AG for NT First Week 2
Chris Williams
Recap Sheaves
Let
X
•
be a topological space.
A presheaf
F
is some collection
F (U )
of abelian group (rings, modules) for each open sets
U ⊆ X,
with some compatibility.
•
A presheaf
F
•
Dene the stalk
•
Local conditions, we have some nice local to global properties.
is a sheaf if it satises additional local conditions.
Fx
for each
x ∈ X.
(i.e., a morphism of sheaf is an
isomorphism if and only if it is an isomorphism of stalks)
Example.
A be a group/ring and dene the constant presheaf to be the presheaf
F (U ) = A for each open U ⊆ X . This is not a sheaf. Let X = {x1 , x2 } with discrete topology, and A = Z
. Then F ({x1 }) = Z, F ({x2 }) = Z. Take 2 ∈ F ({x1 }) and 3 ∈ F ({x2 }). There does not exists s ∈ F (X)
such that s|{x1 } = 2 and s|{x2 } = 3.
L
We x this by instead setting F (U ) =
connect component of U A.
(The constant sheaf ) Let
Schemes
Motivation:
some
K[X]
Let
X
to some irreducible ane variety over an algebraically closed eld
a regular function eld. Hilbert's Nullstellensatz
⇒
K.
This gives
there exists a bijection between
{K−pts of X} ↔ {maximal ideal in K[X]}
by
p 7→ mp = {f (p) = 0}
Denition.
X := Spec(A) = {p ( A : p prime}. We give X
V (f ) = {p ∈ X : hf i ⊂ p} (plus the intersections of some
collection of V (f )), giving rise to the open sets D(f ) = X \ V (f ). The set D(f ) form a basis for the open
sets of X .
h i
h i
1
1
We put a sheaf O of rings on Spec(A). For each D(f ) we dene O(D(f )) = A
. (Here A
f
f =
n
o
1 q
a
f n : a ∈ A, n ∈ Z , e.g., A = Z then Z 5 = 5n : q ∈ Z ). This actually denes a B -sheaf, where B is
the basis consisting of all the D(f ). We can then use the lemma from last week to extend this to a sheaf
O on the whole of X .
Let
A
be a (commutative). Dene a space
the Zariski topology, i.e., the closed sets are
Proposition (Q-L page 42 proposition 3.1 ).
O
is a sheaf.
1
Proof.
B -sheaf.
p
n
First, need compatibility maps. If D(g) ⊂ D(f ), then g ∈
hf
h i
h ii = {a ∈ A : ∃n ∈ N s.t. a ∈ hf i}.
m = af for some a ∈ A, m ∈ N. Dened a map A 1 → A 1 by bf −n 7→ ban g −mn . (Check that
Hence g
f
g
if D(f ) = D(g) then this map is an isomorphism)
So now let {Ui } be a covering of X .
Claim.
Let
By last week lemma, we just need to show that
O
is a
There is a nite subcover
Ui = D(fi ),
P
P
X = ∪D(fi )P
⇒ ∩V (fi ) = ∅P
. Hence V (
hfi i) = ∅ ⇒
hfi i = A.
I such that 1 = I ai fi . Then I (fi ) = A ⇒ X = ∪I D(fi ).
then
exists some nite set
Hence there
To prove the local conditions:
4. Suppose s ∈ O(X) = O(D(1)) = A, with s|Ui = 0 for all i. We want to show s = 0. For each i ∈ I ,
P
mi
s|Ui = 0 ⇒ ∃mP
= 0. Then as ∪D(fim ) P
= ∪D(fi ) = X . In particular (fim ) = A, we
i ∈ Z such that sfi
m
m
m
can write 1 =
i∈I ai fi . So sfi = 0 for all i ∈ I . Hence
I sai fi = s = 0.
5. Let si ∈ O(D(fi )) such that si |Ui ∩Uj = sj |Ui ∩Uj . We want some s ∈ O(X) = A such that s|Ui = si .
Now, D(fi ) ∩ D(fj ) = D(fi fj ) so si |Ui ∩Uj = sj |Ui ∩Uj means that there exists some r ∈ Z such that
h i
(si − sj )(fi fj )r = 0. Each si ∈ A f1i , so si = ci fi−mi for some mi ∈ N. Take m = maxi∈I {mi },
si = bi fi−m for some bi ∈ A. Combined with the above, we get (bi fi−m − bj fj−m )fir fjr = 0 ⇒
P
P
bi fjm+r fir = bj fim+r fjr . We still have some ai ∈ A such that 1 = ai fim+r . Dene s := i∈I ai bi fir , then
P
P
sfjm+r = I ai bi fir fjm+r = I ai bj fjr fim+r = bi fjr = s|D(fi )
then
Fact. The stalk of O at p ∈ X is the local ring Ap .
Denition. An ane scheme is topological space
isomorphic to
(Spec(A), O)
for some ring
X
with a sheaf of rings
OX ,
such that
#
is a pair (f, f ) such that f : X → Y is continuous and
#
that the map fx are local homomorphism.)
Let
Let
(Ane line) Let
•
A = k[x], where k
: OY → f∗ OX
(X, OX ) = (Y, OY )
is a morphism of sheaves, such
is a eld. Then dene the ane line over
k ,o1k = Spec(k[x]).
A = k[x1 , . . . , xn ] we get the ane n-space nk . O(D(fi )) = fgn : g ∈ k[x1 , . . . , xn ]
i
f ∈ k[x1 , . . . , xn ] be irreducible and let A := k[x1 , . . . , xn ]/ hf i then Spec(A) correspond
• A
Spec(A)
={
V (f ).
prime ideal}
gives a sheaf.
Ane scheme, something that is isomorphism to
Exercise.
to
a commutative ring
• U = D(fi ) ⊂ Spec(A), O(U ) = A[ f1 ],
•
f#
n
Summary.
is
A.
Where isomorphism is an isomorphism of Ringed topological space. A morphism of
Example.
(X, OX )
(SpecA, O).
( X be a topological space, p ∈ X , A an abelian
A p∈U
ip (A)(U ) =
.
0 p∈
/U
Show that ip (A) is a sheaf.
(
A q=p
Show that ip (A)q =
0 q 6= p
Show also that ip (A) = i∗ (A) , A constant sheaf i : {p} → X
Let
2
group.
Dene a sheaf
inclusion.
ip (A)
as follows: