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If the category K admits projective limits (in the general sense), and if B, B 0 are bases
for the topologies of X and Y respectively, then to define a ψ-morphism u of sheaves, it
suffices to give uU,V for U ∈ B, V ∈ B 0 , and ψ(U ) ⊆ V , checking compatibility conditions
(3.5.1.1) for U, U 0 ∈ B, V, V 0 ∈ B 0 . Define, then, uW , for any open W ⊆ Y , to be any
projective limit of uU,V where V ∈ B 0 , V ⊆ W and U ∈ B, ψ(U ) ⊆ V .
When the category K admits inductive limits, one has, for all x ∈ X, a morphism
G(V ) → F(ψ −1 (V )) → Fx for every open neighborhood V of ψ(x) in Y . These morphisms form an inductive system which yields a morphism Gψ(x) → Fx by passing to the
limit.
(3.5.2) Under the hypotheses of (3.4.3), let F, G, K be K-valued presheaves of X, Y, Z
respectively, and let u : G → ψ∗ (F), v : K → ψ∗ (G) be a ψ-morphism and a ψ 0 -morphism
v
ψ∗0 (u)
/ ψ 0 (G)
/ ψ ∗0 (ψ∗ (F)) = ψ 00 (F) ,
respectively. One can derive a ψ 00 -morphism w : K
∗
∗
called the composition of u and v. One can thus consider couples (X, F) consisting of
a topological space X and a presheaf F on X (with values in K) to form a category,
the morphisms being couples (ψ, θ) : (X, F) → (Y, G) consisting of a continuous map
ψ : X → Y and a ψ-morphism θ : G → F .
(3.5.3) Let ψ : X → Y be a continuous map, G a presheaf on Y (with values in K).
We define the inverse image of G by ψ to be a couple (G 0 , ρ), where G 0 is a K-valued
sheaf on X and ρ : G → G 0 is a ψ-morphism (ie-a morphism G → ψ∗ (G 0 )) such that, for
every K-valued sheaf F of X, the map
(3.5.3.1) HomX (G 0 , F) → Homψ (G, F) = HomY (G, ψ∗ (F))
taking v to ψ∗ (v) ◦ ρ, is a bijection. This map, being functorial in F, defines an
isomorphism of functors in F. The couple (G 0 , ρ), being universal, is determined up to
unique isomorphism when it exists. We then write G 0 = ψ ∗ (G), ρ = ρG , and by abuse of
language, we call ψ ∗ (G) the inverse image sheaf of G by ψ, which is equipped with a
canonical ψ-morphism ρG : G → ψ ∗ (G) (ie-the canonical morphism
(3.5.3.2) ρG : G → ψ∗ (ψ ∗ (G))
of presheaves on Y ).
For every homomorphism v : ψ ∗ (G) → F (where F is a K-valued sheaf on X), we have
v [ = ψ∗ (v) ◦ ρG . By definition, every morphism of presheaves u : G → ψ∗ (F) is of the
form v [ for one and only one v, which we denote u] . In other words, every morphism of
presheaves u : G → ψ∗ (F) factors uniquely as
(3.5.3.3) u : G
ρG
ψ∗ (u] )
/ ψ∗ (ψ ∗ (G))
/ ψ∗ (F) .
1
(3.5.4) Suppose that the category K is such that every presheaf G on Y with values
in K has an inverse image by ψ, denoted ψ ∗ (G) 1 .
We will see that one can define ψ ∗ (G) to be a covariant functor on G from the category
of K-valued presheaves on Y to the category of K-valued presheaves on X, such that the
isomorphism v → v [ is an isomorphism of bifunctors
→
(3.5.4.1) HomX (ψ ∗ (G), F)−
∼ HomY (G, ψ∗ (F))
on G and F.
