aa r X i v : . [ m a t h . AG ] J a n Constructible sheaves and functions up to infinity
Pierre SchapiraJanuary 6, 2021
Abstract
Posted here for easier accessibility, the first part recalls classical results onconstructible functions and their Euler calculus. Next, we introduce the cate-gory of b-analytic manifolds, a natural tool to define constructible sheaves andfunctions up to infinity. We study the operations on these objects and show inparticular how the γ -topology for constructible functions could be an efficienttool for TDA. Contents R -constructible sheaves and functions 4 γ -constructible sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 γ -constructible functions . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Key words: constructible sheaves, constructible functions, subanalytic, γ -topologyMSC: 55N99, 32B20, 32S60This research was supported by the ANR-15-CE40-0007 “MICROLOCAL”. ntroduction Constructible sheaves on a real analytic manifold play an increasing role in various fieldsof mathematics and are perfectly understood. Constructible functions, which may bededuced from constructible sheaves by taking the local Poincar´e indices of constructiblesheaves, are also of fundamental importance and are also well-understood. However, itoften happens that one is led to consider such sheaves or functions “up to infinity”andin this case no systematic study exists to our knowledge. The aim of this paper is tofill this gap. However, for easier accessibility of the subject, we shall first recall theclassical results on constructible functions.The group of constructible functions on a real analytic manifold is isomorphic tothe Grothendieck group of constructible sheaves, and there is another group isomorphicto this Grothendieck group, namely that of Lagrangian cycles. For an history of thesubject, we refer to [KS90, Ch. IX, Notes]. Simply recall that constructible functionsand/or Lagrangian cycles first appeared in the algebraic setting with Robert McPher-son [McP74] and in the complex analytic setting with Masaki Kashiwara [Kas73]. Inthe complex setting, Lagrangian cycles were studied by several people and in particularby Victor Ginsburg [Gin86] and Claude Sabbah [Sab85] for their functorial properties.The real case was first treated in [Kas85].The Euler calculus of constructible functions has been introduced independently byOleg Viro (see [Vir88]) in the complex setting and by the author in the real analyticsetting (see [Sch89]). It has many applications, particularly to tomography i.e., realRadon transform, see [Sch95] (see also Lars Ernstr¨om [Ern94] for complex projectiveduality) and more generally in Topological Data Analysis (see [CGR12] for a survey).In this paper, we shall restrict ourselves to constructible functions (and sheaves). Aconstructible function ϕ on a real analytic manifold X is mathematically very simple: itis a Z -valued function, the sets ϕ − ( m ) ( m ∈ Z ) being all subanalytic and the family ofsuch sets being locally finite. It is not difficult (with the tools of subanalytic geometry athands) to check that the set CF ( X ) of constructible functions on X is a commutativeunital algebra and that the composition of such a function by a morphism f : Z −→ X is again constructible. Things become more unusual when looking at direct images, inparticular integration. Assume that ϕ has compact support. One may write ϕ as afinite sum P i ∈ I c i K i where c i ∈ Z , K i is a compact subanaytic subset of X and for S ⊂ X , S is the characteristic function of S . Then one defines the integral of ϕ by theformula Z X ϕ = X i ∈ I c i · χ ( K i )where χ ( K i ) denotes the Euler-Poincar´e index of K i . (Of course, one has to check thatthis definition does not depend on the decomposition of ϕ .) For a morphism f : X −→ Y of real analytic manifolds, one defines the integral along f of a function ϕ ∈ CF ( X )whose support is proper with respect to f by setting for y ∈ Y ,( Z f ϕ )( y ) = Z X ϕ · f − ( y ) . (0 , is −
1) and a setreduced to one point has integral 1. In fact, one easily translates all operations onconstructible sheaves to operations on constructible functions. In particular dualitymakes sense for constructible functions and commutes with direct images.As mentioned in the title, we shall extend all these constructions to the case ofconstructibles sheaves and functions “up to infinity”.Let us describe with some details the contents of this paper.In
Section 1 , we recall results already published in [Sch89, Sch91, Sch95] and alsopartly in [KS90, Ch. IX]. We give equivalent definitions of the notion of a constructiblefunction and discuss carefully the sheaf of algebras of such functions and in particularthe operations, such as duality, inverse images and direct images. We give a base changeformula and a projection formula and prove that, under a natural properness condition,the composition of kernels is associative.Then, following [Sch95], we briefly recall how to apply these results to the study ofcorrespondences on real flag manifolds and in particular to the Radon transform ( i.e., projective duality).
Sections 2 and 3 develop the notion of being constructible up to infinity. Forthat purpose we introduce the category of b-analytic manifolds, whose objects aresimply the open embeddings X ⊂ b X (denoted X ∞ ) of smooth real analytic manifoldswith X subanalytic and relatively compact in b X , a morphism being a real analyticmap f : X −→ Y such that the graph of f is subanalytic in b X × b Y . This definitionhas a certain similarity with that of bordered space of [DK16] but is different. Alsonote that the notion of being subanalytic up to infinity is a particular case of thatof definable sets and of o -minimal structures, well-known from the specialists (see inparticular [VdD98, VdDM96]) and constructible sheaves and functions in this generalframework have already been defined in [Sch03]. Nevertheless, our approach for sheaves,based on the notion of micro-support, is of a different nature.In Sections 2 we define a (derived) sheaf constructible up to infinity on X as aconstructible sheaf whose micro-support is subanalytic in the cotangent bundle T ∗ b X .This is equivalent to saying that its (proper or non proper) direct image in b X is againconstructible. (Note that such a property already appeared in [KS20].) We brieflystudy the six operations on the triangulated category of derived constructible sheavesup to infinity and show that, on a real vector space, the non proper convolution isassociative, in contrast with the usual case.Finally, in Sections 3 , we define the space CF ( X ∞ ) of constructible functions upto infinity and study with some care the operations on such functions. Contrarily tothe classical case, we have now two kind of integrals, proper and non proper. We provethat the proper composition of kernels is associative.We give two applications.First, we give a new construction of the Fourier-Sato transform for conic con-structible functions on a vector space V and prove an inversion formula. Our methodis rather different than the classical one since we prove that the composition of theFourier-Sato kernel and its inverse gives a kernel which makes the functions conic and3nduces the identity on such functions.Next, given a closed convex proper cone γ (still on V ), we prove that the non properconvolution γ np ⋆ is a projector from the space of constructible functions up to infinity tothat of γ -constructible functions, that is, constructible functions for the γ -topology. Wealso prove that if { γ j } j is a finite family of cones as above whose union of the interiors ofthe polar cones covers V ∗ \ { } , then the map ϕ
7→ { γ j np ⋆ ϕ } j is injective. This allowsus to endow the space CF ( V ∞ ) with various norms since, contrarily to CF ( V ∞ ), thesupport of any non-zero function of CF ( V γ ) has strictly positive measure. Thses normsmay have some applications in TDA, allowing ones to define new distances on sheaves. Acknowledgement
We warmly thank Fran¸cois Petit for several fruitful and stim-ulating discussions on this subject. R -constructible sheaves andfunctions References to sheaf theory are made to [KS90].
Recall that a topological space is good if it is Hausdorff, locally compact, countable atinfinity and of finite flabby dimension.We consider a field k and a good topological space X . We denote by D b ( k X ) thebounded derived category of sheaves of k -modules on X and simply call an object ofthis category “a sheaf”. We shall freely make use of the six Grothendieck operationson sheaves and refer to [KS90] for an exposition and for notations. In particular, wedenote by ω X the dualizing complex, ω X := a ! X k { pt } where a X : X −→ pt. We have theduality functors D ′ X ( • ) = R H om ( • , k X ) , D X = R H om ( • , ω X ) . For a locally closed subset Z ⊂ X , we denote by k XZ the sheaf on X which is theconstant sheaf with stalk k on Z and 0 elsewhere. If there is no risk of confusion (inparticular when Z is closed), we simply denote it by k Z . If F is a sheaf on X , onesets F Z := F ⊗ k Z . One shall be aware that for Z closed in X , one has RΓ( X ; k ZX ) ≃ RΓ( Z ; k Z ) but this formula is no longer true for Z open in X . However, RΓ c ( X ; k ZX ) ≃ RΓ c ( Z ; k Z ) for Z locally closed in X since k ZX ≃ j ! k Z , j denoting the embedding Z ֒ → X .Now assume that X is a real manifold of class C ∞ and denote by π X : T ∗ X −→ X itscotangent bundle. To F ∈ D b ( k X ), one associates its micro-support SS( F ) (also called singular support ), a closed R + -conic subset of T ∗ X and this set is co-isotropic (in asense that we do not recall here). See [KS90, Th. 6.5.4].4 ernels Given topological spaces X i ( i = 1 , ,
3) we set X ij = X i × X j , X = X × X × X .We denote by q i : X ij −→ X i , q i : X −→ X i and by q ij : X −→ X ij the projections. X q } } ③③③③③③③③ q ! ! ❉❉❉❉❉❉❉❉ X q | | ①①①①①①①① q " " ❋❋❋❋❋❋❋❋ q (cid:15) (cid:15) X X X X X (1.1)For A ⊂ X and B ⊂ X one sets A × B = A × X B = q − A ∩ q − B, A ◦ B = q ( A × B ) . (1.2)For good topological spaces X i ’s as above, one often calls an object K ij ∈ D b ( k X ij ) a kernel . One defines as usual the composition of kernels K ◦ K := R q ( q − K ⊗ q − K ) . (1.3)If there is no risk of confusion, we write ◦ instead of ◦ .It is easily checked, and well-known, that the convolution is associative, namelygiven three kernels K ij ∈ D b ( k X ij ), i = 1 , , j = i + 1 one has an isomorphism( K ◦ K ) ◦ K ≃ K ◦ ( K ◦ K ) , (1.4)this isomorphism satisfying natural compatibility conditions that we shall not makehere explicit.Of course, this construction applies in the particular cases where X i = pt for some i . For example, if K ∈ D b ( k X × Y ) and F ∈ D b ( k X ), one usually sets Φ K ( F ) = F ◦ K .Hence Φ K ( F ) = F ◦ K = R q ( q − F ⊗ K ) . (1.5)We shall also use the right adjoint of the functor Φ K ( · ), namely the functor Ψ K ( · )(see [KS90, § G ∈ D b ( k Y ) by:Ψ K ( G ) = R q ∗ R H om ( K, q !2 G ) . (1.6)The next result is elementary. Denote by K v the image of K by the map X × Y ∼−→ Y × X , ( x, y ) ( y, x ). Lemma 1.1.
