is left-exact, this implies that the bifonctor Hom

Lemma 0.1.1. The presheaf Hom(F,F⁰) defined by

Hom(F,F⁰)(U) = Hom_Sh(U)(F_|U,F_|U⁰ ) is a sheaf.

Proof. Easy exercice.

In general the morphismHom(F,F⁰)_x−→Hom(F_x,F_x⁰) is neither injective nor surjective.

By definition one has Γ(Hom(F,F⁰)) = Hom(F,F⁰). As the bifunctor Hom(·,·) is left-exact, this implies that the bifonctor

Hom(·,·) :Sh(X)^op×Sh(X)−→Sh(X) is left-exact.

Definition 0.1.2. We define F ⊗ F⁰ as the sheaf associated to the presheaf U 7→ F(U)⊗ F⁰(U) .

Since tensor products commute with colimits we get that (F ⊗ F⁰)_x =F_x⊗ F_x⁰ hence

· ⊗ ·:Sh(X)⊗Sh(X)−→Sh(X) is right-exact.

Most classical formulas relating⊗and Hom forZ-modules have their counterpart forSh(X).

In particular one has an adjunction formula :

Hom(F ⊗ F⁰,F⁰⁰)' Hom(F,Hom(F⁰,F⁰⁰)) ' Hom(F⁰,Hom(F,F⁰⁰)) . In particular Hom(F ⊗ F⁰,F⁰⁰)'Hom(F,Hom(F⁰,F⁰⁰)).

The adjunction between direct and inverse images generalizes as Hom(F, f∗G)'f∗Hom(f⁻¹F,G) . 0.2. Direct image with proper support.

Definition 0.2.1. Let F ∈ Sh(X) and U ∈ Op_X. The support of a section s ∈ F(U) is the closed set

supp (s) ={x∈U / sx 6= 0} . Definition 0.2.2. Let f :Y −→X ∈Top.

We define the sub-presheaf f_!G ⊂f∗G by

f!G(V) ={s∈ G(f⁻¹(V))/ f|supps: supps−→V is proper} . Proposition 0.2.3. f!G is a sheaf.

Proof. It is enough to show that given (Ui)i∈I a family of open subsets of X with union U, and S a closed subset of f⁻¹(U) such that f :S∩f⁻¹(U_i) −→ U_i is proper for all i∈ I then f :S −→U is proper. Any fiber off :S −→U is the fiber of somef :S∩f⁻¹(U_i)−→U_i, thus is relatively Hausdorff and compact. Hence it is enough to check that f : S −→ U is closed.

This follows once more from the fact that the mapsf :S∩f⁻¹(U_i)−→U_i,i∈I, are closed.

Remarks 0.2.4. (1) notice that if f is proper (for example if Y and X are compact, or f :Z ,→X is a closed subspace) thenf!=f∗.

1

(2) As f_!is a subfunctor off∗ it is left exact.

(3) One showed that (f ◦g)∗ =f∗◦g∗. Similarly one can check that (f◦g)!=f!◦g!. Definition 0.2.5. The sections of G with proper support are defined by

Γc(Y,G) :=p_Y!G .

Equivalently : Γc(Y,G) ={s∈Γ(Y,G)/supp (s) is compact and relatively Hausdorff} . Remark 0.2.6. In particular Γ_c(X, f_!G) = Γ_c(Y,G).

Last time we proved that the natural morphism

(1) (f∗G)_x−→Γ(f⁻¹(x),G_|f⁻¹_(x)) is an isomorphism if f is proper. Similarly one obtains :

Proposition 0.2.7. Let f :Y −→X∈Top, andG ∈Sh(Y). Assume that XandY are locally compact spaces (in particular Hausdorff ).

Then for any x∈X the natural morphism

(f!G)_x−→Γc(f⁻¹(x),G_|f⁻¹_(x)) is an isomorphism.