Indeed, for every morphism w : G1 → G2 of K-valued presheaves on Y , we consider
w
ρG2
/ G2
/ ψ∗ (ψ ∗ (G2 )) . There corresponds to this a
the composition morphism G1
morphism (ρG2 ◦ w)] : ψ ∗ (G1 ) → ψ ∗ (G2 ), which we denote ψ ∗ (w). One now has, under
the conditions of (3.5.3.3)
(3.5.4.2) ψ∗ (ψ ∗ (w)) ◦ ρG1 = ρG2 ◦ w.
For every morphism u : G2 → ψ∗ (F) where F is a K-valued sheaf on X, one has, via
(3.5.3.3), (3.5.4.2), and the definition of u[ :
(u] ◦ ψ ∗ (w))[ = ψ∗ (u)] ◦ ψ∗ (ψ ∗ (w)) ◦ ρG1 = ψ∗ (u] ) ◦ ρG2 ◦ w = u ◦ w
or
(3.5.4.3) (u ◦ w)] = u] ◦ ψ ∗ (w).
0
ρG
/ 3ψ∗ (ψ ∗ (G3 )) , we have
If, in particular, we take u to be the morphism G2 w / G3
ψ ∗ (w0 ◦ w) = (ρG3 ◦ w0 ◦ w)] = (ρG3 ◦ w)] ◦ ψ ∗ (w) = ψ ∗ (w0 ) ◦ ψ ∗ (w), which proves our
assertion.
Finally, for every K-valued sheaf F on X, let iF be the identity morphism on ψ∗ (F)
and denote by σF : ψ ∗ (ψ∗ (F)) → F the morphism (iF )] . The formula (3.5.4.3) gives, in
particular, the factorization
(3.5.4.4) u] : ψ ∗ (G)
ψ ∗ (u)
/ ψ ∗ (ψ∗ (F))σF
/F
for every morphism u : G → ψ∗ (F). We do not claim that the morphism σF is
canonical.
1
In a book cited in the introduction we give three general conditions on the category K which ensure
the existence of inverse images of K-valued presheaves.
2
(3.5.5) Let ψ 0 : Y → Z be a continuous map, and suppose that every K-valued presheaf
H of Z admits an inverse image ψ 0∗ (H) by ψ 0 . Then (under the hypotheses of (3.5.4))
all K-valued presheaves H of Z admit an inverse image by ψ 00 = ψ 0 ◦ ψ and there is a
canonical functorial isomorphism
→
(3.5.5.1) ψ 00∗ (H)−
∼ ψ ∗ (ψ 0∗ (H)).
This is, indeed, immediate from the definitions, taking into account that ψ 00 = ψ 0 ◦ ψ.
Moreover, if u : G → ψ∗ (F) is a ψ-morphism, v : H → ψ∗0 (G) a ψ 0 -morphism, and
w = ψ∗0 (u) ◦ v their composition (3.5.2), we note immediately that w] is the composition
morphism
ψ ∗ (v ] )
w] : ψ ∗ (ψ 0∗ (H))
]
/ ψ∗G u
/F .
(3.5.6) In particular, take ψ to be the identity map idX : X → X. If the inverse image
by ψ of a K-valued presheaf F on X exists, it is the inverse image of the sheaf associated
to the presheaf F. Every morphism u : F → F 0 from F to a K-valued sheaf F 0 factors
uniquely as F
ρF
]
/ id∗ (F) u
X
/ F0 .
3.6. Simple Sheaves and Locally Simple Sheaves
3.6.1 We will say that a K-valued presheaf F of X is constant if the canonical morphism
F(X) → F (U ) is an isomorphism for every nonempty open U ⊆ X. One notes that F
is not necessarily a sheaf. A sheaf is said to be simple if it is associated (3.5.6) to a
constant presheaf. A sheaf F is said to be locally simple if every x ∈ X has an open
neighborhood U such that F|U is simple.
(3.6.2) Suppose that X is irreducible (2.1.1); then the following are equivalent:
a F is a constant presheaf on X.
b F is a simple sheaf on X.
c F is a locally simple sheaf on X.