Let f : X −→ Y , F ∈ D b ( k X ) and G ∈ D b ( k Y ) . Set for short K f = k Γ f .Then f − G ≃ K f ◦ G = Φ K vf G, R f ∗ F ≃ R q ∗ R H om ( K f , q !1 F ) = Ψ K vf F, R f ! F ≃ F ◦ K f = Φ K f F, f ! G ≃ R q ∗ R H om ( K f , q !2 G ) = Ψ K f ( G ) . Proof.
The first and third isomorphisms are obvious (identify X with Γ f ). The twoothers follow by adjunction. 5 .2 Constructible sheaves From now on and unless otherwise specified, we work on real analytic manifolds. How-ever, many results, if not all, extend to the case of subanalytic spaces, for the definitionof which we refer to [KS16, § Subanalytic subsets
We shall not review here the history of subanalytic geometry, which takes its originin the work of Lojasiewicz, simply mentioning the names of Gabrielov and Hironaka.References are made to [BM88].Let X be a real analytic manifold. Denote by S X the family of subanalytic subsetsof X . Then S X contains the family of semi-analytic subsets (those locally definedby analytic inequalities) and is stable by closure, complement, finite unions and finiteintersections. If f : X −→ Y is subanalytic, then f − S Y ⊂ S X . If A ∈ S X and f isproper on A , then f ( A ) ∈ S Y .Moreover, to be subanalytic in X is a local property on X . More precisely, given X = S a ∈ A U a an open covering, a subset Z ⊂ X is subanalytic in X if and only if Z ∩ U a is subanalytic in U a for all a ∈ A .Note that if Z is a locally closed subanalytic subset of X , then there exist an openset U and a closed subset S both subanalytic in X such that Z = U ∩ S . Indeed, set Y = Z \ Z . Then Y is closed since Z is locally closed. Choose S = Z and U = X \ Y .A subanalytic stratification of X is a locally finite stratification X = F a ∈ A X a whereeach X a is a smooth locally closed real analytic submanifold of X , subanalytic in X . Constructible sheaves
A sheaf F ∈ D b ( k X ) is weakly R -constructible if there exists a subanalytic stratification X = F a ∈ A X a such that for all j ∈ Z , H j ( F ) | X a is locally constant. If moreover, theselocally constant sheaves are of of finite rank, then F is R -constructible. By the resultsof [KS90, Ch. VIII], F is weakly R -constructible if and only if SS( F ) is contained in aclosed conic subanalytic Lagrangian subvariety of T ∗ X and this implies that SS( F ) isequal to such a Lagrangian subset.One denotes by D b R c ( k X ) the full triangulated subcategory of D b ( k X ) consisting of R -constructible sheaves. The subcategories of constructible sheaves are stable by thesix Grothendieck operations with the exception of direct images which should be properon the supports of the constructible sheaves. Euler-Poincar´e indices
Let Z be a locally closed relatively compact subanalytic subset of X . One defines theEuler-Poincar´e indices of Z by χ ( Z ) = χ (RΓ( Z ; k Z )) , χ c ( Z ) = χ (RΓ c ( Z ; k Z )) . (1.7) 6f course, when Z is compact, these two indices coincide. Denote by j Z : Z ֒ → X theembedding and by a Z the map Z −→ pt. One has χ ( Z ) = χ (R a Z ∗ k Z ) = χ (RΓ( X ; R j Z ∗ k Z )) , χ c ( Z ) = χ (R a Z ! k Z ) = χ (RΓ c ( X ; k XZ )) . In this section, we recall without proofs the main constructions and results on con-structible functions. References are made to [Sch91] and [KS90, § §
3, we assume that the field k has characteristic zero . FunctionsDefinition 1.2.
A function ϕ : X −→ Z is constructible if:(i) for all m ∈ Z , ϕ − ( m ) is subanalytic in X ,(ii) the family { ϕ − ( m ) } m ∈ Z is locally finite. Notation 1.3.
For a locally closed subanalytic subset S ⊂ X , we denote by S thecharacteristic function of S (with values 1 on S and 0 elsewhere). For a ∈ X we alsoset δ a = { a } .The next result is well-known. Note that the implication (b) ⇒ (d) follows from thetriangulation theorem for compact subanalytic subsets. Lemma 1.4.
Let ϕ be a Z -valued function on X . The conditions below are equivalent. (a) ϕ is constructible, (b) there exist a locally finite family of subanalytic locally closed subsets { Z a } a ∈ A and c a ∈ Z such that ϕ = P a c a Z a , (c) there exist a subanalytic stratification { Z a } a ∈ A and c a ∈ Z such that ϕ = P a c a Z a , (d) same as (b) assuming moreover each Z a compact and contractible. Notation 1.5.
One denotes by CF ( X ) the group of constructible functions on X andby CF X the presheaf U CF ( U ). Proposition 1.6.
The presheaf CF X is a sheaf on X .Proof. (i) Clearly, the presheaf U CF ( U ) is separated.(ii) Let X = S a ∈ A U a be an open covering of X and let ϕ be a Z -valued fonction on X such that ϕ | U a is constructible on U a . For m ∈ Z , set Z m := ϕ − ( m ) and Z m,a = Z m ∩ U a .Each Z m,a is subanalytic in U a , which implies that Z m is subanalytic in X . Moreover,the family { Z m,a } m being locally finite in U a , the family { Z m } m is locally finite in X .Hence, ϕ is constructible on X . The same argument holds when replacing X with anopen subset U ⊂ X . 7ecall an important theorem (see [KS90, Th. 9.7.1]) which clarifies the notion ofconstructible function. First, denote by χ loc the local Euler-Poincar´e index: χ loc : Ob(D b R c ( k X )) −→ CF ( X ) , χ loc ( F )( x ) = X i ( − i dim H i ( F x ) . (1.8)Denote by K ( C ) the Grothendieck group of either an abelian or a triangulated category C , and recall that if C is abelian then K ( C ) ∼−→ K (D b ( C )). In the sequel, we set forshort K R c ( k X ) := K (D b R c ( k X )) . The tensor product on D b R c ( k X ) defines a ring structure on K R c ( k X ), with unit theimage of the constant sheaf k X . Theorem 1.7.
Let X be a real analytic manifold. Then the map χ loc defines an isomor-phism of commutative unital algebras ( we keep the same notation ) χ loc : K R c ( k X ) ∼−→ CF ( X ) . We have the general “principle”:
The operations on constructible functions are the image by the local Euler-Poincar´e index χ loc of the corresponding operations on constructible sheaves. Internal operations
The sum on CF ( X ) is the image by χ loc of the direct sum for sheaves, the unit X is theimage of the constant sheaf k X , the map ϕ
7→ − ϕ corresponds to the shift F F [+1]and the usual product on CF ( X ) is the image of the tensor product. External product
For two real analytic manifolds X and Y , one defines the morphism ⊠ : CF X ⊠ CF Y −→ CF X × Y , ( ϕ ⊠ ψ )( x, y ) = ϕ ( x ) ψ ( y ) . Inverse image or composition
Let f : X −→ Y be a morphism of real analytic manifolds. One defines the inverse imagemorphism f ∗ : f − CF Y −→ CF X , ( f ∗ ψ )( y ) = ψ ( f ( y )) for ψ ∈ CF ( Y ) . Inverse images are functorial, that is, if f : X −→ Y and g : Y −→ Z are morphismsof manifolds, then: g ∗ ◦ f ∗ = ( f ◦ g ) ∗ . irect image or integral Recall first that, if K is a subanalytic compact subset of X , then the Euler-Poincar´eindex χ ( K ) is defined by χ ( K ) = P j ( − j b j ( K ) where b j ( K ) = dim Q H j ( K ; Q K ). Inparticular, if K is contractible, then χ ( K ) = 1. One sets Z X K = χ ( K ) . (1.9)If ϕ has compact support, one may assume that the sum in Lemma 1.4 (d) is finite,and one checks that the integer P i c i depends only on ϕ , not on its decomposition.One sets: Z X ϕ = X i c i . In particular, if Z is locally closed relatively compact and subanalytic in X , then(see (1.7)): Z Z = χ c ( Z ) . (1.10)One shall be aware that the integral is not positive, that is ϕ ≥ Z X ϕ ≥ . For example, take X = R and ϕ = ( − , . Hence, ϕ ≥ R R ϕ = − f : X −→ Y be a morphism of real analytic manifolds. One defines the directimage morphism Z X : f ! CF X −→ CF Y , (cid:0)Z f ϕ (cid:1) ( x ) = Z X f − ( x ) · ϕ. (1.11)Recall that a section of f ! CF Y on an open subset U ⊂ X is a section of CF X on f − ( U )such that f is proper on its support. Hence the integral makes sense.Direct images are functorial, that is, if f : X −→ Y and g : Y −→ Z are morphisms ofmanifolds, then: Z g ◦ Z f = Z g ◦ f . The next lemma shows that, in some sense, the integral of ϕ ∈ CF ( X ) does notdepend of X . Lemma 1.8.
