Lecture Notes on Algebraic Geometry

Lecture Notes on Algebraic Geometry
Jialong Zhang∗
January 11, 2025
Contents
1 Sheaves
Proposition 1.1. (a) If 0→ F0→ F → F00 →0is an exact sequence of sheaves, and if F0is
asque, then for any open set U, the sequence 0→ F0(U)→ F(U)→ F00(U)→0is also exact.
(b) If 0→ F0→ F → F00 →0is an exact sequence of sheaves, and if F0and Fare asque,
then so is F00.
Proof. (a)
(b) For any open sets V⊂U, and for each s∈ F00(V), since F0is asque, there exists t∈ F(V)
whose image in F00(V)is s. Since Fis asque, tlifts to an element t0∈ F(U). Let s0be the image
of t0in F00(U). Then s0|V=s. Therefore F00 is asque. See the following commutative diagram
with exact rows and columns:
0F0(U)F(U)F00(U) 0
0F0(V)F(V)F00(V) 0
0 0 0
Example 1.2. A constant sheaf on an irreducible topological space is asque.
Example 1.3 (Sheaf of discontinous sections).Let Fbe any sheaf on X. We dene a new sheaf
G, called the sheaf of discontinuous sections of Fas follows. For each open set U⊂X,G(U)
is the set of maps s:U→ ∪P∈UFP, such that for each P∈U,sP∈ FP. Then Gis a asque sheaf
and there is a natural injective morphism of Fto G.
∗[email protected]
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Remark 1.4. Proposition ?? and Example ?? show that in the category of sheaves of abelian groups
on a topological space X, the full subcategory of asque sheaves forms a right adapted class for the
left exact functor Γ(X, −).
Example 1.5 (Skyscraper Sheaves).Let Xbe a topological space and Pa point of X. Let iP(A)
be the sheaf on Xsending an open set Uto Aif P∈Uand to zero if P /∈U. Then it is direct
image of Aunder the inclusion {P} → X, where Ais the constant sheaf on {P}. Therefore its
stalk at Q∈ {P}is Aand is 0 elsewhere.
We now introduce a classical exact sequence for any sheaf. Let Xbe a topological space and
Za closed subset. Let i:Z→Xbe the inclusion, let U=X−Zbe the complementary open
subset, and let j:U→Xbe its inclusion. Let Fbe a sheaf on Z. The stalk (i∗F)Pof the direct
image sheaf on Xis FPif P∈Zand is 0 if P /∈Z. Hence we call i∗Fths sheaf obtained by
extending Fby zero outside Z. By abuse of notation we will sometimes write Finstead of i∗F,
and say “consider Fas a sheaf on X”, when we mean “consider i∗F”.
Now let Fbe a sheaf on U. Let j!Fbe the sheaf on Xassociated to the presheaf V7→ F(V)if
V⊂U,V7→ 0otherwise. The stalk (j!F)Pequal to FPif P∈Uand 0 if P /∈U. For any open
set V⊂U, by denition of sheacation, it is easy to verify
(j!F)(V) = F(V).
We call j!Fthe sheaf obtained by extending Fby zero outside U.
Proposition 1.6. j!Fis the only sheaf on Xwhose restriction on Uis Fsuch that the stalk at P
is FPif P∈U, and 0 if P /∈U.
Proof. Suppose Gis another sheaf with this property. Then there is a natural map from the presheaf
V7 F(V)if V⊂Uand V7→ 0otherwise to G, which induces a morphism of sheaves j!F → G. It is
clear that the restriction to Uis identity. Therefore the morphism of sheaves is identity and hence
j!F=G.
Proposition 1.7. Let Fbe a sheaf on X. There is an exact sequence of sheaves on X,
0→j!(F|U)→ F → i∗(F|Z)→0.
We now dene subsheaf with supports. Let Zbe a closed subset of X, and let Fbe a sheaf on
X. We dene ΓZ(X, F)to be the subgroup of Γ(X, F)consisting of all sections whose support is
contained in Z.
Proposition 1.8. The presheaf V7→ ΓZ∩V(V, F|V)is a sheaf. It is called the subsheaf of F
with supports in Z, and is denoted by H0
Z(F).
Proof. Trivial.
Proposition 1.9. Let U=X−Z, and let j:U→Xbe the inclusion. There is an exact sequence
of sheaves on X
0→H0
Z(F)→ F → j∗(F|U).
Furthermore, if Fis asque, the map F → j∗(F|U)is surjective.
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Proof. For P∈U, we have an exact sequence
0→0 = (H0
Z(F))P→ FP→(j∗(F|U))P.