Proof. First notice that the natural morphism

(2) (f_!G)_x−→^α Γ_c(f⁻¹(x),G_|f−1(x)) is nothing else than the restriction of the morphism (1) :

(f!G)_{ _ x}

//___ Γ_c(f⁻¹(x),G_|f−1(x))

 _

(f∗G)_x ^//Γ((f⁻¹(x),G_|f−1(x)) .

As we already showed in full generality that the morphism (1) is injective, the morphism (2) too.

Next we show that α is surjective. Let s∈ Γ_c(f⁻¹(x),G_|f−1(x)) and put K = supp (s). As K is compact and Y is Hausdorff, by proposition ??(ii) there exists an open neighbourhood U of K in Y and a section t ∈ Γ(U,G) such that t_|K = s_|K. By shrinking U we may assume t_|U∩f⁻¹_(x) =s_|U∩f⁻¹_(x). Let V be a relatively compact open neigbourhood of K with V ⊂ U. Since x6∈f(V ∩supp (t)\V) there exists an open neighbourhood W of x such thatf⁻¹(W)∩ V ∩supp (t)⊂V. One definesu∈Γ(f⁻¹(W),G) by

u_|f−1(W)\(supp (t)∩V)= 0 , u_|f⁻¹_(W)∩V =t_|f⁻¹_(W)∩V .

Since supp (u) is contained inf⁻¹(W)∩supp (t)∩V the mapf is proper on this set. Moreover u_|f⁻¹_(x)=s.

Recall that a subspace of a locally compact space is locally compact for the induced topology if and only if it is locally closed, where :

Definition 0.2.8. A subspace Z ,→X is said locally closed if any pointz∈Z has a neighbour- hood U in X such that U∩Z is closed relative to U.

EquivalentlyZ =U∩W withU open in X andW closed in X; orZ is open in its closureZ.

Proposition 0.2.7 has the following consequence :

Corollary 0.2.9. If X is locally compact andj :Y ,→X is a locally closed subspace then j_! is exact.

Proof. Proposition 0.2.7 implies in this case that (j_!G)_x=

(G_x ifx∈Y , 0 otherwise

and the result.

Notice that proposition 1 can be read asi⁻¹_x f_!G=f_|f⁻¹_(x)!i⁻¹_f−1(x)G for the diagram f⁻¹(x)

f_|f−1(x)//

 _

i_f−1(x)

{x}_{ _}

ix

Y f //X We generalize it to the following :

Proposition 0.2.10 (proper base change). Let Y^{0 f}

0 //

g⁰

X⁰

g

Y _f ^//X

be a Cartesian square of locally compact spaces. Then g⁻¹◦f_!=f_!⁰◦g⁰⁻¹ . Proof. We first construct a canonical morphism

f!◦g⁰_∗ −→g∗◦f_!⁰ .

LetG⁰∈Sh(Y⁰). LetV ∈Op_X. A sectiont∈Γ(V, f_!◦g⁰_∗(G⁰)) is a sectiont∈Γ(f⁻¹(V), g⁰_∗G⁰) such thatf : supp (t)−→V is proper. Equivalently this is a sections∈Γ((f◦g⁰)⁻¹(V),G) with supp (s) ⊂g⁰⁻¹(Z) for f⁻¹(V) ⊃Z ^fproper−→ V. But then g⁰⁻¹(Z) −→ g⁻¹(V) is proper thus s defines a section ofg∗f⁰_!G⁰.

By adjunction :

Hom(g⁻¹f_!G, f⁰_!g⁰⁻¹G) = Hom(f_!G, g∗f⁰_!g⁰⁻¹G) . The morphism

f!−→f!g⁰_∗g⁰⁻¹−→g∗f⁰_!g⁰⁻¹ ,

where the first map comes from the adjunction 1 −→ g⁰_∗g⁰⁻¹, gives the required canonical morphism

g⁻¹◦f!=f_!⁰◦g⁰⁻¹ .