Let f : X −→ Y be a morphism of real analytic manifolds and let S ⊂ X and Z ⊂ Y be two closed subanalytic subsets. Assume that f induces a topologicalisomorphism f | S : S ∼−→ Z . Then for ψ ∈ CF ( Y ) such that supp( ψ ) ∩ Z is compact,one has Z X S · f ∗ ψ = Z Y Z · ψ. roof. One may assume that ψ = A for A a closed subanalytic subset of Y . Then Z Y Z · A = Z Y Z ∩ A = χ ( Z ∩ A )= χ ( f | − S ( Z ∩ A )) = Z X f | − S ( Z ∩ A ) = Z X S · f − A . Duality On X , the dual of a constructible functions is the image by χ loc of the duality functorD X for sheaves. One defines the dual of a constructible function ϕ on X as follows. Let x ∈ X , and choose a local chart in a neighborhood of x . Let B ε ( x ) denote the openball with center x and radius ε > X : CF X −→ CF X , (D X ϕ )( x ) = Z X ϕ · B ε ( x ) . (1.12)The integral R X ϕ · B ε ( x ) neither depends on the local chart nor on ε , for 0 < ε ≤ ε ( x ),for some ε ( x ) > x .Duality is an involution, that is,D X ◦ D X = id X . Moreover, duality commutes with integration:D Y ( Z f ϕ ) = Z f D X ( ϕ ) . (1.13)By mimicking a classical formula for constructible sheaves, one sets hom ( ϕ, ψ ) := D X (D X ψ · ϕ ) . (1.14) Example 1.9.
Let Z be a closed subanalytic subset of X and assume that Z is a C -manifold of dimension d with boundary ∂Z . Set A = Z \ ∂Z . Hence, locally on X , Z ⊂ X is topologically isomorphic to U ⊂ R n where U is a convex open subset of R d ⊂ R n and A ≃ U . We thus haveD X Z = ( − d A (1.15)Moreover Z X ∂Z = Z X Z − Z X A = (1 − ( − d ) Z X Z . When Z is a closed convex polyhedron, one recovers the classical Euler formula.10 ther operations In fact, most (if not all) operations on constructible sheaves admit a counterpart in thelanguage of constructible functions. In [KS90, Def. 9.7.8] one defines the specializa-tion ν M along a submanifold M , its Fourier-Sato transform, the microlocalization µ M (see § µhom : ν M : CF ( X ) −→ CF R + ( T M X ) , µ M : CF ( X ) −→ CF R + ( T ∗ M X ) µhom : CF ( X ) × CF ( X ) −→ CF R + ( T ∗ X ) . One can also define the micro-support of ϕ ∈ CF ( X ) by settingSS( ϕ ) = supp( µhom ( ϕ, ϕ )) . Base change formula
Consider a Cartesian square of morphisms of real analytic manifolds: X ′ f ′ / / g ′ (cid:15) (cid:15) Y ′ g (cid:15) (cid:15) X f / / Y. (1.16)Recall that the square is Cartesian means that X ′ is topologically isomorphic to thespace { ( x, y ′ ) ∈ X × Y ′ ; f ( x ) = g ( y ′ ) } . Proposition 1.10.
Consider the square (1.16) , let ϕ ∈ CF ( X ) and assume that f isproper on supp( ϕ ) . Then: g ∗ Z f ϕ = Z f ′ ( g ′∗ ϕ ) . (1.17)We shall prove a more general statement in Proposition 3.14. The projection formulaProposition 1.11.
Let f : X −→ Y , ϕ ∈ CF ( X ) , ψ ∈ CF ( Y ) and assume that f isproper on supp( ϕ ) . Then Z f ( ϕ · f ∗ ψ ) = ψ Z f ϕ. (1.18)We shall prove a more general statement in Proposition 3.15.11 omposition of kernels Recall Diagram (1.1). Let λ ∈ CF ( X ) and λ ∈ CF ( X ). Assume that theprojection q : X −→ X is proper on supp( λ ) × X supp( λ ). One defines thecomposition (also called “convolution”) of λ and λ by λ ◦ λ := Z q q ∗ λ · q ∗ λ . (1.19)When there is no risk of confusion, one writes ◦ instead of ◦ .The convolution is in particular defined when X or X is a point. For example, let X and Y be two real analytic manifolds and let λ ∈ CF ( X × Y ), ϕ ∈ CF ( X ), ψ ∈ CF ( Y ).One gets (the integrals being defined under suitable properness hypotheses) λ ◦ ψ = Z q λ · q ∗ ψ, ϕ ◦ λ = Z q λ · q ∗ ϕ. Note that the convolution is associative under suitable hypotheses. In order to get aprecise statement, we consider four (non empty) manifolds X i , i = 1 , . . . , q ji instead of q i or q kij instead of q ij and q klij for the projections q ji : X ij −→ X i , q kij : X ikj −→ X ij , q klij : X −→ X ij . For example, we have the maps X q ①①① | | ①①① q (cid:15) (cid:15) q ❋❋❋ " " ❋❋❋ X q ①①① | | ①①① q (cid:15) (cid:15) q ❋❋❋ " " ❋❋❋ X X X X X (1.20)and the Cartesian squares X q PPPPP ' ' PPPPP q / / q (cid:15) (cid:15) X q (cid:15) (cid:15) X q PPPPP ' ' PPPPP q / / q (cid:15) (cid:15) X q (cid:15) (cid:15) X q / / X X q / / X . (1.21)For i = 1 , , j = i + 1, let λ ij ∈ CF ( X ij ). Set Z ij = supp( λ ij ) and set Z = Z × Z , Z = Z × Z , Z = Z × Z ⊂ X . Consider the assertions: q is proper on Z × Z = Z , (1.22) q is proper on q ( Z ) × Z = q Z , (1.23) q is proper on Z × Z = Z , (1.24) q is proper on Z × q ( Z ) = q Z , (1.25) q is proper on Z . (1.26) 12ote that (1.22) and (1.24) are respectively equivalent to q is proper on ( q ) − Z ∩ ( q ) − Z = ( q ) − Z , (1.27) q is proper on ( q ) − Z ∩ ( q ) − Z = ( q ) − Z . (1.28)Also note that (1.24)–(1.25) or (1.22)–(1.23) implies (1.26). Theorem 1.12.
Assume (1.22) , (1.23) , (1.24) and (1.25) . Then ( λ ◦ λ ) ◦ λ = λ ◦ ( λ ◦ λ ) ∈ CF ( X ) . Proof.
First, note that ( λ ◦ λ ) ◦ λ is well-defined thanks to hypotheses (1.22)–(1.23) and λ ◦ ( λ ◦ λ ) is well-defined thanks to hypotheses (1.24)–(1.25)Consider the Cartesian square (1.21) on the right. Using hypothesis (1.24), we mayapply the base change formula and we get q ∗ Z q q ∗ λ · q ∗ λ = Z q q ∗ (cid:0) q ∗ λ · q ∗ λ (cid:1) = Z q q ∗ λ · q ∗ λ . (1.29)We have λ ◦ ( λ ◦ λ ) = Z q ( q ∗ λ · q ∗ Z q q ∗ λ · q ∗ λ ) = Z q ( q ∗ λ Z q q ∗ λ · q ∗ λ )= Z q Z q q ∗ λ · q ∗ λ · q ∗ λ = Z q q ∗ λ · q ∗ λ · q ∗ λ . Here, we have used (1.29) for the 2nd equality and the projection formula for the 3rdequality using (1.28).By the same argument, using hypothesis (1.22) and (1.27), one gets( λ ◦ λ ) ◦ λ = Z q q ∗ λ · q ∗ λ · q ∗ λ . In practice, we will encounter the situation in which either X of X is reduced toa point. Then we change our notations for simplicity and consider the diagram X q ①①①① | | ①①①① q (cid:15) (cid:15) q ❋❋❋❋ " " ❋❋❋❋ X q ③③③ } } ③③③ q ❋❋❋❋ " " ❋❋❋❋ X p ❘❘❘❘❘❘❘❘ ( ( ❘❘❘❘❘❘❘❘ p ❧❧❧❧❧❧❧❧ v v ❧❧❧❧❧❧❧❧ X r ①①①① | | ①①①① r ❉❉❉ ! ! ❉❉❉ X X X (1.30)Let λ ∈ CF ( X ), λ ∈ CF ( X ) and let Z ij = supp( λ ij ). Consider one of thehypothesis q is proper on Z and r is proper on Z , (1.31) q is proper on Z and r is proper on Z . (1.32) 13 orollary 1.13. (a) Assume (1.31) and let ϕ ∈ CF ( X ) . Then ( ϕ ◦ λ ) ◦ λ = ϕ ◦ ( λ ◦ λ ) ∈ CF ( X ) . (b) Assume (1.32) and let ψ ∈ CF ( X ) . Then λ ◦ ( λ ◦ ψ ) = ( λ ◦ λ ) ◦ ψ ∈ CF ( X ) . Proof.