Indeed, let F be a constant presheaf on X. If U, V are nonempty open subsets of X
and U ∩ V is nonempty, then, as F(X) → F (U ) → F(U ∩ V ) and F(X) → F (U ) are
isomorphisms, F(U ) → F(U ∩ V ) and F(V ) → F (U ∩ V ) are also isomorphisms. One
immediately concludes that axiom (F) of (3.1.2) holds, F is isomorphic to its associated
sheaf, and a) implies b).
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Now let (Uα ) be a cover of X by nonempty open sets and let F be a sheaf on X
such that F|Uα is simple for all α. Since Uα is irreducible, FUα is a constant presheaf
on X by the preceding paragraph. As Uα ∩ Uβ is nonempty, F(Uα ) → F (Uα ∩ Uβ )
and F(Uβ ) → F(Uα ∩ Uβ ) are isomorphisms from which one can derive a canonical
isomorphism θαβ : F(Uα ) → F (Uβ ) for each pair of indices. But then, applying condition
(F) for U = X, we see that for every index α0 , F(Uα0 ) and θα0 α are universal, which (by
unicity) implies that F(X) → F (Uα0 ) is an isomorphism, showing c) implies a).
Inverse Images of Presheaves of Groups or Rings
(3.7.1) We will show that when one takes K to be the category of sets, the inverse
image by ψ of every K-valued presheaf G must exist (the notation and hypotheses for X,
Y , ψ being those of (3.5.3)). Indeed, for every open U ⊆ X, G 0 (U ) is defined as follows:
an element s0 of G 0 (U ) is a family (s0x )x∈U where s0x ∈ Gψ(x) for every x ∈ U and where,
for every x ∈ U , the following condition is satisfied: there exists an open neighborhood
V of ψ(x) in Y , a neighborhood W ⊆ ψ −1 (V ) ∩ U of x, and an element s ∈ G(V ) such
that s0z = sψ(z) for every z ∈ W . One verifies immediately that U 7→ G 0 (U ) satisfies the
sheaf axioms.
Let F be a sheaf of sets on X with morphisms u : G → ψ∗ (F), v : G 0 → ψ∗ (F). One
defines u] and v [ in the following way: if s0 is a section of G 0 over a neighborhood U of
x ∈ X, V is an open neighborhood of ψ(x), and s ∈ G(V ) satisfies s0z = sψ(z) for all z in
some neighborhood of x contained in ψ −1 (V ) ∩ U , then one defines u]x (s0x ) = uψ(x) (sψ(x) ).
Similarly, if s ∈ G(V ) (V open in Y ), v [ (s) is the section of F over ψ −1 (V ) whose image
under v is the section s0 of G 0 such that s0x = sψ(x) for every x ∈ ψ −1 (V ). Moreover,
the canonical homomorphism (3.5.3) ρ : G → ψ∗ (ψ ∗ (G)) is defined in the following way:
for every open V ⊆ Y and every section s ∈ Γ(V, G), ρ(s) is the section (sψ(x) )x∈ψ−1 (V )
of ψ ∗ (G) over ψ −1 (V ). The verification of the relations (u] )[ = u, (v [ )] = v, and v [ =
ψ∗ (v) ◦ ρ is immediate, proving our assertion.
One checks that, if w : G1 → G2 is a homomorphism of presheaves of sets on Y , ψ ∗ (w)
may be made explicit in the following way: if s0 = (s0x )x∈U is a section of ψ ∗ (G1 ) over
an open subset U of X, (ψ ∗ (w))(s0 ) is the family (wψ(x) (s0x ))x∈U . Finally, it is immediate
that, for every open V of Y , the inverse image of G|V by the restriction of ψ to ψ −1 (V )
is equal to the sheaf induced by ψ ∗ (G)|ψ−1 (V ) .
When ψ is the identity idX , the definition coincides with that of a sheaf of sets associated to a presheaf (G, II, 1.2). The preceding considerations apply without change when
K is the category of groups or (not necessarily commutative) rings.
