Download 3. Sheaves of groups and rings.

ALGBOOK
SHEAVES 3.1
19. januar 2002
3. Sheaves of groups and rings.
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(3.1) Definition. Let X be a topological space and let B be a basis for the topology.
Moreover let !F! and !A! be presheaves defined on B. We say that F takes values in
groups, or is a presheaf of groups, on B if for every subset U of X belonging to B we
have that F(U ) is a group, and for every inclusion U ⊆ V of subsets of X belonging
to B the map (ρF )VU : F(V ) → F(U ) is a group homomorphism. Similarly we say
that A takes values in rings, or is a presheaf of rings on B, if A(U ) is a ring, and
(ρA )VU is a homomorphism of rings for all inclusions U ⊆ V of sets belonging to B.
When F and A are sheaves on B we say that F is a sheaf of groups, respectively
that A is a sheaf of rings, on B if they are presheaves of groups, respectively rings,
when considered as presheaves on B.
(3.2) Remark. It follows from Remark (?) that for a sheaf of groups F we have
that F(∅) = (0).
(3.3) Stalks. Let F and A be presheaves of groups, respectively rings, defined on
a basis B of the topological space X. For every point x in X the stalk Fx has a
natural structure as a group in such a way that the map (ρF )U
x : F(U ) → Fx is a
group homomorphism, and Ax has a natural structure as a ring in such a way that
the map (ρA )U
x is a ring homomorphism for all neighbourhoods U of x belonging to
B. In order to define the addition on Fx , and the multiplication on Ax we let sx
and tx in Fx be the classes of pairs (V, s) and (W, t) where V and W belong to B,
and s ∈ F(V ) and t ∈ F(W ). Then there is a neighbourhood U of x belonging to
B contained in V ∩ W . We define the sum sx + tx of sx and tx as the class in Fx
of the pair (U, (ρF )VU (s) + (ρF )W
U (t)). It is clear that the definition is independent of
the choice of the representatives (V, s) and (W, t) of the classes sx and tx and of U .
Moreover it is clear that Fx with the addition becomes a group in such a way that
(ρF )U
x is a homomorphism of groups. When s ∈ A(V ) and t ∈ A(W ) we define the
product sx tx of sx and tx as the class of the pair (U, (ρA )VU (s)(ρA )W
U (t)) in Ax . It
is clear that the definition is independent of the choice of the representatives (V, s),
(W, t) of sx and tx , and of U . Moreover it is clear that Ax with the given addition
and multiplication becomes a ring in such a way that (ρA )U
x is a homomorphism of
rings.
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n
(3.4) Definition. Let F and !G! be presheaves of groups on a basis B of the
topological space X. A homomorphism u : F → G of presheaves is a homomorphism
of presheaves of groups on B if for every subset U of X belonging to B we have
that uU : F(U ) → G(U ) is a group homomorphism. Let A and !B! be presheaves of
rings defined on B. We have that a homomorphism ϕ : A → B of presheaves is a
homomorphism of presheaves of rings on B if for every subset U of X belonging to
B we have that ϕU : A(U ) → B(U ) is a homomorphism of rings.
When A, B, F and G are sheaves of rings respectively groups we say that the
homomorphisms are homomorphisms of sheaves of groups respectively rings. A sheaf
of rings B together with a homomorphism ϕ : A → B we call an A-algebra.
sheaves3
ALGBOOK
SHEAVES 3.2
19. januar 2002
(3.5) Remark. We easily see that if u and ϕ are homomorphisms of groups, respectively rings, then the map of stalks ux : Fx → Gx , respectively ϕx : Ax → Bx ,
are homomorphisms of groups, respectively rings for all x ∈ X.
(3.6) The direct image. Let ψ : X → Y be a continuous map of topological
spaces and F a presheaf of groups on X. The direct image ψ∗ (F) is then a presheaf
of groups on Y . In fact, for every inclusion V ⊆ W of open subsets of Y we have
ψ −1 (W )
that ψ∗ (F)(V ) = F(ψ −1 (V )) is a group, and (ρF )ψ−1 (V ) is a group homomorphism.