By Theorem 1.12, it is enough to check that each of the hypothesis (1.31)or (1.32) implies (1.22), (1.23), (1.24) and (1.25), which is clear.The proof of Theorem 1.12 in case X = pt may be visualized as follows. X X q ✺✺✺✺✺✺ (cid:26) (cid:26) ✺✺✺✺✺✺ q ✠✠✠✠✠✠ (cid:4) (cid:4) ✠✠✠✠✠✠ X q (cid:15) (cid:15) X q ❄❄ (cid:31) (cid:31) ❄❄ q ✂✂ (cid:1) (cid:1) ✂✂ X q ❁❁ (cid:29) (cid:29) ❁❁ q ⑧⑧ (cid:127) (cid:127) ⑧⑧ X X q ❃❃ (cid:30) (cid:30) ❃❃ q (cid:0)(cid:0) (cid:0) (cid:0) (cid:0)(cid:0) X X X X X X X Z q λ · ( q ∗ Z q λ · q ∗ ϕ ) = Z q q ∗ λ · q ∗ λ · q ∗ ϕ = Z q ( Z q q ∗ λ · q ∗ λ ) · q ∗ ϕ. Consider the situation of Diagram (1.30). Assume to be given two locally closed sub-analytic subsets S ⊂ X , S ⊂ X . We set, for ϕ ∈ CF ( X ) R S ( ϕ ) = ϕ ◦ S = Z q q ∗ ϕ · S . Consider the hypothesis q is proper on S and r is proper on S .(1.33)Set λ := S ◦ S ∈ CF ( X ) . (1.34)Applying Corollary 1.13 (a), we get that λ is well defined and moreover R S ◦ R S ( ϕ ) = ϕ ◦ λ. (1.35) 14ow we assume that X = X and we change our notations, setting X = X = X, X = Y. For ( x, x ′ ) ∈ X × X , let S ( x, x ′ ) = { y ∈ Y ; ( x, y ) ∈ S , ( y, x ′ ) ∈ S } = ( S × Y S ) ∩ q − ( x, x ′ ) . (1.36)Then λ ( x, x ′ ) = Z q S × Y S · { q − ( x,x ′ ) } = Z Y S ( x,x ′ ) . (1.37)We now consider the hypothesis there exists a = b ∈ Z such that, for ( x, x ′ ) ∈ X × X : λ ( x, x ′ ) = (cid:26) a if x = x ′ ,b if x = x ′ . (1.38)Writing λ ( x, x ′ ) = ( b − a ) ∆ + a X × X , we get: Corollary 1.14 ([Sch95, Th. 3.1]) . Assume (1.33) and (1.38) . Let ϕ ∈ CF ( X ) . Then: R S ◦ R S ( ϕ ) = ( b − a ) ϕ + a Z X ϕ. Here, a R X ϕ ∈ Z is identified with the constant function ( a R X ϕ ) · X .Let W be a real ( n + 1)-dimensional vector space (with n >
0) and denote by F n +1 ( p, q ), with 0 ≤ p ≤ q ≤ n + 1, the set of pairs { ( l, h ) } of linear subspaces of W with l ⊂ h and dim l = p , dim h = q . One sets F n +1 ( p, p ) = F n +1 ( p ) and denotes as usualby q and q the two projections defined on F n +1 ( p ) × F n +1 ( q ). Then F n +1 ( p, q ) is a realcompact submanifold of F n +1 ( p ) × F n +1 ( q ), called the incidence relation. We denote by F n +1 ( q, p ) its image by the map F n +1 ( p ) × F n +1 ( q ) −→ F n +1 ( q ) × F n +1 ( p ) , ( x, y ) ( y, x ).In the sequel, we set X = F n +1 ( p ) , Y = F n +1 ( q ) , S = F n +1 ( p, q ) ⊂ X × Y, S ′ = F n +1 ( q, p ) ⊂ Y × X. Now we shall assume p = 1 and q >
1. Recall that F n +1 (1) = P n , the n -dimensionalreal projective space.In order to apply Corollary 1.14, it is enough to calculate λ ( x, x ′ ) given by (1.37)and (1.36) with S = S and S = S ′ . Set µ n +1 ( q ) = χ ( F n +1 ( q )) . Proposition 1.15.
Let ϕ ∈ CF ( P n ) . Then: R S ′ ◦ R S ( ϕ ) = ( µ n ( q − − µ n − ( q − ϕ + µ n − ( q − Z P n ϕ. Proof.
Let us represent x and x ′ by lines in W and y ∈ F n +1 ( q ) by a q -dimensionallinear subspace. Then the set S ( x, x ′ ) is the set of q -dimensional linear subspaces of W containing both the line x and the line x ′ . This set is isomorphic to F n − ( q −
2) if x = x ′ and and to F n ( q −
1) if x = x ′ .Of course, this formula is interesting only when µ n ( q − = µ n − ( q − .5 The Radon transform We now treat the case p = 1, q = n . See [Sch95] for other situations such as the X-raystransform.We have F n +1 (1) = P n , the n -dimensional projective space and F n +1 ( n ) = P ∗ n , thedual projective space.With the preceding notations, the incidence relation S is given by S = F n +1 (1 , n ) = { ( x, y ) ∈ P n × P ∗ n ; h x, y i = 0 } . The Radon transform of ϕ ∈ CF ( P n ), an element of CF ( P ∗ n ), is defined by R S ( ϕ ) = Z P n S · q ∗ ϕ = ϕ ◦ S . (1.39)For y ∈ P ∗ n , we shall denote by h y its image in P n by the incidence relation: h y = { x ∈ P n , h x, y i = 0 } . Therefore, ( R S ( ϕ ))( y ) = Z P n ϕ · h y . Recall that the Euler-Poincar´e index of P n is given by the formula: χ ( P n ) = (cid:26) n is even,0 if n is odd.(1.40)Applying Proposition 1.15 together with (1.40), we get: Corollary 1.16.
Let ϕ ∈ CF ( P n ) . Then: R S ′ ◦ R S ( ϕ ) = (cid:26) ϕ if n is odd, − ϕ + R P n ϕ if n is even and n > . Now assume dim V = 3 and let us calculate the Radon transform of the characteristicfunction K of a compact subanalytic subset K of V (see (1.9)). First, consider acompact subanalytic subset L of a two dimensional affine vector space W . By Poincar´e’sduality, there is an isomorphism H L ( W ; Q W ) ≃ H ( L ; Q L ) and moreover there is a shortexact sequence:0 −→ H ( W ; Q W ) −→ H ( W \ L ; Q W ) −→ H L ( W ; Q W ) −→ , from which one deduces that: b ( L ) = b ( W \ L ) − . Note that b ( W \ L ) is the number of connected components of W \ L , hence b ( L ) isthe “number of holes” of the compact set L . We may sumarize: Corollary 1.17.
The value at y ∈ P ∗ of the Radon transform of K is the number ofconnected components of K ∩ h y minus the number of its holes. The inversion formula of the Radon transform tells us how to reconstruct the set K from the knowledge of the number of connected components and holes of all its affineslices. 16 Constructible sheaves up to infinity
In order to define subanalytic subsets up to infinity, we introduce the category ofb-analytic manifolds, inspired by (but rather different from) that of bordered spaceof [DK16]. As mention in the introduction, the notion of being subanalytic up toinfinity is a particular case of that of definable set, well-known from the specialists(see [VdD98, VdDM96]), and constructible sheaves in this framework have already beendefined in [Sch03]. However, our approach is direct and quite different since it is basedon the notion of micro-support.
Definition 2.1.
The category of b-analytic manifolds is the category defined as follows.(a) An object X ∞ is a pair ( X, b X ) with X ⊂ b X an open embedding of real analyticmanifolds such that X is relatively compact and subanalytic in b X ,(b) a morphisms f : X ∞ = ( X, b X ) −→ Y ∞ = ( Y, b Y ) of b-analytic manifolds is a mor-phism of real analytic manifolds f : X −→ Y such that the graph Γ f of f in X × Y is subanalytic in b X × b Y .(c) The composition ( X, b X ) f −→ ( Y, b Y ) g −→ ( Z, b Z ) is given by g ◦ f : X −→ Z and theidentity id ( X, b X ) is given by id X (see Lemma 2.3 below).If there is no risk of confusion, we shall often denote by j X : X ֒ → b X the open embed-ding. Remark 2.2.
Instead of requiring b X to be a smooth real analytic manifold and X relatively compact in it, one could ask b X to be a compact subanalytic space in thesense of [KS16, § X, X )nor ( b X, b X ) are b-analytic manifolds. However, if X is compact, ( X, X ) is a b-analyticmanifold.
Lemma 2.3. (a)
The identity id ( X, b X ) is a morphism of b-analytic manifolds. (b) Let f : ( X, b X ) −→ ( Y, b Y ) and g : ( Y, b Y ) −→ ( Z, b Z ) be morphisms of b-analytic mani-folds. Then the composition g ◦ f is a morphism of b-analytic manifolds.Proof. (a) Since X is subanalytic in b X , X × X is subanalytic in b X × b X , and ∆ X = X × X ∩ ∆ b X is subanalytic in b X × b X .(b) By the hypothesis, Γ g is subanalytic and relatively compact in b Y × b Z and Γ f issubanalytic and relatively compact in b X × b Y . It follows that Γ f × b Y Γ g is subanalyticand relatively compact in b X × b Y × b Z . Therefore, its projection Γ f ◦ Γ g is subanalyticin b X × b Z . Since Γ f ◦ Γ g = Γ g ◦ f , the proof is complete.17 efinition 2.4. Let X ∞ = ( X, b X ) be a b-analytic manifold and let Z be a subset of X . We say that Z is subanalytic in X ∞ if Z is subanalytic in b X . If there is no risk ofconfusion (that is, b X has been already defined) we also say that Z is subanalytic up toinfinity.Note that the family of subsets subanalytic up to infinity inherits all propertiesof the family of subanalytic subsets. In particular, it is stable by finite union, finiteintersection, interior, closure, difference and X itself is subanalytic up to infinity. Proposition 2.5.
