PMATH 945: Modern Algebraic Geometry

Ben Webster

Estimated study time: 1 hr 18 min

Table of contents

Sources and References

Hartshorne, Robin. Algebraic Geometry. Graduate Texts in Mathematics 52, Springer, 1977. (Primary reference; Chapters II–IV.)
Vakil, Ravi. The Rising Sea: Foundations of Algebraic Geometry. Publicly available draft, math.stanford.edu/~vakil/216blog/. (Comprehensive modern treatment with excellent motivation.)
Mumford, David. The Red Book of Varieties and Schemes. Lecture Notes in Mathematics 1358, Springer, 1999.
Eisenbud, David and Joe Harris. The Geometry of Schemes. Graduate Texts in Mathematics 197, Springer, 2000.
Sernesi, Edoardo. Deformations of Algebraic Schemes. Grundlehren der Mathematischen Wissenschaften 334, Springer, 2006.
Atiyah, M.F. and I.G. Macdonald. Introduction to Commutative Algebra. Addison-Wesley, 1969.

Chapter 1: Sheaves and Ringed Spaces

1.1 Presheaves and Sheaves

The language of sheaves is the foundational toolkit of modern algebraic geometry. A sheaf encodes the idea that local data can be consistently patched together to yield global data, and that global data is completely determined by its local restrictions. This principle — locality of data and locality of agreement — underpins everything from the structure sheaf of a scheme to cohomological machinery.

Definition (Presheaf). Let \( X \) be a topological space. A presheaf of abelian groups \( \mathcal{F} \) on \( X \) consists of:

For each open set \( U \subseteq X \), an abelian group \( \mathcal{F}(U) \), whose elements are called sections of \( \mathcal{F} \) over \( U \).
For each inclusion \( V \subseteq U \) of open sets, a restriction homomorphism \( \rho_{UV} : \mathcal{F}(U) \to \mathcal{F}(V) \), satisfying \( \rho_{UU} = \mathrm{id} \) and \( \rho_{UW} = \rho_{VW} \circ \rho_{UV} \) whenever \( W \subseteq V \subseteq U \).

We write \( s|_V \) for \( \rho_{UV}(s) \) when \( s \in \mathcal{F}(U) \). The group \( \mathcal{F}(\emptyset) \) is conventionally taken to be the trivial group.

One can analogously define presheaves of rings, sets, modules over a fixed ring, and so on. The presheaf axioms amount to saying that \( \mathcal{F} \) is a contravariant functor from the category \( \mathbf{Open}(X) \) (objects: open sets of \( X \); morphisms: inclusions) to the target category.

A presheaf is not, by itself, a satisfactory geometric object: it may fail to detect when local sections agree, and it may fail to guarantee that compatible local sections glue. These two conditions define a sheaf.

Definition (Sheaf). A presheaf \( \mathcal{F} \) on \( X \) is a sheaf if for every open cover \( \{U_i\} \) of an open set \( U \), the following two conditions hold:

(Identity axiom) If \( s, t \in \mathcal{F}(U) \) satisfy \( s|_{U_i} = t|_{U_i} \) for all \( i \), then \( s = t \).
(Gluing axiom) If sections \( s_i \in \mathcal{F}(U_i) \) satisfy \( s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j} \) for all \( i, j \), then there exists \( s \in \mathcal{F}(U) \) with \( s|_{U_i} = s_i \) for all \( i \).

Together, these say that \( \mathcal{F}(U) \) is the equalizer of the two maps

\[ \prod_i \mathcal{F}(U_i) \rightrightarrows \prod_{i,j} \mathcal{F}(U_i \cap U_j). \]

Example. The sheaf \( \mathcal{O}_X \) of continuous real-valued functions on a topological space \( X \) is a sheaf of rings: \( \mathcal{O}_X(U) = \{ f : U \to \mathbb{R} \mid f \text{ is continuous} \} \), with the usual pointwise ring structure. The identity and gluing axioms are immediate from the definition of continuity. Similarly, on a smooth manifold, the sheaf of smooth functions \( C^\infty_X \) is a sheaf of rings, and the sheaf of smooth \( k \)-forms is a sheaf of abelian groups.