ψ −1 (W )
Hence (ρψ∗ (F ) )W
V = (ρF )ψ −1 (V ) is a group homomorphism. When A is a presheaf of
rings we have that the direct image ψ∗ (A) is a presheaf of rings. In fact ψ∗ (A)(V ) =
ψ −1 (W )
A(ψ −1 (V )) is a ring and (ρψ∗ (A) )W
V = (ρA )ψ −1 (V ) is a homomorphism of rings for
all open subsets V , and all inclusions V ⊆ W of open subsets of Y .
For every point x of X the map (ψF )x : ψ∗ (F)ψ(x) → Fx is a homomorphism of
groups, and the map (ψA )x : ψ∗ (A)ψ(x) → Ax is a homomorphism of rings.
When u : F → G is a homomorphism of presheaves of groups on X we have that
ψ∗ (u) : ψ∗ (F) → ψ∗ (G) is a homomorphism of presheaves of groups. In fact, for
every open subset U of V , the map ψ∗ (u)U comes from a homomorphism of groups
uψ−1 (U ) : F(ψ −1 (U )) → G(ψ −1 (U )).
When ϕ : A → B is a homomorphism of presheaves of rings we have correspondingly that ψ∗ (ϕ) : ψ∗ (A) → ψ∗ (B) is a homomorphism of presheaves of rings.
(3.7) The inverse image. Let ψ : X → Y be a continuous map of topological
spaces, and let B be a basis for Y . Moreover let G be a presheaf of groups on B
and B a presheaf of rings on B. The inverse image ψ ∗ (G) of G is a sheaf of groups
on X and ψ ∗ (B) is a sheaf of rings on X.QIn fact, we shall verify that for every
a group structure on
open subset U of X the group structure on x∈U Gψ(x) induces
Q
∗
ψ (G)(U ). Let (sψ(x) )x∈U and (tψ(x) )x∈U be elements of x∈U Gψ(x) that belong to
the subset ψ ∗ (G)(U ). Then there is an open nieghbourhood Vψ(x) of ψ(x) belonging
to B and s(x), t(x) in G(Vψ(x) ) such that for all y in a neighbourhood of x contained
in U ∩ ψ −1 (Vψ(x) ) we have that sψ(x) = s(x)y and tψ(y) = t(x)y . We have that
s(x) + t(x) ∈ G(Vψ(x) ) and (s(x) + t(x))y = s(x)y + t(x)y = sψ(y) + tψ(y) . Hence
(sψ(x) )x∈U + (tψ(x) )x∈U = (sψ(x) + tψ(x) )x∈U = (s(x)x + t(x)x ) = (s(x) + t(x))x .
Hence we have that (sψ(x) )x∈U + (tψ(x) )x∈U is contained in ψ ∗ (G)(U ) as we wanted
to verify.
Moreover,
U ⊆ V of open subsets of X we have that the
Q for every inclusion
Q
projection x∈V Gψ(x) → x∈U Gψ(x) is a group homomorphism that induces a group
homomorphism
(ρψ∗ (G) )VU : ψ ∗ (G)(V ) → ψ ∗ (G)(U ). Similarly the ring
Q structure on
Q
∗
B
induces
a
ring
structure
on
ψ
(B)(U
),
and
the
projection
x∈V Bψ(x) →
Qx∈U ψ(x)
V
x∈U Bψ(x) is a ring homomorphism inducing a ring homomorphism (ρψ ∗ (B) )U :
ψ ∗ (B)(V ) → ψ ∗ (B)(U ), for every inclusion U ⊆ V of sets belonging to B. For every
x ∈ X we have that the map (ιG )x : Gψ(x) → ψ ∗ (G)x is a group homomorphism, and
the map (ιB )x : Bψ(x) → ψ ∗ (B)x is a ring homomorphism.