Let X i ∞ = ( X i , b X i ) ( i = 1 , , be three b-analytic manifolds. (a) Setting b X = b X × b X , the pair ( X , b X ) is a b-analytic manifold. Moreover,if S and S are two subsets of X and X respectively, subanalytic up to infinity,then S × S is subanalytic up to infinity in X . (b) Let S and S be subanalytic up to infinity in X and X respectively, then S ◦ X S is subanalytic up to infinity in X . (c) In particular, let f : X ∞ −→ Y ∞ be a morphism of b-analytic manifolds. If Z ⊂ Y is subanalytic up to infinity, then f − ( Z ) is subanalytic up to infinity in X and if S ⊂ X is subanalytic up to infinity, then f ( S ) is subanalytic up to infinity in Y . We shall denote by ( X × Y ) ∞ the b-analytic manifold ( X × Y, b X × b Y ). Proof. (a) is obvious.(b) S × X S is subanalytic and relatively compact in b X . Therefore, its image by q is subanalytic and relatively compact in b X .(c) By the hypothesis, Γ f is subanalytic up to infinity in b X × b Y . By (b), f − ( Z ) = Γ f ◦ Y Z is subanalytic up to infinity in X and f ( S ) = S ◦ X Γ f is subanalytic up to infinity in Y . Although this paper is mainly concerned with constructible functions, we briefly exposethe theory of constructible sheaves up to infinity since it does not seem to appearelsewhere.In this section, we consider b-analytic manifolds X ∞ = ( X, b X ) and Y ∞ = ( Y, b Y ). Lemma 2.6 (See [KS20, Th.2.2]) . Let F ∈ D b R c ( k X ) . The following conditions areequivalent. (a) The micro-support
SS( F ) is contained in a closed R + -conic Lagrangian subset Λ of T ∗ X which is subanalytic in T ∗ b X , (b) j X ! F ∈ D b R c ( k b X ) , (c) R j X ∗ F ∈ D b R c ( k b X ) . roof. For the reader’s convenience, we recall the proof of loc. cit.(i) The implication (b) or (c) ⇒ (a) is obvious.(ii) (a) ⇒ (b). The set Λ is a locally closed subanalytic subset of T ∗ b X and is isotropic.By [KS90, Cor. 8.3.22], there exists a µ -stratification b X = F a ∈ A Y a such that Λ ⊂ F a ∈ A T ∗ Y a b X .Set X a = X ∩ Y a . Then X = F a ∈ A X a is a µ -stratification and one can applyloc. cit. Prop. 8.4.1. Hence, for each a ∈ A , F | X a is locally constant of finite rank.Hence ( j X ! F ) | X a as well as ( j X ! F ) b X \ X ≃ j X ! F ∈ D b R c ( k b X ).(iii) (a) ⇒ (c). Set G = j X ! F . Then G ∈ D b R c ( k b X ) by (ii) and so does R j ∗ F ≃ R H om ( k X , G ) (apply [KS90, Prop. 8.4.10]). Definition 2.7.
Let F ∈ D b R c ( k X ). One says that F is constructible up to infinityif it satisfies one of the equivalent conditions in Lemma 2.6. We denote by D b R c ( k X ∞ )the full triangulated subcategory of D b R c ( k X ) consisting of sheaves constructible up toinfinity.It follows that if F ∈ D b R c ( k b X ), then j − X F ∈ D b R c ( k X ∞ ). Proposition 2.8.
Let X ∞ and Y ∞ be two b-analytic manifolds. (i) Let F ∈ D b R c ( k X ∞ ) and G ∈ D b R c ( k Y ∞ ) . Then F ⊠ G ∈ D b R c ( k ( X × Y ) ∞ ) . (ii) Let F and F belong to D b R c ( k X ∞ ) . Then F ⊗ F and R H om ( F , F ) belong to D b R c ( k X ∞ ) . In particular, the dual D X F of F ∈ D b R c ( k X ∞ ) belongs to D b R c ( k X ∞ ) .Proof. (i) One has SS( F ⊠ G ) ⊂ Λ × Λ with Λ i an R + -conic Lagrangian subset of T ∗ X i subanalytic in T ∗ b X i ( i = 1 , × Λ has the same property in T ∗ ( b X × b X ).(One could also use the fact that j X × Y ! ( F ⊠ G ) ≃ j X ! F ⊠ j Y ! G .)(ii) follows from [KS90, Cor. 8.3.18 (i) and 6.4.5]. Proposition 2.9.
Let f : X ∞ −→ Y ∞ be a morphism of b-analytic manifolds. (i) Let G ∈ D b R c ( k Y ∞ ) . Then f − ( G ) and f ! G belong to D b R c ( k X ∞ ) . (ii) Let F ∈ D b R c ( k X ∞ ) . Then R f ! F and R f ∗ F belong to D b R c ( k Y ∞ ) .Proof. By Proposition 2.8 and Lemma 1.1, we are reduced to prove that(a) if H ∈ D b R c ( k ( X × Y ) ∞ ), then R q H and R q ∗ H belong to D b R c ( k X ∞ ),(b) if F ∈ D b R c ( k X ∞ ), then q − F and q !1 F belong to D b R c ( k ( X × Y ) ∞ ).The assertion (b) follows from Proposition 2.8 since q − F ≃ F ⊠ k Y and q !1 F ≃ F ⊠ ω Y .To prove (a), denote by ˆ q the projection b X × b Y −→ b X . Then R q H ≃ j − X R ˆ q R j X × Y ! H and similarly R q ∗ H ≃ j − X R ˆ q ∗ R j X × Y ∗ H . Corollary 2.10.
Let f : X ∞ −→ Y ∞ be a morphism of b-analytic manifolds and let F ∈ D b R c ( k X ∞ ) and G ∈ D b R c ( k Y ∞ ) . Then R f ∗ F ≃ D Y R f ! D X F and f ! G ≃ D X f − D Y G .Proof. (i) Both D X F and R f ! D X F are R -constructible. Then apply [KS90, Exe. VIII.3].(ii) Similarly, both D Y G and f − D Y G are R -constructible. Then apply loc. cit.19 .3 Convolution In this subsection, we consider a real n -dimensional vector space V . We consider itsprojective compactification P ( V ⊕ R ) of V . The pair ( V , P ( V )) is a b-analytic manifoldthat we denote by V ∞ , or simply V if there is no risk of confusion.We denote by s the addition map. s : V × V −→ V , ( x, y ) x + y, and by a : V −→ V the antipodal map x
7→ − x . For A and B be two subsets of V , wedenote by A a te image of A by ( · ) a and by A + B the image of A × B by s . We also set˙ V = V \ { } . Lemma 2.11.
Let A and B be two subsets of V both subanalytic up to infinity. Then A a and A + B are subanalytic up to infinity.Proof. The map ( · ) a and the map s are morphisms of b-analytic manifolds. Hence, theresults follow from Proposition 2.5 (c).A subset A of V is called a cone if R > · A = A . Lemma 2.12.
Let Z ⊂ ˙ V be a subanalytic cone. Then Z is subanalytic in V ∞ .Proof. (a) The set Z is subanalytic in V by [KS90, Prop. 8.3.8 (i)].(b) Choose a subanalytic norm k · k on V and consider the real analytic isomorphism f : ˙ V −→ ˙ V , f ( x ) = x/ k x k . The map f defines an automorphism of the b-analyticmanifold V ∞ . It is thus enough to check that f ( Z ) is subanalytic in V . Since this setis a subanalytic cone, this follows from (a).We define the convolution and the non-proper convolution as follows. For F, G ∈ D b R c ( k V ∞ ), we set F ⋆ G := R s ! ( F ⊠ G ) , F np ⋆ G := R s ∗ ( F ⊠ G ) . By Proposition 2.9, both
F ⋆ G and F np ⋆ G belong to D b ( V ∞ ). Note that F ⋆ G = G ⋆ F, F np ⋆ G = G np ⋆ F. Proposition 2.13.
Let F i ∈ D b ( V ∞ ) , i = 1 , , . Then F ⋆ F = D X (D X F ⋆ D X F ) , ( F ⋆ F ) np ⋆ F = F ⋆ ( F ⋆ F ) . In other words, non proper convolution is commutative and associative in the cate-gory of constructible sheaves up to infinity.
Proof. (i) The first equality follows from Corollary 2.10.(ii) follows from (i).
Remark 2.14.
Proposition 2.13 is remarkable since, in general, the operation np ⋆ is notassociative. 20 .4 γ -constructible sheaves References to the γ -topology and its links with sheaf theory are made to [KS90, KS18].Here again, we consider a real n -dimensional vector space V . γ -subanalytic subsets Recall that a subset A of V is called a cone if R > A = A . A convex cone A is proper if A ∩ A a = { } .We consider a cone γ ⊂ V and we assume: γ is a closed convex proper subanalytic cone with non-empty interior. (2.1)Note that such a cone γ is subanalytic up to infinity by Lemma 2.12.The family of γ -invariant open subsets of V defines a topology, which is called the γ -topology on V . One denotes by V γ the space V endowed with the γ -topology and onedenotes by(2.2) ϕ γ : V −→ V γ the continuous map associated with the identity. Note that the closed sets for thistopology are the γ a -invariant closed subsets of V and that a subset is γ -locally closedif it is the intersection of a γ -closed subset and a γ -open subset. Lemma 2.15.
Let A ⊂ V . The conditions below are equivalent: (a) A = ( U + γ ) ∩ ( U + γ a ) with U open and subanalytic up to infinity. (b) A is the intersection of a γ -closed subset S and a γ -open subset U , both S and U being subanalytic up to infinity. (c) A is γ -locally closed and A is subanalytic up to infinity.Proof. (a) ⇒ (b). It remains to check that U being subanalytic up to infinity, U + γ a is subanalytic up to infinity. It is enough to check that U + γ a is subanalytic up toinfinity which follows from Lemma 2.11.(b) ⇒ (c) is obvious.(c) ⇒ (a). By [KS18, Prop. 3.4], we may write A = ( U + γ ) ∩ ( U + γ a ) with U = Int( A ).Therefore, U is subanalytic up to infinity. Definition 2.16.