Example (Skyscraper sheaf). Let \( x \in X \) and \( A \) an abelian group. The skyscraper sheaf at \( x \) with value \( A \) is defined by \[ \mathcal{F}(U) = \begin{cases} A & \text{if } x \in U, \\ 0 & \text{if } x \notin U. \end{cases} \]

This is a sheaf. It will later appear as the structure sheaf of a closed point.

1.2 Stalks

The stalk of a sheaf at a point captures all local information near that point.

Definition (Stalk). Let \( \mathcal{F} \) be a presheaf on \( X \) and \( x \in X \). The stalk of \( \mathcal{F} \) at \( x \) is the direct limit (colimit) \[ \mathcal{F}_x = \varinjlim_{U \ni x} \mathcal{F}(U), \]

where the colimit is taken over all open sets \( U \) containing \( x \), ordered by reverse inclusion.

Concretely, an element of \( \mathcal{F}_x \) is an equivalence class \( (U, s) \) where \( U \ni x \) is open and \( s \in \mathcal{F}(U) \), with \( (U, s) \sim (V, t) \) if there exists an open \( W \subseteq U \cap V \) with \( x \in W \) and \( s|_W = t|_W \). Such an equivalence class is called a germ of \( \mathcal{F} \) at \( x \).

Stalks are essential for checking whether a morphism of sheaves is an isomorphism: a morphism \( \phi : \mathcal{F} \to \mathcal{G} \) is an isomorphism of sheaves if and only if the induced map on stalks \( \phi_x : \mathcal{F}_x \to \mathcal{G}_x \) is an isomorphism for every \( x \in X \).

1.3 Sheafification

Not every presheaf is a sheaf, but every presheaf has a universal sheafification.

Theorem (Sheafification). For any presheaf \( \mathcal{F} \) on \( X \), there exists a sheaf \( \mathcal{F}^+ \) and a morphism \( \theta : \mathcal{F} \to \mathcal{F}^+ \) of presheaves such that for any sheaf \( \mathcal{G} \) and any morphism \( \phi : \mathcal{F} \to \mathcal{G} \) of presheaves, there is a unique morphism \( \psi : \mathcal{F}^+ \to \mathcal{G} \) with \( \phi = \psi \circ \theta \). The sheaf \( \mathcal{F}^+ \) is called the sheafification of \( \mathcal{F} \).

The sheafification can be constructed explicitly: sections of \( \mathcal{F}^+(U) \) are functions assigning to each \( x \in U \) a germ \( s_x \in \mathcal{F}_x \), subject to the condition that these germs are locally represented by actual sections of \( \mathcal{F} \). The sheafification has the same stalks as the original presheaf: \( (\mathcal{F}^+)_x \cong \mathcal{F}_x \) for all \( x \).

1.4 Morphisms of Sheaves and Exact Sequences

Definition. A morphism of presheaves (or sheaves) \( \phi : \mathcal{F} \to \mathcal{G} \) on \( X \) consists of a group homomorphism \( \phi_U : \mathcal{F}(U) \to \mathcal{G}(U) \) for each open \( U \), compatible with restrictions: \( \phi_V \circ \rho_{UV}^\mathcal{F} = \rho_{UV}^\mathcal{G} \circ \phi_U \) for all \( V \subseteq U \).

The category \( \mathbf{Sh}(X) \) of sheaves of abelian groups on \( X \) is an abelian category. The kernel of \( \phi \) is the sheaf \( U \mapsto \ker(\phi_U) \); the image and cokernel require sheafification of the corresponding presheaves. A sequence \( \mathcal{F} \to \mathcal{G} \to \mathcal{H} \) is exact in \( \mathbf{Sh}(X) \) if and only if the induced sequence of stalks is exact at every point.

1.5 Locally Ringed Spaces

Definition (Ringed Space). A ringed space is a pair \( (X, \mathcal{O}_X) \) where \( X \) is a topological space and \( \mathcal{O}_X \) is a sheaf of commutative rings on \( X \), called the structure sheaf.