When X is an arbitrary subset of a topological space Y and j is the canonical inclusion
X → Y , for all K-valued sheaves G on Y , we define the sheaf on X induced by G to
be the inverse image j ∗ (G) (when it exists); for sheaves of sets (or groups or rings) this
coincides with the usual definition.
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(3.7.2) Using the notation and hypotheses of (3.5.3), let G be a sheaf of groups [resp.
rings] on Y . The definition of sections of ψ ∗ (G) (3.7.1) shows (taking into account
(3.4.4)) that the homomorphism of fibres ψx ◦ ρψ(x) : Gψ(x) → (ψ ∗ (G))x is a functorial isomorphism in G, which allows us to identify the two fibres. Via this identification, u]x is precisely the homomorphism defined in (3.5.1) and, in particular, we have
Supp(ψ ∗ (G)) = ψ −1 (Supp(G)).
An immediate consequence of this result is that the functor ψ ∗ (G) is exact on G in the
abelian category of sheaves of abelian groups.
Sheaves of Pseudo-Discrete Spaces
3.8.1 Let X be a topological space whose topology admits a basis B consisting of
quasi-compact open sets. Let F be a sheaf of sets on X. If each F(U ) is given the
discrete topology then U 7→ F(U ) is a presheaf of topological spaces. We will see that
there is a sheaf of topological spaces F 0 associated to F (3.5.6) such that Γ(U, F 0 ) is the
dicrete space F(U ) for every open, quasi-compact U . It will suffice to show that the
presheaf U 7→ F (U ) of discrete topological spaces on B satisfies condition (F0 ) of (3.2.2)
and, more generally, if U is a quasi-compact open and if (Ux ) is a cover of U by sets
in B then the coarsest topology T on Γ(U, F) making the maps Γ(U, F) → Γ(Uα , F)
continuousSis the discrete topology. However, there are a finite number of indices αi such
that U = Uαi . Let s ∈ Γ(U, F) and let si be its image in Γ(Uαi , F). The intersection of
the inverse images of the sets {si } is, by definition, a neighborhood of s for T , but since
F is a sheaf of sets and the Uαi cover U , this intersection is {s}, proving our assertion.
One notes that if U is an open, non quasi-compact subset of X then topological space
Γ(U, F 0 ) still has the underlying set of Γ(U, F), but that the topology is not necessarily
discrete; it is the coarsest making the maps Γ(U, F) → Γ(V, F) continuous for V ∈ B
and V ⊆ U (Γ(V, F) being discrete).
The preceding considerations apply without modification for sheaves of groups and
(not necessarily commutative) rings with their respective sheaves of topological groups
and topological rings. In the interest of brevity, we say that the sheaf F 0 is the pseudo −
discrete sheaf of spaces (resp. groups, rings) associated to the sheaf of sets (resp.
groups, rings) F.
(3.8.2) Let F, G be sheaves of sets (resp. groups, rings) on X and u : F → G a
homomorphism. Then u is also a continuous homomorphism F 0 → G 0 , where F and G 0
are the pseudo-discrete sheaves associated to F and G. This results from (3.2.5).
(3.8.3) Let F be a sheaf of sets, H a subsheaf of F, F 0 and H0 the pseudo-discrete
sheaves associated to F and H respectively. Then, for all open U ⊆ X, Γ(U, H0 ) is
closed in Γ(U, F 0 ). Indeed, it is the intersection of the inverse images of the Γ(V, H) (for
5
V ∈ B, V ⊆ U ) under the continuous maps Γ(U, F) → Γ(V, F), and Γ(V, H) is closed in
the discrete space Γ(V, F).
4. Ringed Spaces
4.1. Ringed Spaces, A-Modules, A-Algebras
(4.1.1) A ringed (resp. topologically ringed) space is a couple (X, A) consisting of a
topological space X and a sheaf of (not necessarily commutative) rings (resp. topological
rings) A on X. X is said to be the underlying topological space of the ringed space
(X, A) and A the strcture sheaf . This last is denoted OX and the fibre at a point x ∈ X
is denoted OX,x , or simply Ox when there is no risk of confusion.