Let A be a subset of V . One says that A is subanalytic γ -locallyclosed if A satisfies one of the equivalent conditions in Lemma 2.15. γ -constructible sheaves Let γ be a cone satisfying (2.1). Consider the full triangulated subcategories of thecategory D b ( k V ): ( D b γ ◦ a ( k V ) := { F ∈ D b ( k V ); SS( F ) ⊂ V × γ ◦ a } , D b R c ,γ ◦ a ( k V ∞ ) := D b R c ( k V ∞ ) ∩ D b γ ◦ a ( k V ) . (2.3)We call an object of the category D b R c ,γ ◦ a ( k V ∞ ) a γ -constructible sheaf.21 heorem 2.17. Let F ∈ D b R c ,γ ◦ a ( k V ∞ ) . Then there exists a finite covering V = S a ∈ A Z a where the Z a ’s are subanalytic γ -locally closed and F | Z a is constant.Proof. There exists a stronger result. Indeed, one may find a finite stratification V = F a ∈ A Z a where the Z a ’s are in the statement, and F | Z a is constant. This is proved byEzra Miller in [Mil20]. If we make the extra-hypothesis that F is PL and the cone γ ispolyhedral, then this result is proved in [KS18, Th. 3.18]. Lemma 2.18.
The endofunctor k γ a np ⋆ of D b ( k V ) defines a projector D b R c ( k V ∞ ) −→ D b R c ,γ ◦ a ( k V ∞ ) .Proof. We know by [KS90, Prop. 5.2.3] that the functor ϕ − γ R ϕ γ ∗ : D b ( k V ) −→ D b γ ◦ a ( k V )is a projector and we know by [KS90, Pro. 3.5.4] that the two functors ϕ − γ R ϕ γ ∗ and k γ a np ⋆ are isomorphic. Moreover, the functor k γ a np ⋆ sends D b R c ( k V ∞ ) to itself by Propo-sition 2.9. A Koszul complex
Recall that dim V = n . We denote for short by I the interval [0 , n ] of the totally orderedset Z . For J ⊂ I , we denote by | J | its cardinal. For each i ∈ I , let λ i be an opensubanalytic convex cone of V such that λ i is a closed subanalytic convex proper cone.We assume [ i λ i = V ∗ \ { } . (2.4)For ∅ / ∈ J ⊂ I , set λ J = T j ∈ J λ j and consider the co-Koszul complex (the subjectbeing classical, we do not describe the differentials): K • : 0 −→ M | J | = n k λ J d −→ · · · d −→ M | J | =1 k λ J −→ k V ∗ \{ } −→ . (2.5)It is well-known that this complex is exact. Applying the functor Hom ( • , k V ) to thetruncated complex τ ≤ K • where L | J | =1 is in degree 0, we get the compelx0 −→ M | J | =1 Γ( λ J ; k V ) d −→ · · · d −→ M | J | = n Γ( λ J ; k V ) −→ V ∗ \ { } ; k V ∗ ) and hence is concentrated in degree 0 and n −
1. Itis extracted from the exact complex0 −→ k −→ M | J | =1 Γ( λ J ; k V ) d −→ · · · d −→ M | J | = n Γ( λ J ; k V ) −→ k −→ { u i } ni =0 of k with u i = id k for all i .This implies that Γ( λ J ; k V ) = 0 for all J with | J | = n and hence λ J = ∅ for such a J .22ow set γ J = λ ◦ aJ , a closed convex proper cone with non-empty interior. TheFourier-Sato transform of k λ J is isomorphic to k γ J [ − n ] and the distinguished triangle k V ∗ \{ } −→ k V ∗ −→ k { } +1 −→ gives rise to the distinguished triangle F ∧ ( k V ∗ \{ } ) −→ k { } [ − n ] −→ k V +1 −→ Therefore, we get the exact complex0 −→ k V −→ M | J | = n k γ J d −→ · · · d −→ M | J | =1 k γ J −→ k { } −→ . (2.6)This implies: Proposition 2.19.
Let F ∈ D b ( k V ) . Then F is isomorphic to the complex −→ k V np ⋆ F −→ M | J | = n k γ J np ⋆ F d −→ · · · d −→ M | J | =1 k γ J np ⋆ F −→ where L | J | =1 ( · ) is in degree . We shall apply this result in § From now on, we assume that the field k has of characteristic zero . Definition 3.1.
Let X ∞ be a b-analytic manifold. A function ϕ : X −→ Z is con-structible up to infinity if:(i) for all m ∈ Z , ϕ − ( m ) is subanalytic up to infinity,(ii) the family { ϕ − ( m ) } m ∈ Z is finite.We denote by CF ( X ∞ ) the space of constructible functions up to infinity.For any function ϕ on X , we denote by j X ! ϕ the function on b X obtained as thefunction ϕ on X extended by 0 on b X \ X . Lemma 3.2.
Let ϕ ∈ CF ( X ) . The conditions below are equivalent. (a) The function ϕ is constructible up to infinity, (b) The function j X ! ϕ belongs to CF ( b X ) . (c) There exists ψ ∈ CF ( b X ) such that ϕ = ψ | X . There exist a finite family of locally closed subsets subanalytic up to infinity { Z a } a ∈ A and c a ∈ Z such that ϕ = P a c a Z a .Proof. (a) ⇒ (b). By the hypothesis, one may write ϕ = P a c a Z a where the sum isfinite and the Z a ’s are subanalytic up to infinity. Therefore, Z a ∈ CF ( b X ) and theresult follows from Lemma 1.4.(b) ⇒ (c) is obvious.(c) ⇒ (d) and (c) ⇒ (a). By definition, for each m ∈ Z , Z m := ψ − ( m ) is subanalytic in b X and the family { Z m } m is locally finite. Therefore, Z m ∩ X is subanalytic in X and X being relatively compact, the family { X ∩ Z m } m is finite.(d) ⇒ (b) is obvious.Clearly, CF ( X ∞ ) is a subalgebra of CF ( X ).Recall Theorem 1.7 and denote now by K R c ( k X ∞ ) the Grothendieck group of thecategory D b R c ( k X ∞ ). Theorem 3.3.
The isomorphism of commutative unital algebras χ loc : K R c ( k X ) ∼−→ CF ( X ) induces an isomorphism χ loc : K R c ( k X ∞ ) ∼−→ CF ( X ∞ ) .Proof. (i) The map χ loc takes its values in CF ( X ∞ ). Indeed, for F ∈ D b R c ( k X ∞ ), χ loc ( F ) = j ∗ X ( χ loc ( j X ! F )).(ii) The map χ loc : K R c ( k X ∞ ) −→ CF ( X ∞ ) is injective by the same arguments as in theproof of [KS90, Th. 9.7.1].(iii) The map χ loc is surjective since for Z locally closed and subanalytic up to infinity, Z = χ loc ( k Z ) and k Z is constructible up to infinity. Remark 3.4.
In the sequel, it would be possible to use Theorem 3.3 in order to provevarious results on constructible functions. However, most of the time, we shall givedirect proofs.
Lemma 3.5. If ϕ ∈ CF ( X ∞ ) , then D X ϕ ∈ CF ( X ∞ ) .Proof. The result follows from Lemma 3.2 (c) since duality commutes with restrictionto an open subset.Let ϕ ∈ CF ( X ∞ ). One sets j X ∗ ϕ = D b X j X ! D X ϕ. (3.1)The next result follows from the corresponding result for sheaves. Lemma 3.6. If ϕ ∈ CF ( X ∞ ) has compact support in X , then j X ∗ ϕ = j ! ϕ . Applying Lemma 3.2 and Proposition 2.5, we get:
Proposition 3.7.
Let X ∞ and Y ∞ be two b-analytic manifolds. Let ϕ ∈ CF ( X ∞ ) and ψ ∈ CF ( Y ∞ ) . Then the function ϕ ⊠ ψ defined by ( ϕ ⊠ ψ )( x, y ) = ϕ ( x ) ψ ( y ) belongs to CF (( X × Y ) ∞ ) . (b) Let f : X ∞ −→ Y ∞ be a morphism of b-analytic manifolds and let ψ ∈ CF ( Y ∞ ) .Then the function f ∗ ψ defined by f ∗ ψ ( x ) = ψ ( f ( x )) belongs to CF ( X ∞ ) . Although we shall not use it, let us mention that one can also define the internalhom and the exceptional inverse image by the formulas hom ( ϕ, ψ ) := D X (D X ψ · ϕ ) , ϕ, ψ ∈ CF ( X ∞ ) ,f ! ψ := D X f ∗ (D Y ψ ) , ψ ∈ CF ( Y ∞ ) . (3.2)Now we study the integrals of constructible functions up to infinity. One can definetwo integrals of ϕ ∈ CF ( X ∞ ). One sets Z X ϕ := Z b X j X ! ϕ, Z np X ϕ := Z b X j X ∗ ϕ. (3.3)Recall notations (1.7). Lemma 3.8. (a)
Let Z be a locally closed subset of X subanalytic up to infinity. Then R X Z = χ c ( Z ) . (b) One has R np X ϕ = R X D X ϕ . (c) The integrals R X ϕ and R np X ϕ do not depend on the choice of b X .Proof. (a) Recall (1.10). Denote by a Z the map Z −→ pt and similarly with b X and by j Z the embedding Z ֒ → b X . Then Z X Z = χ (RΓ c ( b X ; Q b XZ )) = χ (R a b X ! R j Z ! Q Z ) = χ (R a Z ! Q Z )) = χ c ( Z ) . (b) follows from Z b X j X ∗ ϕ = Z b X D b X j X ! D X ϕ = Z b X j X ! D X ϕ where the last equality follows from (1.13) applied with Y = pt.(c) follows from (a) and (b). Example 3.9.