To show this is an isomorphism we compute the stalks at a pointx⁰ ∈X⁰: Then (g⁻¹f_!G)_x⁰ = (f_!G)_g(x0)

= Γ_c(f⁻¹(g(x⁰)),G) by proposition 0.2.7.

The mapg⁰ induces a homeomorphism f⁰⁻¹(x⁰)'f⁻¹(g(x⁰)) and an isomorphism Γc(f⁻¹(g(x⁰)),G)'Γc(f⁰⁻¹(x⁰), g⁰⁻¹G)'(f⁰_!g⁰⁻¹G)_x⁰ .

0.3. Locally closed subspaces : the functors (·)_Z, Γ_Z, and j^!. Let j :Z ,→ X a locally closed subspace of a locally compact spaceX. We already noticed that j_! is exact in this case.

Moreover the definition ofj_! is equivalent to :

Γ(U, j_!G) ={s∈Γ(Z∩U,G)/supp (s) is closed relative to U} . Definition 0.3.1. Let F ∈Sh(X). The support of F is

suppF ={x∈X /F_x6= 0} .

Proposition 0.3.2. The functorj!:Sh(Z)−→Sh(X) defines an equivalence of categories j_!:Sh(Z)−→^←− Sh_Z(X) :j⁻¹ ,

where ShZ(X) denotes the full subcategory of Sh(X) of sheaves with support in Z.

Proof. First notice that for any sheafG ∈Sh(Z) thenj⁻¹j_!G =G.

Next, let F ∈ ShZ(X). One easily checks that the adjunction morphism F −→ j∗j⁻¹F factorizes throughj_!j⁻¹F, yielding an isomorphism F 'j_!j⁻¹F.

0.3.1. Restriction functor (·)_Z.

Definition 0.3.3. We define the functor (·)_Z :Sh(X)−→Sh(X) by

∀F ∈Sh(X), F_Z :=j!j⁻¹F .

As both functors j_! and j⁻¹ are exact, the functor (·)_Z is exact too. If Z is closed in X then F_Z = j∗j⁻¹ thus one has a morphism F −→ F_Z. If Z is open one easily checks that F_Z = ker(F −→ F_X\Z) thus one has a morphismF_Z −→ F.

Proposition 0.3.4. (i) The sheaf F_Z is uniquely characterized by the properties :

(3) (F_Z)_|Z =F_|Z and (F_Z)_|X\Z = 0 .

(ii) For any other locally closed subspace Z⁰ ,→X one has : (F_Z)_Z⁰ ' F_Z∩Z⁰ . (iii) If Z⁰ ⊂Z is closed one has an exact sequence :

0−→ F_Z\Z⁰ −→ F_Z −→ F_Z⁰ −→0 .

(iv) If Z₁, Z₂ are two closed subsets of X the following sequence is exact : 0−→ F_Z1∪Z2 ^(α−→ F^1,α2) _Z1 ⊕ F_Z2 ^(β1−→^,−β2)F_Z1∩Z2 −→0 . (v) If U₁, U₂ are two open subsets of X the following sequence is exact :

0−→ F_U1∩U₂ −→ F_U1 ⊕ F_U2 −→ F_U1∪U₂ −→0 .

0.3.2. The functor Γ_Z. Let U ⊂X be open andZ ⊂U be closed. One defines : ΓZ(U,F) :={s∈ F(U)/supp (s)⊂Z}(= ker(F(U)−→ F(U\Z))) .

If Z ⊂ V ⊂U then the canonical morphism Γ_Z(U,F) −→ Γ_Z(V, F) is an isomorphism. Thus for Z locally closed we may define ΓZ(X,F) as ΓZ(U,F) where U is any open subset of X containingZ as a closed subset.

Definition 0.3.5. One denotes by Γ_Z(F) the sheaf U 7→ ΓZ∩U(U,F) and calls it the sheaf of sections of F supported by Z.

