Lemma 0.1.1. The presheaf Hom(F,F^{0}) defined by

Hom(F,F^{0})(U) = Hom_{Sh(U)}(F_{|U},F_{|U}^{0} )
is a sheaf.

Proof. Easy exercice.

In general the morphismHom(F,F^{0})_{x}−→Hom(F_{x},F_{x}^{0}) is neither injective nor surjective.

By definition one has Γ(Hom(F,F^{0})) = Hom(F,F^{0}). 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 ⊗ F^{0} as the sheaf associated to the presheaf
U 7→ F(U)⊗ F^{0}(U) .

Since tensor products commute with colimits we get that
(F ⊗ F^{0})_{x} =F_{x}⊗ F_{x}^{0}
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 ⊗ F^{0},F^{00})' Hom(F,Hom(F^{0},F^{00}))
' Hom(F^{0},Hom(F,F^{00})) .
In particular Hom(F ⊗ F^{0},F^{00})'Hom(F,Hom(F^{0},F^{00})).

The adjunction between direct and inverse images generalizes as
Hom(F, f∗G)'f∗Hom(f^{−1}F,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 //

g^{0}

X^{0}

g

Y _{f} ^{//}X

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

f!◦g^{0}_{∗} −→g∗◦f_{!}^{0} .

LetG^{0}∈Sh(Y^{0}). LetV ∈Op_{X}. A sectiont∈Γ(V, f_{!}◦g^{0}_{∗}(G^{0})) is a sectiont∈Γ(f^{−1}(V), g^{0}_{∗}G^{0})
such thatf : supp (t)−→V is proper. Equivalently this is a sections∈Γ((f◦g^{0})^{−1}(V),G) with
supp (s) ⊂g^{0−}^{1}(Z) for f^{−1}(V) ⊃Z ^{f}^{proper}−→ V. But then g^{0−}^{1}(Z) −→ g^{−1}(V) is proper thus s
defines a section ofg∗f^{0}_{!}G^{0}.

By adjunction :

Hom(g^{−1}f_{!}G, f^{0}_{!}g^{0−1}G) = Hom(f_{!}G, g∗f^{0}_{!}g^{0−1}G) .
The morphism

f!−→f!g^{0}_{∗}g^{0−1}−→g∗f^{0}_{!}g^{0−1} ,

where the first map comes from the adjunction 1 −→ g^{0}_{∗}g^{0−}^{1}, gives the required canonical
morphism

g^{−1}◦f!=f_{!}^{0}◦g^{0−1} .

To show this is an isomorphism we compute the stalks at a pointx^{0} ∈X^{0}: Then
(g^{−1}f_{!}G)_{x}^{0} = (f_{!}G)_{g(x}0)

= Γ_{c}(f^{−1}(g(x^{0})),G)
by proposition 0.2.7.

The mapg^{0} induces a homeomorphism f^{0−1}(x^{0})'f^{−1}(g(x^{0})) and an isomorphism
Γc(f^{−1}(g(x^{0})),G)'Γc(f^{0−1}(x^{0}), g^{0−1}G)'(f^{0}_{!}g^{0−1}G)_{x}^{0} .

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_{x}6= 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^{−1}j_{!}G =G.

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

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^{−1}F .

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 Z^{0} ,→X one has :
(F_{Z})_{Z}^{0} ' F_{Z∩Z}^{0} .
(iii) If Z^{0} ⊂Z is closed one has an exact sequence :

0−→ F_{Z\Z}^{0} −→ F_{Z} −→ F_{Z}^{0} −→0 .

(iv) If Z_{1}, Z_{2} are two closed subsets of X the following sequence is exact :
0−→ F_{Z}_{1}_{∪Z}_{2} ^{(α}−→ F^{1}^{,α}^{2}^{)} _{Z}_{1} ⊕ F_{Z}_{2} ^{(β}^{1}−→^{,−β}^{2}^{)}F_{Z}_{1}_{∩Z}_{2} −→0 .
(v) If U_{1}, U_{2} are two open subsets of X the following sequence is exact :

0−→ F_{U}_{1}∩U_{2} −→ F_{U}_{1} ⊕ F_{U}_{2} −→ F_{U}_{1}∪U_{2} −→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 Z^{0} be another locally closed subset of X. Then :

Γ_{Z}^{0}(·)◦ΓZ(·) = Γ_{Z∩Z}^{0}(·) .
(iii) Assume that Z is open in X. Then

ΓZ =j∗◦j^{−1} .

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

0−→Γ_{U}_{1}∪U2(F)^{(α}−→^{1}^{,α}^{2}^{)}Γ_{U}_{1}(F)⊕Γ_{U}_{2}(F)^{(β}−→^{1}^{,−β}^{2}^{)}Γ_{U}_{1}∩U2(F)
is exact.

(vi) Let Z_{1}, Z_{2} be two closed subsets ofX. Then the sequence

0−→Γ_{Z}_{1}∩Z_{2}(F)^{(α}−→^{1}^{,α}^{2}^{)}Γ_{Z}_{1}(F)⊕Γ_{Z}_{2}(F)^{(β}−→^{1}^{,−β}^{2}^{)}Γ_{Z}_{1}∩Z_{2}(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 ⊗ F^{0})_{Z}' F_{Z}⊗ F^{0} ' F ⊗ F_{Z}^{0}
(6)

Γ_{Z}(Hom(F,F^{0}))' Hom(F,Γ_{Z}(F^{0}))' Hom(F_{Z},F^{0})
(7)

0.3.4. The functor j^{!}.

Proposition 0.3.7. Let j^{!}F :=j^{−1}Γ_{Z}F. 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_{Z}F)

= 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^{−1}Γ_{Z}F. Using the inclusion Γ_{Z}F −→ F this yields a monomor-
phismj^{!}F −→j^{−1}F. 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