Let X = R . Then:(i) One has R R R = − R np R R = 1.(ii) Let U = ( −∞ , b ) with −∞ < b < ∞ . Then R R U = − R np R U = 0.(iii) Let Z = ( −∞ , b ] with −∞ < b < + ∞ . Then R R Z = 0, R np R Z = 1.(iv) Let S = [ a, b ] with −∞ < a ≤ b < + ∞ . Then R R S = R np R S = 1.(v) Let Z = [ a, b ) with −∞ < a ≤ b < + ∞ . Then R R Z = R np R Z = 0.Indeed, (i) is obvious. Let U be as in (ii). Then U is topologically isomorphic to R andwe get R R U = −
1. By the additivity of the integral, we deduce that for Z as in (iii), R R Z = 0. By Lemma 3.8, we get R np R U = 0 and by additivity, R np R Z = 1. Finally,(iv) and (v) are obvious. 25ow consider the case of a projection q : X × Y −→ X and denote by b q the projection b X × b Y −→ b X . For θ ∈ CF ( b X × b Y ) and x ∈ X , one sets( Z q θ )( x ) := Z Y θ ( x , y ) . Lemma 3.10.
In the preceding situation, R q θ belongs to CF ( X ∞ ) .Proof. One knows that R b q j X × Y ! θ belongs to CF ( b X ). Therefore it is enough to checkthat for x ∈ X ( Z Y θ )( x ) = ( Z b q j X × Y ! θ )( x ) . This follows from (3.3) and the formula ( j X × Y ! θ )( x , · ) = j X ! θ ( x , · ).Let f : X ∞ −→ Y ∞ be a morphism of b-analytic manifolds and let ϕ ∈ CF ( X ∞ ).One sets Z f ϕ = Z q q ∗ ϕ · Γ f , Z np f ϕ = D Y Z f D X ϕ. (3.4)The next lemma is a particular case of Proposition 3.14 as well as of Proposition 3.15below. Lemma 3.11.
Let y ∈ Y . Then ( R f ϕ )( y ) = R X ϕ · f − ( y ) Proof.
One has ( Z f ϕ )( y ) = ( Z q q ∗ ϕ · Γ f )( y ) = Z X ϕ · Γ f ( · , y ) . Moreover, for y ∈ Y given, the two functions on X , x Γ f ( x, y ) and x f − ( y ) ( x )are the same. Proposition 3.12.
Let ϕ ∈ CF ( X ∞ ) . (a) The notation R f ϕ in (3.4) is in accordance with the previous definition ( see (1.11)) when f is proper on supp( ϕ ) . (b) The notation R np f ϕ in (3.4) is in accordance with (3.3) when Y = pt . (c) Both R f ϕ and R np f ϕ belong to CF ( Y ∞ ) . Moreover, if f is proper on supp( ϕ ) , then R f ϕ = R np f ϕ . (d) Let F ∈ D b R c ( k X ∞ ) and let ϕ = χ loc ( F ) . Then R f ϕ = χ loc (R f ! F ) and R np f ϕ = χ loc (R f ∗ F ) . roof. (a) follows from Lemmas 3.6 and 3.11.(b) follows from Lemma 3.8 (b).(c) follows from Lemma 3.10.(d) In order to prove the first equality, we are reduced by Lemma 3.11 to treat the casewhere Y = pt. In this case, by (3.3) and using the fact that j ! commutes with χ loc , wemay assume that F has compact support.(ii) The second equality is deduced from the first one by [KS90, Exe. VIII.3].We need an analogue of Lemma 1.8. Lemma 3.13.
Let f : X ∞ −→ Y ∞ be a morphism of b-analytic manifolds and let S ⊂ X and Z ⊂ Y be two closed subsets subanalytic up to infinity. Assume that f induces atopological isomorphism f | S : S ∼−→ Z . Then for ψ ∈ CF ( Y ∞ ) , one has Z X S · f ∗ ψ = Z Y Z · ψ. Proof.
One may assume that ψ = A for A a closed subset of Y subanalytic up toinfinity. Then Z Y Z · A = Z Y Z ∩ A = χ c ( Z ∩ A )= χ c ( f − ( Z ∩ A )) = Z X f − ( Z ∩ A ) = Z X S · f − A . Base change formula
Consider the diagrams below X ′ f ′ / / g ′ (cid:15) (cid:15) Y ′ g (cid:15) (cid:15) X ′∞ f ′ / / g ′ (cid:15) (cid:15) Y ′∞ g (cid:15) (cid:15) (cid:31) (cid:127) j / / X f / / Y X ∞ f / / Y ∞ , (3.5)where the diagram on the left is a diagram in the category of real analytic manifoldsand the diagram on the right is a diagram in the category of b-analytic manifolds. Itfollows from the definitions that the diagram on the right is Cartesian if and only ifthat on the left is. Proposition 3.14.
Consider the square (3.5) and let ϕ ∈ CF ( X ∞ ) . Then g ∗ Z f ϕ = Z f ′ ( g ′∗ ϕ ) . (3.6) 27 roof. We shall use Lemma 3.11. Let y ′ ∈ Y ′ . Then( g ∗ Z f ϕ )( y ′ ) = ( Z f ϕ )( g ( y ′ )) = Z X f − g ( y ′ ) · ϕ. The map g ′ induces a topological isomorphsm f − g ( y ′ ) ≃ f ′− ( y ′ ). Therefore, applyingLemma 3.13, Z X f − g ( y ′ ) · ϕ = Z X ′ f ′− ( y ′ ) · g ′∗ ϕ = ( Z f ′ g ′∗ ϕ )( y ′ ) . The projection formulaProposition 3.15.
Let f : X ∞ −→ Y ∞ be a morphism of b-analytic manifolds, let ϕ ∈ CF ( X ∞ ) and ψ ∈ CF ( Y ∞ ) . Then Z f ( ϕ · f ∗ ψ ) = ψ Z f ϕ. (3.7) Proof.
Let y ∈ Y . Then( Z f ϕ · f ∗ ψ )( y ) = Z X f − ( y ) · ϕ · f ∗ ψ = ψ ( y ) Z X f − ( y ) · ϕ = ( ψ Z f ϕ )( y ) . Example 3.16.
Equality (3.7) is no more true when replacing R f with R np f . Set X = R with coordinates ( x, y ) and Y = R , f being the first projection. Let ϕ = S with S = { ( x, y ); y = 1 / (1 − x ) , − < x < } and let ψ = Z with Z = ( − , ϕ is subanalytic up to infinity when choosing for example for b X theprojective compactification of R . We have S · f ∗ Z = S , D X S = − S , Z f S = Z . Therefore, Z np f S · f ∗ Z = Z np f S = D Y Z f D X S = − D Y Z = − [ − , , Z · Z np f S = − ( − , · Z np f S = − ( − , . Composition of kernels
Recall Diagram 1.1. Let λ ∈ CF ( X ∞ ) and λ ∈ CF ( X ∞ ). It follows fromProposition 3.7 that the function (see (1.19)) λ ◦ λ := Z q q ∗ λ · q ∗ λ . (3.8)is well-defined and belongs to CF ( X ∞ ). Moreover28 heorem 3.17. Let λ ij ∈ CF ( X ij ∞ ) ( i = 1 , , , , j = i + 1) . One has ( λ ◦ λ ) ◦ λ = λ ◦ ( λ ◦ λ ) ∈ CF ( X ∞ ) . The proof is the same as that of Theorem 1.12, using Propositions 3.14 and 3.15.In the sequel, we shall skip the parentheses and simply write λ ◦ λ ◦ λ . Corollary 3.18.