We denote by 1 or e the unit section of OX over X (the unit element of Γ(X, OX )).
Since, in this volume, we will need to consider commutative rings we will assume that,
unless otherwise specified, that the structure sheaf A is a sheaf of commutative rings.
The ringed spaces (resp. topologically ringed spaces) with structure sheaf not necessarily commutative form a category, when one defines a morphism (X, A) → (Y, B) to
be a couple (ψ, θ) = Ψ consisting of a continuous map ψ : X → Y and a ψ-morphism
θ : B → A (3.5.1) of rings (resp. topological rings). The composition of Ψ and second
morphism Ψ0 = (ψ 0 , θ0 ) : (Y, B) → (Z, C), denoted Ψ00 = Ψ0 ◦ Ψ, is the morphism (ψ 00 , θ00 )
where ψ 00 = ψ 0 ◦ ψ and θ00 is the composition of θ and θ0 (ie-ψ∗0 (θ) ◦ θ0 , cf. 3.5.2). For
ringed spaces, recall that we then have θ00] = θ] ◦ ψ ∗ (θ0] ) (3.5.5). Thus, if θ0] and θ]
are injective (resp. surjective) homomorphisms, then the same is true of θ00] taking into
account that ψx ◦ ρψ(x) is an isomorphism for all x ∈ X (3.7.2). One verifies immediately,
thanks to the preceding, that, when ψ is a continuous injective map and θ] is a surjective
homomorphism of sheaves of rings, the morphism (ψ, θ) is a monomorphism (T, 1.1) in
the category of ringed spaces.
By abuse of language, one often replaces ψ by Ψ in notation. For example, we write
Ψ (U ) in place of ψ −1 (U ) for a subset U of Y when there is no risk of confusion.
−1
(4.1.2) For every subset M of X the couple (M, A|M ) is clearly a ringed space, induced
on M by the ringed space (X, A) (and still called the restriction of (X, A) to M ). If
j is the canonical injection M → X and w the identity map on A|M then (j, w[ ) is a
monomorphism (M, A|M ) → (X, A) of ringed spaces called the canonical injection. The
composition of a morphism Ψ : (X, A) → (Y, B) and this injection is call the restriction
of Ψ to M .
6
(4.1.3) We will not revisit the definitions of A-modules or algebraic sheaves on a ringed
space (X, A) (G,II,2.2). When A is a sheaf of (not necessarily commutative) rings, when
we say A-module we always mean a left A-module unless otherwise specified. The subA-modules of A will be called sheaves of ideals (left, right, or double sided) or A-ideals.
When A is a sheaf of commutative rings and, in the definition of A-modules, one
replaces everywhere the structure of a module with that of an algebra, one obtains the
definition of an A-algebra on X. That is to say, a (not necessarily commutative) Aalgebra is an A-module, C, endowed with a homomorphism of A-modules φ : C ⊗A C → C
and a section e over X such that the diagram
C ⊗A C ⊗A C
1⊗φ
φ⊗1
φ
²
C ⊗A C
/ C ⊗A C
²
φ
/C
commutes and, for every open U ⊆ X and every section s ∈ Γ(U, C), one has φ((e|U )⊗
s) = φ(s⊗(e|U )) = s. To say that C is a commutative A-algebra is to say that the diagram
σ
/C⊗ C
C ⊗A GC
A
GG
GG
G
φ GGG
#
ww
ww
w
ww φ
w{ w
C
commutes where σ is the canonical symmetry map on the tensor product C ⊗A C.
The homomorphisms of A-algebras are defined in the same way as homomorphisms of
A-modules were defined in (G,II,2.2), but these do not naturally form an abelian group.
