227–235. [8] M. Coste and M.-F. Roy La topologie du spectre réel in Ordered fields and real algebraic geometry, Contemporary Mathematics 8 (1982) 27–59. [9] A. D’Agnolo On the Laplace transform for tempered holomorphic functions Int. Math. Res. Not., Advance Access (2013). [10] A. D’Agnolo and M. Kashiwara Riemann-Hilbert correspondence for holonomic D-modules arXiv:1311.2374 (2013), pp. 114. [11] H. Delfs The homotopy axiom in semi-algebraic sheaf cohomology J. Reine Angew. Math. 355 (1985) 108–128. [12] H. Delfs Homology of locally semialgebraic spaces Lecture Notes in Math. 1484, Springer-Verlag, Berlin (1991). [13] H. Delfs and M. Knebush Semi-algebraic topology over a real closed field II: Basic theory of semi-algebraic spaces Math. Z. 178 (1981) 175–213. [14] J. Denef and L. van den Dries p-adic and real subanalytic sets Annals Math. 128 (1988) 79–138. [15] L. van den Dries A generalization of Tarski-Seidenberg theorem and some nonde- finability results Bull. Amer. Math. Soc. (N.S) 15 (1986) 189–193. [16] L. van den Dries Tame topology and o-minimal structures London Math. Soc. Lec- ture Note Series 248, Cambridge University Press, Cambridge (1998). [17] L. van den Dries, A. Macintyre, D. Marker The elementary theory of restricted analytic fields with exponentiation Ann. Math. 140 (1994) 183–205. [18] L. van den Dries and C. Miller Geometric categories and o-minimal structures Duke Math. J. 84 (1996) 497–540. [19] M. Edmundo Structure theorems for o-minimal expansions of groups Ann. Pure Appl. Logic 102 (1-2) (2000) 159–181. [20] M. Edmundo, G. Jones and N. Peatfield Sheaf cohomology in o-minimal structures J. Math. Logic 6 (2) (2006) 163–179. [21] M. Edmundo, M. Mamino and L. Prelli On definably proper maps arXiv: 1404.6634. [22] M. Edmundo, M. Mamino, L. Prelli, J. Ramakrishnan and G. Terzo On the o- minimal Hilbert’s fifth problem Preprint, April 2014 (http://www.ciul.ul.pt/ ~edmundo/oh5p_20_07_14.pdf) [23] M. Edmundo and M. Otero Definably compact abelian groups. J. Math. Logic 4 (2) (2004) 163–180. [24] M. Edmundo and L. Prelli Poincaré- Verdier duality in o-minimal structures Ann. Inst. Fourier Grenoble 60 (4) (2010) 1259–1288. 44 MARIO´ J. EDMUNDO AND LUCA PRELLI

[25] M. Edmundo and L. Prelli Invariance of o-minimal cohomology with definably compact supports arXiv: 1205.6124. [26] M. Edmundo and G. Terzo A note on generic subsets of definable groups Fund. Math. 215 (1) (2011) 53–65. [27] P. Eleftheriou and S. Starchenko Groups definable in ordered vector spaces over ordered division rings J. Symb. Logic 72 (2007) 1108–1140. [28] P. Eleftheriou, Y. Peterzil and J. Ramakrishnan Interpretable groups are definable arXiv: 1110.6581v1. [29] R. Godement Théoriedes faisceaux Hermann, Paris (1958). [30] A. Grothendieck and J. Dieudonné Elements de géometriealgébriqueI Inst. Hautes Etudes´ Sci. Publ. Math. 4 (1960) 5–228. [31] A. Grothendieck and J. Dieudonné Elements de géometrie algébrique II Inst. Hautes Etudes´ Sci. Publ. Math. 8 (1961) 5–222. [32] R. Hartshorne Algebraic geometry Graduate Texts in Mathematics 52, Springer Verlag, New York (1977). [33] E. Hrushovski, Y. Peterzil A question of van den Dries and a theorem of Lipshitz and Robinson; not everything is standard J. Symb. Logic 72 (2007) 119–122. [34] E. Hrushovski, Y. Peterzil and A. Pillay Groups, measures and the NIP J. Amer. Math. Soc. 21 (2) (2008) 563 – 596. [35] B. Iversen Cohomology of sheaves Universitext, Springer Verlag, Berlin (1986). [36] T. Kayal and G. Raby Ensemble sous-analytiques: quelques propriétésglobales C. R. Acad. Sci. Paris Ser. I Math. 308 (1989) 521–523. [37] M. Kashiwara and P. Schapira Sheaves on manifolds Grundlehren der Math. 292, Springer-Verlag, Berlin (1990). [38] M. Kashiwara and P. Schapira Ind-sheaves Astérisque 271 (2001). [39] M. Kashiwara and P. Schapira Categories and sheaves Grundlehren der Math. 332, Springer-Verlag, Berlin (2006). [40] L. Lipshitz, Z. Robinson Overconvergent real closed quantifier elimination Bull. London Math. Soc. 38 (2006) 897–906. [41] M. Otero A survey on groups definable in o-minimal structures in Model Theory with Applications to Algebra and Analysis, vol. 2, Editors: Z. Chatzidakis, D. Macpherson, A. Pillay and A. Wilkie, LMS LNS 350 Cambridge Univ. Press (2008) 177–206. [42] Y. Peterzil Returning to semi-bounded sets J. Symb. Logic 74 (2) (2009) 597–617. [43] Y. Peterzil and S. Starchenko Uniform definability of the Weierstrass ℘-functions and generalized tori of dimension one Selecta Math. (N.S.) 10 (4) (2004) 525–550. [44] Y. Peterzil and C. Steinhorn Definable compactness and definable subgroups of o- minimal groups J. London Math. Soc. 59 (2) (1999) 769–786. n [45] J. Pila O-minimality and the André-Oortconjecture for C Ann. Math. 173 (3) (2011) 1779–1840. [46] J. Pila and U. Zannier Rational points in periodic analytic sets and the Manin- Mumford conjecture Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 19 (2) (2008) 149–162. [47] J. Pila and A. J. Wilkie The rational points of a definable set Duke Math. J. 133 (3) (2006) 591–616. [48] A. Pillay Sheaves of continuous definable functions J. Symb. Logic 53 (4) (1988) 1165–1169. [49] A. Pillay On groups and fields definable in o-minimal structures J. Pure Appl. Algebra 53 (1988) 239–255. [50] A. Pillay Type-definability, compact Lie groups and o-minimality J. Math. Logic 4 (2) (2004) 147–162. [51] A. Pillay and C. Steinhorn Definable sets in ordered structures I Trans. Amer. Math. Soc. 295 (2) (1986) 565–592. [52] L. Prelli Sheaves on subanalytic sites Rend. Sem. Mat. Univ. di Padova 120 (2008) 167–216. [53] N. Schwartz The basic theory of real closed spaces Memoirs of the AMS 77 (397) (1989). THE SIX GROTHENDIECK OPERATIONS ON O-MINIMAL SHEAVES 45

[54] J. L. Verdier Dualitédans la cohomologie des espaces localment compact Seminaire Bourbaki 300 (1965). [55] A. Wilkie Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function J. Amer. Math. Soc. 9 (1996) 1051–1094.

Universidade Aberta, Campus do Tagus Park, Edif´ıcio Inovac¸ao˜ I, Av. Dr. Jaques Delors, 2740-122 Porto Salvo, Oeiras, Portugal, and, CMAF Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal E-mail address: [email protected]

Universita` di Padova, Dipartimento di Matematica Pura ed Applicata, Via Trieste 63, 35121 Padova, Italy E-mail address: [email protected]