For P∈Z, and for any (V, s)∈ FPwhose image at (j∗(F|U))Pis zero, then there is an open
neighborhood Wof Psuch that s|W= 0 in (j∗F|U)(W), i.e., s|U∩W= 0. Therefore the support of
s|Wis contained in Zand thus s|W∈(H0
Z(F))(W). Hence we have an exact sequence
0→(H0
Z(F))P→ FP→(j∗(F|U))P.
We now see how to glue sheaves.
Proposition 1.10. Let Xbe a topological space, let U={Ui}be an open cover of X, and suppose
we are given for each ia sheaf Fion Ui, and for each i, j an isomorphism ϕij :Fi|Uij → Fj|Uij
such that
(1) for each i,ϕii = Id,
(2) for each i, j, k,ϕik =ϕjk ◦ϕij on Uijk.
Then there exists a unique sheaf Fon X, together with isomorphisms ψi:F|Ui→ Fisuch that
for each i, j,ϕj=ϕij ◦ψion Uij .
At last, we give several examples of sheaves on varieties. Let Xbe a variety over an algebraically
closed eld k. Let OXbe the sheaf of regular functions on X.
Let Ybe a closed subset of X. For each open set U⊂X, let IY(U)be the ideal in the ring
OX(U)consisting of those regular functions which vanish at all points of Y∩U.
Proposition 1.11. The presheaf U7→ IY(U)is a sheaf on X, called the sheaf of ideals IYof
Y.
Proof. Trivial.
Proposition 1.12. If Yis a subvariety, i.e., an irreducible locally closed subset of X, then the
quotient sheaf OX/IYis isomorphic to i∗OY, where i:Y→Xis the inclusion, and OYis the
sheaf of regular functions on Y.
Proof.
2 Schemes
We show that the notion of schemes does in fact generalize the notion of variety. It is not quite
true that a variety is a scheme: the underlying topological space of a scheme such as A1
kor A2 khas
more points than the corresponding variety. However, we will show that there is a natural way of
adding generic points for every irreducible subset of a variety so that the variety becomes a scheme.
Proposition 2.1. Let kbe an algebraically closed eld. There is a natural fully faithful functor
t:Var(k)→Sch(k)from the category of varieties over kto schemes over k. For any variety V,
its topological space is homemorphic to the set of closed points of the underlying topological space
t(V), and its sheaf of regular functions is obtained by restricting the structure sheaf of t(V)via this
homeomorphism.
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Since the proof is complicated, we divide it into several parts.
To begin with, for any topological space X, we let t(X)be the set of all nonempty irreducible
closed subset of X. If Yis a closed subset of X, then t(Y)⊂t(X). Furthermore, t(Y1)∪t(Y2) =
t(Y1∪Y2)and ∩it(Yi) = t(∩iYi). So we can dene a topology on t(X)by taking as closed sets
the subset of the form t(Y). Let f:X1→X2be a continuous map of topological spaces. We let
t(f) : t(X1)→t(X2)be the map sending an irreducible closed subset of X1to the closure of its
image in X2.It is easy to verify that t(f)is continuous. Therefore we have a fnuctor tfrom the
category of topological spaces to the category of topological spaces. Furthermore, let α:X→t(X)
be the map P7→ {P}. Note that α−1(t(Y)) = Y. Therefore αis continuous and its induces a
bijection between the set of closed subsets of t(X)and the set of closed subsets of X.
Now for any variety Vover the algebraically closed eld k, let OVbe the sheaf of regular
functions on V. We rst assume that Vis an ane variety. Let Abe its coordinate ring and
X= Spec A. We dene a morphism of ringed spaces β: (V, OV)→Xas follows: for each point
P∈V, let β(P) = mP. Then βis continuous. For any open set Uof Spec A, and for any section
s∈ OX(U), let β♭
U(s) : β−1(U)→kbe the function sending a point P∈β−1(U)to the image of
s(mP)in kunder the map AmP→AmP/mP∼
=k. It is easy to check that β♭
U(s)is a regular function
on β−1(U). Therefore we get a morphism of ringed spaces. Note that βis homemorphism from V
to the set of all closed points of X.
Now we dene γ: (t(V), α∗OV)→Xas follows: for each irreducible closed subset Zof V,
Z=V(p)for some prime ideal pof Aand we let γ(Z) = p. Then γis a homeomorphism.
Proposition 2.2. Let Vbe a projective variety with homogeneous coordinate ring S. Then t(V)∼
=
Proj S.
Proposition 2.3. (1) Let Vbe a variety over the algebraically closed eld k. A point P∈t(V)is
a closed point if and only if its residue eld is k.
(2) If f:X→Yis a morphism of schemes over k, and if P∈Xis a point with residue eld
k, then f(P)∈Yalso has residue eld k.