If M is a sub-A-module of an A-algebra C then the sub-A-algebra of C generated by
M is the sum of the images of the homomorphisms ⊗n M → C (for n ≥ 0). It is also
the sheaf associated to the presheaf U 7→ B(U ) where B(U ) denotes the subalgebra of
Γ(U, C) generated by the submodule Γ(U, M).
(4.1.4) We say that a sheaf of rings on a topological space X is reduced at a point
x ∈ X if the fibre Ax is a reduced ring (1.1.1). We say that A is reduced if it is reduced
at every point of X. We recall that a ring A is said to be regular if every local ring Ap (as
p transverses the set of prime ideals of A) is a regular local ring. We say that a sheaf of
rings A on X is regular at a point x (resp. regular) if the fibre Ax is a regular ring (resp.
if A is regular at every point). Finally, we say that s sheaf of rings A on X is normal
at a point x (resp. normal) if the fibre Ax is integral and integrally closed (resp. if A is
normal at every point). We say that the ringed space (X, A) has one of the preceding
properties if the sheaf of rings A has that property.
A sheaf of graded ring is by definition a sheaf of rings which is the direct sum (G,II,2.7)
of a family (An )n∈Z of sheaves of abelian groups such that Am An ⊆ Am+n . A graded
A-module F is an A-module that is the direct sum of a family (Fn )n∈Z of abelian groups
7
satisfying Am Fn ⊆ Fm+n . It is equivalent to say that (Am )x (An )x ⊆ (Am+n )x (resp.
(Am )x (Fn )x ⊆ (Fm+n )x ) for every point x.
(4.1.5) Given a (not necessarily commutative) ringed space (X, A), we will not recall the definitions of the bifunctors HomA (F, G) and HomA (F, G) in the categories of
left and right A-modules with values in the category of sheaves of abelian groups (or
more generally, of C-modules, if C is the center of A). The fibre (F ⊗A G)x is canonically identified with Fx ⊗Ax Gx and one can define a canonical functorial homomorphism
(HomA (F, G))x → HomAx (Fx , Gx ) which is not, in general, injective or surjective. The
bifunctors considered above are additive and, in particular, commute with finite direct
sums. F ⊗A G is right exact in F and G and commutes with inductive limits, and A ⊗A G
(resp. F ⊗A A) is canonically identified with G (resp. F). The functors HomA (F, G)
and HomA (F, G) are left exact in F and G. More precisely, given an exact sequence of
the form 0 → G 0 → G → G 00 , the sequence
0 → HomA (F, G 0 ) → HomA (F, G) → HomA (F, G 00 )
is exact, and given an exact sequence of the form F 0 → F → F 00 → 0, the sequence
0 → HomA (F 00 , G) → HomA (F, G) → HomA (F 0 , G)
is exact, with the analogous properties for the functor Hom. Moreover, HomA (A, G)
is canonically identified with G. Finally, for all open U ⊆ X, one has
Γ(U, HomA (F, G)) = HomAU (F|U , G|U ).
For every left (resp. right) A-module F, one calls the right (resp. left) A-module
HomA (F, A) the dual of F, denoted F̌.
Finally, if A is a sheaf of commutative rings and F is an A-module then U 7→ ∧p Γ(U, F)
is a presheaf whose associated sheaf is an A-module which we denote ∧p F and call the
pth exterior power of F. One easily verifies that the canonical map of presheaves from
U 7→ ∧p Γ(U, F) to the associated sheaf ∧p F is injective and that, for all x ∈ X, one has
(∧p F)x = ∧p (Fx ). It is clear that ∧p F is a covariant functor in F.
(4.1.6) Suppose that A is a sheaf of (not necessarily commutative) rings, J is a sheaf
of left ideals of A, and F is a left A-module. One denotes by J F the sub-A-module
of F that is the image of J ⊗Z F (where Z is the sheaf associated to the constant
presheaf U 7→ Z) under the canonical map J ⊗Z F → F. It is clear that, for all
x ∈ X, we have (J F)x = Jx Fx . When A is commutative J F is also the canonical
image of J ⊗A F → F . It is immediate that J F is also the A-module associated to
the presheaf U 7→ Γ(U, J )Γ(U, F). If J1 , J2 are two sheaves of left ideals of A, one has
J1 (J2 F) = (J1 J2 )F.