Definition (Locally Ringed Space). A ringed space \( (X, \mathcal{O}_X) \) is locally ringed if for every point \( x \in X \), the stalk \( \mathcal{O}_{X,x} \) is a local ring. A morphism of locally ringed spaces \( (f, f^\#) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y) \) is a continuous map \( f : X \to Y \) together with a morphism \( f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X \) of sheaves of rings such that for each \( x \in X \), the induced ring map \( \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X,x} \) is a local ring homomorphism (i.e., it maps the maximal ideal into the maximal ideal).

Here \( f_* \mathcal{O}_X \) is the pushforward sheaf: \( (f_* \mathcal{O}_X)(V) = \mathcal{O}_X(f^{-1}(V)) \) for open \( V \subseteq Y \). The locality condition on stalk maps is what distinguishes morphisms of locally ringed spaces from mere morphisms of ringed spaces. It encodes the idea that a regular function that vanishes at a point pulls back to a function that vanishes at the preimage.

Chapter 2: Affine Schemes

2.1 The Spectrum of a Ring

Let \( A \) be a commutative ring with unity. The prime spectrum of \( A \) is the fundamental geometric object associated to \( A \).

Definition (Spec). The spectrum \( \mathrm{Spec}(A) \) is the set of all prime ideals of \( A \), equipped with the Zariski topology, in which the closed sets are of the form \[ V(\mathfrak{a}) = \{ \mathfrak{p} \in \mathrm{Spec}(A) \mid \mathfrak{a} \subseteq \mathfrak{p} \} \]

for ideals \( \mathfrak{a} \subseteq A \). The complement of \( V(\mathfrak{a}) \) is the union of open sets \( D(f) = \mathrm{Spec}(A) \setminus V(f) \) for \( f \in A \), which form a basis for the topology.

Note that \( V(A) = \emptyset \), \( V(0) = \mathrm{Spec}(A) \), \( V(\mathfrak{a}) \cup V(\mathfrak{b}) = V(\mathfrak{a} \cap \mathfrak{b}) \), and \( \bigcap_i V(\mathfrak{a}_i) = V(\sum_i \mathfrak{a}_i) \). The points of \( \mathrm{Spec}(A) \) correspond to prime ideals, not necessarily maximal: this is a crucial difference from classical algebraic geometry over algebraically closed fields, where one usually considers only maximal ideals.

Example. \( \mathrm{Spec}(\mathbb{Z}) \) has points \( (0), (2), (3), (5), (7), \ldots \) — the zero ideal (the generic point) and the prime ideals \( (p) \) for each prime \( p \). The closed points are \( (p) \) for primes \( p \), and the generic point \( (0) \) is dense. This illustrates how the spectrum interpolates between the geometric and arithmetic.

For a field \( k \), \( \mathrm{Spec}(k) \) is a single point, corresponding to the zero ideal. For \( k[x] \), the points are the irreducible polynomials and the generic point \( (0) \).

2.2 The Structure Sheaf of an Affine Scheme

Definition (Structure Sheaf). For \( A \) a commutative ring, define the structure sheaf \( \mathcal{O}_{\mathrm{Spec}(A)} \) on the basis of distinguished open sets \( \{ D(f) \}_{f \in A} \) by \[ \mathcal{O}_{\mathrm{Spec}(A)}(D(f)) = A_f = A\left[\frac{1}{f}\right], \]

the localization of \( A \) at the multiplicative set \( \{1, f, f^2, \ldots\} \). This extends uniquely to a sheaf of rings on all of \( \mathrm{Spec}(A) \).

For general open sets, \( \mathcal{O}_{\mathrm{Spec}(A)}(U) = \varinjlim_{D(f) \subseteq U} A_f \). A key property is:

Theorem. Let \( A \) be a commutative ring and \( X = \mathrm{Spec}(A) \). Then:

For any \( f \in A \), \( \mathcal{O}_X(D(f)) \cong A_f \).
The stalk at a prime \( \mathfrak{p} \in X \) is \( \mathcal{O}_{X, \mathfrak{p}} \cong A_\mathfrak{p} \), the localization at \( \mathfrak{p} \).
The global sections \( \mathcal{O}_X(X) \cong A \).

In particular, each stalk is a local ring (the local ring \( A_\mathfrak{p} \) has unique maximal ideal \( \mathfrak{p} A_\mathfrak{p} \)), so \( (X, \mathcal{O}_X) \) is a locally ringed space.