ALGBOOK
SHEAVES 3.3
19. januar 2002
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(3.8) Adjunction. Let F and A be sheaves of groups, respectively rings, on X, and
let G and B be presheaves of groups, respectively rings, on Y . The adjunction maps
ρG : G → ψ∗ (ψ ∗ (G)) and σF : ψ ∗ (ψ∗ (F)) → F are both homomorhism of presheaves
of groups, and ρB and σA are homomorphisms of presheaves of rings. Consequently
the bijections HomX (ψ ∗ (G), F) → HomY (G, ψ∗ (F)) of (?) and HomX (ψ ∗ (B), A) →
HomY (B, ψ∗ (A)) induce bijections between the subset consisting of homomorphisms
of presheaves of groups, respectively of presheaves of rings.
(3.9) Definition. A ringed space is a pair consisting of a topological space X and
a sheaf of rings A. We shall often denote a ringed space by !(X, OX )! where OX
is the sheaf of rings on the topological space X. The stalk of OX at a point x of
X we denote by OX,x . A homomorphism (ψ, θ) : (X, A) → (Y, B) of ringed spaces
consists of a continuous map ψ : X → Y of topological spaces and a homomorphism
θ : B → ψ∗ (A) of sheaves of rings. We say that (X, A) is a local ringed space if Ax
is a local ring for all x ∈ X. A local homomorphism of local ringed spaces (ψ, θ) :
(X, A) → (Y, B) is a homomorphism of ringed spaces such that (θ)]x : ψ ∗ (B)x → Ax
maps the maximal ideal in ψ ∗ (B)x = Bψ(x) to the maximal ideal in Ax for all x ∈ X.
(3.10) Remark. The ringed spaces with morphism form a category, as does the
locally ringed spaces with local homomorphisms.
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(3.17) Exercises.
1. Let X be a topological space and G an abelian group. For every non-empty open
subset U of X we let F(U ) = GU be all maps U → G. Let F(∅) = {0}. For every
inclusion U ⊆ V of open subsets of X we define ρVU : F(V ) → F(U ) to be the map
that takes a section s : V → G to its restriction s|U : U → G. Show that F with the
maps ρVU has a natural structure as a sheaf of groups.
2. Let X be a topological space and A a ring. For every non-empty open subset U of
X we let A(U ) = AU be all maps U → A. For every inclusion U ⊆ V of open subsets
of X we define ρVU : A(V ) → A(U ) to be the map that takes a section s : V → A to
its restriction s|U : U → A. Let A(∅) = {∅}. Show that A with the maps ρVU has a
natural structure as a sheaf of rings.
3. Let X = {x0 , x1 } have the topology with open sets ∅, X, {x0 }. We let A(x) = Z(p) ,
A(x0 ) = Q, and F(∅) = {0}. Moreover, we let ρX
x0 : Z(p) → Q be the inclusion map.
(1) Show that A is a sheaf and that the pair (X, A) is a local ringed space.
(2) Let Y = {y0 } and let B be the sheaf B(Y ) = Q on Y . Moreover let ψ : Y → X
be the map that takes y0 to x0 . Show that there is a unique homomorphism
of sheaves of algebras A → ψ∗ (B) which is the identity on Q over {x0 }, and
the inclusion Z(p) → Q on X.
(3) Show that the map of part (2) is a local map of local ringed spaces.
(4) Let ψ : Y → X be the map that takes y0 to x1 . Show that there is a map of
sheaves of rings A → ψ∗ (B) which is the inclusion Z(p) → Q on X and the
zero map Q → {0} on {x0 }. Show that this is not a local homomorphism of
local ringed spaces.
ALGBOOK
SHEAVES 3.4
19. januar 2002
4. Let X be a topological space and let F be a sheaf of groups on X. Show that for
every open subset U of X, and every section s ∈ F(U ) we have that the set consisting
of x ∈ U such that sx = 0 is open in X.