Proposition 0.3.6. Let j:Z ,→X be a locally closed subset of X andF ∈Sh(X). Then : (1) The functors ΓZ(X,·) : Sh(X) −→ Ab and ΓZ : Sh(X) −→ Sh(X) are left exact.

Moreover

ΓZ(X,·) = Γ(X,·)◦ΓZ(·) . (ii) Let Z⁰ be another locally closed subset of X. Then :

Γ_Z⁰(·)◦ΓZ(·) = Γ_Z∩Z⁰(·) . (iii) Assume that Z is open in X. Then

ΓZ =j∗◦j⁻¹ .

(iv) Let Z⁰ ⊂Z be closed. Then the following sequence is exact : 0−→Γ_Z⁰(F)−→ΓZ(F)−→ΓZ\Z⁰(F) . (v) Let U1, U2 be two open subsets of X. Then the sequence

0−→Γ_U1∪U2(F)^(α−→^1,α2)Γ_U1(F)⊕Γ_U2(F)^(β−→^1,−β2)Γ_U1∩U2(F) is exact.

(vi) Let Z₁, Z₂ be two closed subsets ofX. Then the sequence

0−→Γ_Z1∩Z₂(F)^(α−→^1,α2)Γ_Z1(F)⊕Γ_Z2(F)^(β−→^1,−β2)Γ_Z1∩Z₂(F) is exact.

Proof. Easy exercice.

0.3.3. Link with Hom and ⊗. The functors (·)_Z and Γ_Z can be obtained usingZZ := (ZX)_Z : F_Z 'ZZ⊗ F

(4)

ΓZ(F)' Hom(ZZ,F) (5)

(F ⊗ F⁰)_Z' F_Z⊗ F⁰ ' F ⊗ F_Z⁰ (6)

Γ_Z(Hom(F,F⁰))' Hom(F,Γ_Z(F⁰))' Hom(F_Z,F⁰) (7)

0.3.4. The functor j^!.

Proposition 0.3.7. Let j^!F :=j⁻¹Γ_ZF. Then the pair (j_!, j^!) defines an adjunction : j!:Sh(Z)−→^←− Sh(X) :j^! .

Proof. LetU ∈Op_X. Then

Γ(U, j_!j^!F) = Γ(U,(j_!j⁻¹)Γ_ZF)

= Γ(U,ΓZF)

={s∈Γ(U,F), /supp (s)⊂Z} .

This formula defines a monomorphism j_!j^! −→ 1. Moreover it implies that any morphism G −→ F where G ∈ Sh_Z(X) factorizes through j_!j^!F −→ F. For a sheaf G ∈ Sh(Z) we get accordingly

Hom(j!G,F)'Hom(j!G, j_!j^!F) . From proposition 0.3.2 we have an isomorphism :

(j_!)⁻¹: Hom(j_!G, j_!j^!F)'Hom(G, j^!F) . By composing the two previous isomorphisms we get the result.

Corollary 0.3.8. The functorj^!:Sh(X)−→Sh(Z) is left exact and carries injective sheaves into injective sheaves.

Proof. Once more it follows formally from the fact thatj^! is right adjoint to the exact functor

j!.

Lemma 0.3.9. If j:Z −→X is open then j^!=j⁻¹.

Proof. By definition j^!F = j⁻¹Γ_ZF. Using the inclusion Γ_ZF −→ F this yields a monomor- phismj^!F −→j⁻¹F. One checks this is an isomorphism by localization.

Corollary 0.3.10. The inclusion j:U −→X of an open subset transform an injective sheaf I onX into an injective sheaf j^!I on U.

References

[1] Godement R., Th´eorie des faisceaux, Hermann, 1973 [2] Iversen B., Cohomology of sheaves, Springer 1986

[3] Kashiwara M., Schapira P., Sheaves on Manifolds, Grundlehren der mathematischen Wissenschaften 292, Springer, 1990