Let f : X ∞ −→ Y ∞ and g : Y ∞ −→ Z ∞ be morphisms of b-analyticmanifolds and let ϕ ∈ CF ( X ∞ ) . Then Z g ◦ f ϕ = Z g Z f ϕ, Z np g ◦ f ϕ = Z np g Z np f ϕ. Proof. (i) One has R f ϕ = Γ f ◦ ϕ and similarly with g and with g ◦ f . Then Z g ◦ f ϕ = Γ g ◦ f ◦ ϕ = Γ g ◦ Γ f ◦ ϕ = Z g Γ f ◦ ϕ = Z g Z f ϕ. (ii) The second formula follows from the first one in view of (3.4). The Fourier-Sato transform of constructible functions is already performed in [KS90].However, we present here a slightly different approach based on a new kernel of “coni-fication”.Let V be a n -dimensional real vector space as above. SetΓ µ = { ( x, λ, x ′ ) ∈ V × R + × V ; x ′ = λ · x } , ∆ µ = q (Γ µ ) = { ( x, x ′ ) ∈ V × V ; there exists λ ∈ R + , x ′ = λ · x } . The set Γ µ is the graph of the map µ : V × R + −→ V , ( x, λ ) λ · x . It is a closedsubanalytic subset of V × R + × V .Set Γ ◦ µ = Γ µ \ ( { } × R + × { } ) and ∆ ◦ µ = ∆ µ \ { (0 , } . Then q : Γ ◦ µ −→ ∆ ◦ µ isproper. Hence, ∆ ◦ µ is a subanalytic cone in V × V and its closure, ∆ µ is a closed cone,subanalytic up to infinity. Definition 3.19. (a) We say that a function ϕ : V −→ Z is homogeneous if ϕ ( λx ) = ϕ ( x ) for all x ∈ V and all λ ∈ R + .(b) We denote by CF R + ( V ) the subspace of CF ( V ) consisting of homogeneous func-tions.(c) We denote by κ conic ∈ CF R + ( V × V ) the function defined as follows: κ conic ( x , x ) = x = x = 0 , − x = 0 , x = λ · x for some λ > ϕ ∈ CF R + ( V ). Then for each m ∈ Z , ϕ − ( m ) is both subanalytic and conic.Applying Lemma 2.12, we get that CF R + ( V ) ⊂ CF ( V ∞ ) . Lemma 3.20. (a)
Let ϕ ∈ CF ( V ∞ ) . Then ϕ ◦ κ conic ∈ CF R + ( V ) . (b) Let ϕ ∈ CF R + ( V ) . Then ϕ ◦ κ conic = ϕ .Proof. (i) Since κ conic belongs to CF R + ( V × V ), the composition belongs to CF ( V ∞ ).It is homogeneous since κ conic ( x , x ) = κ conic ( x , λ · x ) for λ > ψ ( x ) = R q κ conic ( x , x ) · ϕ ( x ). If x = 0, then κ conic ( x ,
0) = (0) andthus ψ (0) = ϕ (0). If x = 0, then κ conic ( x , x ) = − x = λ · x for some λ > ϕ ( x ) = ϕ ( λ · x ), we get ψ ( x ) = − ϕ ( x ) · R V R + · x = ϕ ( x ).We consider the incidence relations P = { ( x, y ) ∈ V × V ∗ ; h x, y i ≥ } ,P ′ = { ( x, y ) ∈ V × V ∗ ; h x, y i ≤ } . (3.9)As usual, we denote by q and q the first and second projections defined on V × V ∗ .Let ϕ ∈ CF ( V ∞ ) and ψ ∈ CF ( V ∗∞ ). Since P and P ′ are subanalytic up to infinity, itis natural to define the Fourier-Sato transforms by the formulas F ∧ ( ϕ ) = ϕ ◦ P = Z q q ∗ ϕ · P ∈ CF ( V ∗∞ ) , F ∨ ( ψ ) = ( − n P ′ ◦ ψ = ( − n Z q q ∗ ψ · P ′ ∈ CF ( V ∞ ) . However, there is no chance to recover ϕ from F ∧ ( ϕ ) since this last function is homo-geneous. Lemma 3.21.
One has P ◦ P ′ = ( − n κ conic .Proof. Let us keep the notations of (1.36) and (1.37). We have S ( x, x ′ ) = { y ∈ V ∗ ; h x, y i ≥ , h x ′ , y i ≤ } . For ( x, x ′ ) / ∈ ∆ + ∪ { (0 , } , the complementary of S ( x, x ′ ) in V ∗ is topologicallyisomorphic to V ∗ and thus RΓ c ( S ( x, x ′ ); k V ∗ ) ≃
0. For ( x, x ′ ) ∈ ∆ + \ { , } , S ( x, x ′ )is a vector space of dimension n − x = x ′ = 0, S ( x, x ) = V . The resultfollows. Proposition 3.22.
Let ϕ ∈ CF ( V ∞ ) . Then: F ∨ ◦ F ∧ ( ϕ ) = ϕ ◦ κ conic . (3.10) In particular, if ϕ ∈ CF R + ( V ) , then F ∨ ◦ F ∧ ( ϕ ) = ϕ .Proof. This follows immediately from Lemmas 3.21 and 3.20.
Remark 3.23.
A similar argument could be applied to sheaves instead of constructiblefunctions. 30 .4 γ -constructible functions In this section, we consider a real finite dimensional vector space V and its projectivecompactification P as in § ϕ, ψ ∈ CF ( V ∞ ), we set ϕ ⋆ ψ := Z s ϕ ⊠ ψ, ϕ np ⋆ ψ := Z np s ϕ ⊠ ψ. By Proposition 3.12, both ϕ ⋆ ψ and ϕ np ⋆ ψ belong to CF ( V ∞ ). Note that ϕ ⋆ ψ = ψ ⋆ ϕ, ϕ np ⋆ ψ = ψ np ⋆ ϕ. Lemma 3.24.
Let ϕ i ∈ CF ( V ∞ ) , i = 1 , , . Then ϕ ⋆ ϕ = D X (D X ϕ ⋆ D X ϕ ) , ( ϕ ⋆ ϕ ) np ⋆ ϕ = ϕ ⋆ ( ϕ ⋆ ϕ ) . Proof.
The first equality follows from the definition of R np (see (3.4)) and the secondequality follows from the first one. Definition 3.25.
Let ϕ ∈ CF ( V ∞ ). We say that ϕ is γ -constructible if there exists afinite covering V = S a Z a such that ϕ = P a c a Z a and the Z a ’s are subanalytic γ -locallyclosed subsets of V . We denote by CF ( V γ ) the space of γ -constructible functions on V . By construction, we have CF ( V γ ) ⊂ CF ( V ∞ ).We denote by K R c ,γ ◦ a ( k V ∞ ) the Grothendieck group of D b R c ,γ ◦ a ( k V ∞ ). Theorem 3.26.
The isomorphism of commutative unital algebras χ loc : K R c ( k V ) ∼−→ CF ( V ) induces an isomorphism χ loc : K R c ,γ ◦ a ( k X ∞ ) ∼−→ CF ( V γ ) .Proof. (i) It follows from Proposition 2.17 that the map χ loc takes its values in CF ( V γ ).(ii) The map χ loc is injective by Lemma 2.18. Indeed, one checks easily that if A isa full triangulated category of a triangulated category T and if there is a projector T −→ A , then the natural map of the corresponding Grothendieck groups is injective.(iii) The map χ loc is surjective since for Z subanalytic γ -locally closed, Z = χ loc ( k Z )and k Z ∈ D b R c ( V ∞ ). Moreover, SS( k Z ) ⊂ V × γ ◦ a by [KS18, Cor. 1.8].We shall construct a projector CF ( V ∞ ) −→ CF ( V γ ). Theorem 3.27. (a)
Let ϕ ∈ CF ( V ∞ ) . Then ϕ np ⋆ γ a belongs to CF ( V γ ) . (b) If ϕ ∈ CF ( V γ ) , then ϕ np ⋆ γ a = ϕ .Proof. The result follows from Theorem 3.26, Lemma 2.18 and the fact that the oper-ation np ⋆ commutes with χ loc . 31et ϕ ∈ CF ( V ∞ ). One setsΓ( ϕ ) = X ≤| J |≤ n +1 ( − | J | +1 ( γ J np ⋆ ϕ ) , with γ J = V for | J | = n + 1 . (3.11) Proposition 3.28.
With Notation (3.11) , let ϕ ∈ CF ( V ∞ ) . Then ϕ = Γ( ϕ ) . (3.12) Proof.
By (2.6), we have { } = P ≤| J |≤ n +1 ( − | J | +1 γ J . Then apply · np ◦ ϕ to bothsides of this equality. Corollary 3.29.
Let
Γ := { γ j } j ∈ [0 ,n ] be closed convex proper cones with non emptyinterior such that, setting λ j = Int( γ ◦ aj ) , hypothesis (2.4) is satisfied. Then the map CF ( V ∞ ) −→ Y j ∈ [0 ,n ] CF ( V γ j ) , ϕ
7→ { γ j ⋆ ϕ } j ∈ [0 ,n ] , (3.13) is injective.Proof. Assume that γ j np ⋆ ϕ = 0 for all j ∈ [0 , n ]. Applying Lemma 3.24, we get that γ J np ⋆ ϕ = γ J np ⋆ γ j np ⋆ ϕ = 0 for J ⊂ [0 , n ] with 1 ≤ | J | ≤ n and also for γ J = V .Therefore, the result follows from Proposition 3.28. Example 3.30.
Let V = R and let Γ = { ( R ≥ , R ≤ ) , R } . Let ϕ ∈ CF ( R ∞ ). Then ϕ = R ≥ np ⋆ ϕ + R ≤ np ⋆ ϕ − R np ⋆ ϕ. Moreover, R ≥ np ⋆ ϕ = R ≤ np ⋆ ϕ = 0 implies ϕ = 0.Choose ϕ = { } , or ϕ = [0 , as exercises. Remark 3.31.
The advantage of CF ( V γ ) on CF ( V ∞ ) is that the support of a non-zerofunction in the first space has always strictly positive measure.Let Γ := { γ j } j ∈ [0 ,n ] be as in Corollary 3.29 and let a be some strictly positive con-tinuous function with R V a ( x ) dx < ∞ where dx is the Lebesgue measure. One can thendefine a norm on CF ( V ∞ ) by setting for ϕ ∈ CF ( V ∞ ) k ϕ k Γ a = n X i =0 k γ j np ⋆ ϕ k a (3.14)where k · k a is the L -norm on the space CF ( V ∞ ) for the measure a ( x ) dx .Of course, a norm on CF ( V ∞ ) defines a quasi-norm on the objects of D b R c ( k V ∞ )and this quasi-norm satisfies the triangular inequality: if F ′ −→ F −→ F ′′ +1 −→ is adistinguished triangle, then k F k Γ a ≤ k F ′ k Γ a + k F ′′ k Γ a . Unfortunately, these quasi-norms do not satisfy the stability property. For example,consider the two maps (with n ∈ N ): f, g : S −→ R , f ( t ) = sin( nt ) , g ( t ) = 0 . Then k f − g k ∞ = 1. However, R f S = n [ − , + n ( − , and R g S = 0.32 eferences [BM88] E Bierstone and P. D. Milman, Semi-analytic and subanalytic sets , Publ. Math. I.H.E.S (1988), 5-42.[CGR12] Justin Curry, Robert Ghrist, and Michael Robinson, Euler Calculus with Applications toSignals and Sensing , Proc. Sympos. Appl. Math., AMS (2012), available at arXiv:1202.0275 .[DK16] Andrea D’Agnolo and Masaki Kashiwara,
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