(3) If V, W are any two varieties over k, then the natural map
HomVar(V, W )→HomSch/k(t(V), t(W))
is bijective.
Denition 2.4. A scheme (X, OX)is reduced if for every open set U⊂X, the ring OX(U)has
no nilpotent elements.
Proposition 2.5. (1) A scheme is reduced if and only if all its stalk has no nilpotent elements.
(2) Let (X, OX)be a scheme. Let (OX)red be the sheaf associated to the presheaf U7→ OX(U)red,
where for any ring A, we denote by Ared the quotient of Aby its ideal of nilpotent elements. Then
(X, (OX)red)is a scheme. We call it the reduced scheme associated to X, and denote it by
Xred. There is a morphism of schemes Xred →X, which is a homeomorphism on the underlying
topological spaces.
(3) Let f:X→Ybe a morphism of schemes, and assume that Xis reduced. There is a unique
morphism g:X→Yred such that fis obtained by composing gwith the natural map Yred →Y.
Proof. (1) is trivial.
(2) For the rst statement, we can assume X= Spec Ais ane and we claim that (X, (OX)red)∼
=
Spec A/√0. It is easy to verify that for any x=∈X,((OX)red)x=OX,x/√0. Therefore
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(X, (OX)red)is a locally ringed space. The ring map A→A/√0induces a homeomorphism
Spec A/√0→Spec A. We dene a morphism of sheaves f♯: (OX)red →f∗OSpec A/√0as follows:
for any principal
Denition 2.6. Let Xbe a scheme. For any x∈X, we dene the residue eld of xon Xto be
the eld k(x) = OX,x/mx.
Proposition 2.7. Let Kbe any eld. To give a morphism of Spec Kto Xis equivalent to give a
point x∈Xand an inclusion map k(x)→K.
Proof. For any morphism f: Spec K→X, let x=f(0). Then we have a local ring map OX,x →K.
Therefore we have a map of elds k(x)→K. On the other hand, for any point x∈Xand any
morphism k(x)→K, choose an ane open neigborhood U= Spec Aof xand let Spec K→U
be the map induced by Γ(U, OX)→ OX,x →k(x)→K. Let Spec K→Xbe the composition
Spec K→U→X.
Denition 2.8. Let Xbe a scheme. For any point x∈X, we dene the Zariski tangent space
Txto Xat xto be the dual of the k(x)-vector space mx/m2
x.
Proposition 2.9. Assume Xis a scheme over a eld k, and let k[ε]/ε2be the ring of dual
numbers over k. To give a k-morphism of Spec k[ε]/ε2to Xis equivalent to giving a point x∈X,
rational over k(i.e., such that k(x) = k), and an element of Tx. In particular, for any k-rational
point x, there is a bijection between TxXand the set
{ϕ∈Homk(Spec k[ε]/ε2, X)|ϕ|Spec k=x}.
Proof. Note that Spec k[ε]/ε2is a singleton. For any k-morphism ϕ: Spec k[ε]/ε2→X, let xbe the
image of ϕ. Let f: Spec k→Spec k[ε]/ε2be the morphism induced by the ring map k[ε]/ε2→k,
ε7→ 0. Then xis the image of ϕ◦fand hence xis a k-rational point. Note that m(ε)/m2
(ε)∼
=k.
Therefore ϕinduces a morphism mx/m2
x→m(ε)/m2
(ε)∼
=k.
On the other hand, given any k-rational point xand ϕ∈TxX, choose an ane open neighbor-
hood U= Spec Aof x. Let A→k[ε]/ε2be the composition
A→mx→mx/m2
x→k→k[ε]/ε2,
which induces a morphism Spec k[ε]/ε2→U→X.
If Xis a topological space, and Zan irreducible closed subset of X, a generic point for Zis
a point ξsuch that Z={ξ}.
Proposition 2.10. If Xis a scheme, then every irreducible closed subset has a unique generic
point.
Proof. Let Zbe an irreducible closed subset of X. Choose an ane open set U= Spec Aof X
such that Z∩U6=∅. Then Z∩Uis an irreducible closed subset of U. Therefore Z∩U=V(p)for
some prime ideal pof A. Let x∈Ube the corresponding point. Let {x}be the closure of {x}in
X. Then Z∩U={x} ∩ U. Considering the closure in X, we obtain
Z=Z∩U={x} ∩ U={x}.
We now prove the uniqueness. Suppose x, y ∈Xare two points such that {x}={y}. Choose an
ane open set U= Spec Asuch that U∩ {x} 6=∅. Then x, y ∈U. Let p,qbe the corresponding
prime ideals. Since {x} ∩ U=V(p) = {y} ∩ U=V(q),p=qand hence x=y.
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