Definition (Affine Scheme). An affine scheme is a locally ringed space \( (X, \mathcal{O}_X) \) that is isomorphic (as a locally ringed space) to \( (\mathrm{Spec}(A), \mathcal{O}_{\mathrm{Spec}(A)}) \) for some commutative ring \( A \).

2.3 Morphisms of Affine Schemes and Ring Homomorphisms

The category of affine schemes is contravariantly equivalent to the category of commutative rings. This is the algebraic backbone of scheme theory.

Theorem. For commutative rings \( A \) and \( B \), there is a natural bijection \[ \mathrm{Hom}_{\mathbf{Rings}}(A, B) \xrightarrow{\sim} \mathrm{Hom}_{\mathbf{LRS}}(\mathrm{Spec}(B), \mathrm{Spec}(A)). \]

In other words, the functor \( A \mapsto \mathrm{Spec}(A) \) is a contravariant equivalence from the category of commutative rings to the category of affine schemes.

Proof sketch. Given a ring homomorphism \( \phi : A \to B \), define \( f = \mathrm{Spec}(\phi) : \mathrm{Spec}(B) \to \mathrm{Spec}(A) \) by \( f(\mathfrak{q}) = \phi^{-1}(\mathfrak{q}) \). (The preimage of a prime under a ring map is prime.) The sheaf map \( f^\# : \mathcal{O}_{\mathrm{Spec}(A)} \to f_* \mathcal{O}_{\mathrm{Spec}(B)} \) comes from \( \phi \) itself, extended by universal properties of localization. The locality condition on stalks is satisfied because \( \phi^{-1}(\mathfrak{q}) \cdot A_{\phi^{-1}(\mathfrak{q})} \) maps into \( \mathfrak{q} B_\mathfrak{q} \). Conversely, any morphism of locally ringed spaces \( \mathrm{Spec}(B) \to \mathrm{Spec}(A) \) induces a ring map \( A = \mathcal{O}(X) \to \mathcal{O}(Y) = B \) on global sections, and this is inverse to the above construction.

2.4 The Functor of Points

A powerful perspective on schemes — especially useful for functorial constructions — is the functor of points.

Definition (Functor of Points). For a scheme \( X \), its functor of points is the functor \( h_X : \mathbf{Sch}^{\mathrm{op}} \to \mathbf{Set} \) defined by \( h_X(T) = \mathrm{Hom}_{\mathbf{Sch}}(T, X) \). An element \( T \to X \) is called a \( T \)-valued point of \( X \). By Yoneda's lemma, \( X \) is determined up to isomorphism by \( h_X \).

For an affine scheme \( X = \mathrm{Spec}(A) \), we have \( h_X(\mathrm{Spec}(R)) = \mathrm{Hom}(A, R) \) for any ring \( R \). For instance, \( \mathbb{A}^n = \mathrm{Spec}(\mathbb{Z}[x_1, \ldots, x_n]) \), and an \( R \)-valued point of \( \mathbb{A}^n \) is an \( n \)-tuple \( (r_1, \ldots, r_n) \in R^n \), by the universal property of polynomial rings. This makes the functor of points extremely concrete and computation-friendly.

Remark. The functor-of-points perspective allows one to define and work with group schemes purely in terms of functors: a group scheme over \( S \) is a scheme \( G \to S \) such that \( h_G(T) \) is a group for every \( S \)-scheme \( T \), functorially in \( T \). This is the right setting for algebraic groups.

Chapter 3: Schemes

3.1 The Definition of a Scheme

Definition (Scheme). A scheme is a locally ringed space \( (X, \mathcal{O}_X) \) that is locally isomorphic to an affine scheme: there exists an open cover \( X = \bigcup_i U_i \) such that each \( (U_i, \mathcal{O}_X|_{U_i}) \) is an affine scheme.

Schemes form a category: a morphism of schemes is simply a morphism of locally ringed spaces between them. Every affine scheme is a scheme. The category of schemes contains the category of affine schemes as a full subcategory.

3.2 Open and Closed Subschemes

Definition (Open Subscheme). An open subscheme of \( X \) is an open subset \( U \subseteq X \) with the restricted structure sheaf \( \mathcal{O}_X|_U \). Any open subscheme of a scheme is a scheme.