8
(4.1.7) Let (Xλ , Aλ )λ∈L be a family of ringed spaces. For every couple (λ, µ) suppose we
have an open subset Vλµ of X and an isomorphism of ringed spaces ϕλµ : (Vµλ , Aµ|Vµλ ) →
(Vλµ , Aλ|Vλµ ) with Vλλ = Xλ and ϕλλ the identity. Suppose moreover that, for every
triplet (λ, µ, ν), if one denotes by ϕ0µλ the restriction of ϕµλ to ϕλµ ∩ ϕλν then ϕ0µλ
is an isomorphism of (Vλµ ∩ Vλν , Aλ|Vλµ ∩Vλν ) and (Vµν ∩ Vµλ , Aµ|Vµν ∩Vµλ ) and one has
ϕ0λν = ϕ0λµ ◦ϕ0µν (the gluing condition for the ϕλµ ). One first of all considers the topological
space obtained by gluing the Xλ along the Vλµ (using the ϕλµ ). If one identifies Xλ with its
corresponding open subset Xλ0 of X, the assumptions imply that the three sets Vλµ ∩ Vλν ,
Vµν ∩ Vµλ , and Vνλ ∩ Vνµ are identified with Xλ0 ∩ Xµ0 ∩ Xν0 . One can then transfer the
ringed space structure of Xλ to Xλ0 and, if A0λ is the sheaf of rings transferred from Aλ
then A0λ satisfies the gluing conditions (3.3.1) and defines a sheaf of rings A on X. We
call (X, A) the ringed space obtained by gluing the (Xλ , Aλ ) along the Vλµ via the ϕλµ .
4.2. Direct Image of an A-module
(4.2.1) Let (X, A) and (Y, B) be ringed spaces and Ψ = (ψ, θ) a morphism (X, A) →
(Y, B). ψ∗ (A) is then a sheaf of rings on Y and θ is a homomorphism B → ψ∗ (A) of
sheaves of rings. Let F be an A-module. Its direct image ψ∗ (F) is a sheaf of abelian
groups on Y . Moreover, for every open U ⊆ Y ,
Γ(U, ψ∗ (F)) = Γ(ψ −1 (U ), F)
is endowed with a module structure via Γ(U, ψ∗ (A)) = Γ(ψ −1 (U ), A). The bilinear
maps defining these structures are compatible with the restriction maps, defining on
ψ∗ (F) the structure of a ψ∗ (A)-module. The homomorphism θ : B → ψ∗ (A) then allows
us to define the structure of a B-module on ψ∗ (F). We will call this B-module the direct
image of F by the morphism Ψ and denote it Ψ∗ (F). If F1 , F2 are A-modules on X
and u : F1 → F2 is an A-module homomorphism, it is immediate (considering sections
over opens of Y ) that ψ∗ (u) is a ψ∗ (A)-homomorphism ψ∗ (F1 ) → ψ∗ (F2 ) and, a fortiori,
a B-homomorphism Ψ∗ (F1 ) → Ψ∗ (F2 ) which we denote Ψ∗ (u). Thus, Ψ∗ is a covariant
functor from the category of A-modules to the category of B-modules. Moreover, it is
immediate that this functor is left exact (G,II,2.12).
We define the structure of a B-algebra on ψ∗ (A) via its structure as a B-module and
its structure as a sheaf of rings. We denote this B-algebra Ψ∗ (A).
(4.2.2) Let M and N be A-modules. For every open U of Y , there is a canonical map
Γ(ψ −1 (U ), M) × Γ(ψ −1 (U ), N ) → Γ(ψ −1 (U ), M ⊗A N )
which is bilinear over the ring Γ(ψ −1 (U ), A) = Γ(U, ψ∗ (A)) and, a fortiori, over
Γ(U, B). It thus defines a homomorphism
Γ(U, Ψ∗ (M)) ⊗Γ(U,B) Γ(U, Ψ∗ (N )) → Γ(U, Ψ∗ (M ⊗A N ))
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and, as one verifies immediately that these homomorphisms are compatible with the
restriction maps, they give a canonical functorial homomorphism of B-modules
(4.2.2.1) Ψ∗ (M) ⊗B Ψ∗ (N ) → Ψ∗ (M ⊗A N )
which is not, in general, injective or surjective. If P is a third A-module, one verifies
immediately that the diagram
(4.2.2.2) Ψ∗ (M) ⊗B Ψ∗ (N ) ⊗B Ψ ∗ (P)
/ Ψ∗ (M ⊗A N ) ⊗B Ψ∗ (P)
²
/ Ψ∗ (M ⊗A N ⊗A P)
²
Ψ∗ (M) ⊗B Ψ∗ (N ⊗A P)
commutes.
(4.2.3) Let M and N be A-modules. For every open U ⊆ Y , one has by definition
Γ(ψ −1 (U ), HomA (M, N )) = HomA|V (M|V , N|V ) and one can claim V = ψ −1 (U ). The
map u 7→ Ψ∗ (u) is a homomorphism
HomA|V (M|V , N|V ) → HomB|U (Ψ∗ (M)|U , Ψ∗ (N )|U )
of Γ(U, B)-modules. These homomorphisms, being compatible with the restriction
maps, define a canonical functorial homomorphism of B-modules
(4.2.3.1) Ψ∗ (HomA (M, N )) → HomB (Ψ∗ (M), Ψ∗ (N )).
(4.2.4) If C is an A-algebra then the composition
Ψ∗ (C) ⊗B Ψ∗ (C) → Ψ∗ (C ⊗A C) → Ψ∗ (C)
gives Ψ∗ (C) the structure of a B-algebra as a result of (4.2.2.2). Similarly, one sees
that, if M is a C-module, then Ψ∗ (M) is (canonically) a Ψ∗ (C)-module.
(4.2.5) Consider, in particular, the case where X is a closed subspace of Y and ψ is the
canonical injection j : X → Y . If B 0 = B|X = j ∗ (B) is the restriction of the sheaf of rings
B to X then an A-module M can be regarded as a B 0 -module via the homomorphism
θ] : B 0 → A. Ψ∗ (M) is then a B-module which induces M on X and 0 elsewhere. If
N is a second A-module then Ψ∗ (M) ⊗B Ψ∗ (M) is identified with Ψ∗ (M ⊗B0 N ) and
HomB (Ψ∗ (M), Ψ∗ (N )) with Ψ∗ (HomB0 (M, N )).
(4.2.6) Let (Z, C) be a third ringed space, Ψ0 = (ψ 0 , θ0 ) a morphism (Y, B) → (Z, C). If
Ψ00 is the composition Ψ0 ◦ Ψ then clearly Ψ00∗ = Ψ0∗ ◦ Ψ∗ .
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4.3. The Inverse Image of a B-Module
(4.3.1) Under the hypotheses of (4.2.1), let G be a B-module and ψ ∗ (G) its inverse
image (3.7.1), which is a sheaf of abelian groups on X. The definition of the sections of
ψ ∗ (G) and ψ ∗ (B) (3.7.1) shows that ψ ∗ (G) is canonically endowed with the structure of
a ψ ∗ (B)-module. In addition, the homomorphism θ] : ψ ∗ (B) → A provides A with the
structure of a ψ ∗ (B)-module, which is denoted A[θ] when necessary to avoid confusion.
The tensor product ψ ∗ (G) ⊗ψ∗ (B) A[θ] is then endowed with the structure of an A-module.
We will say that this module is the inverse image of G by the morphism Ψ, denoted
Ψ∗ (G).
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