Definition (Closed Subscheme). A closed subscheme of \( X \) is a locally ringed space \( (Z, \mathcal{O}_Z) \) together with a morphism \( i : Z \to X \) such that: (a) the underlying map of topological spaces is a homeomorphism onto a closed subset of \( X \), and (b) the map \( \mathcal{O}_X \to i_* \mathcal{O}_Z \) is surjective. On an affine piece \( \mathrm{Spec}(A) \subseteq X \), a closed subscheme corresponds to a quotient ring \( A/\mathfrak{a} \) for an ideal \( \mathfrak{a} \subseteq A \).

Closed subschemes encode not just the zero set of functions but also their multiplicities and intersections. For instance, the closed subschemes of \( \mathbb{A}^1_k = \mathrm{Spec}(k[x]) \) are in bijection with ideals of \( k[x] \), i.e., with polynomials up to unit — including non-reduced ones like \( (x^2) \), which corresponds geometrically to the “double origin.”

3.3 Gluing Constructions

Schemes can be constructed by gluing affine schemes along open subsets, much as manifolds are built by gluing open sets of \( \mathbb{R}^n \).

Example (The Projective Line \( \mathbb{P}^1_k \)). Let \( k \) be a field. Cover \( \mathbb{P}^1_k \) by two affine lines \( U_0 = \mathrm{Spec}(k[t]) \) and \( U_1 = \mathrm{Spec}(k[t^{-1}]) \). Their overlap is \( U_{01} = \mathrm{Spec}(k[t, t^{-1}]) \). The gluing isomorphism \( \phi : U_{01} \cap U_0 \xrightarrow{\sim} U_{01} \cap U_1 \) corresponds to the ring isomorphism \( k[s, s^{-1}] \to k[t, t^{-1}] \) given by \( s \mapsto t^{-1} \). The resulting scheme \( \mathbb{P}^1_k \) has underlying topological space consisting of the closed points (isomorphism classes of irreducible polynomials over \( k \), together with \( \infty \)) and a generic point.

3.4 Reduced, Integral, and Irreducible Schemes

Definition.

A scheme \( X \) is reduced if for every open \( U \subseteq X \), the ring \( \mathcal{O}_X(U) \) has no nilpotent elements.
\( X \) is irreducible if its underlying topological space is irreducible (i.e., cannot be written as a union of two proper closed subsets).
\( X \) is integral if it is both reduced and irreducible. Equivalently, \( X \) is integral if every \( \mathcal{O}_X(U) \) is an integral domain for nonempty open \( U \).

Integral schemes are the scheme-theoretic analogue of irreducible algebraic varieties. For an affine integral scheme \( \mathrm{Spec}(A) \), the ring \( A \) is a domain, and there is a unique generic point \( \eta \in X \) whose closure is all of \( X \). The residue field \( k(\eta) = \mathrm{Frac}(A) \) is the function field of \( X \).

3.5 Schemes of Finite Type and Varieties

Definition. A scheme \( X \) is of finite type over a field \( k \) (or a variety over \( k \), when also integral) if \( X \) is covered by finitely many affine open sets \( U_i = \mathrm{Spec}(A_i) \), each with \( A_i \) a finitely generated \( k \)-algebra.

The classical varieties of algebraic geometry (affine algebraic sets, projective varieties) correspond to integral schemes of finite type over an algebraically closed field \( k \), possibly with additional conditions (separated, proper). The passage to schemes allows for a much richer theory: non-algebraically closed fields, arithmetic rings like \( \mathbb{Z} \), and non-reduced schemes (with their “thickened” geometry).

Chapter 4: Projective Schemes

4.1 Graded Rings and the Proj Construction

Definition (Graded Ring). A commutative ring \( S = \bigoplus_{d \geq 0} S_d \) is called a graded ring if \( S_d \cdot S_e \subseteq S_{d+e} \) for all \( d, e \geq 0 \). Elements of \( S_d \) are called homogeneous of degree \( d \). We call \( S_+ = \bigoplus_{d > 0} S_d \) the irrelevant ideal.

Definition (\( \mathrm{Proj} \)). For a graded ring \( S \), define \( \mathrm{Proj}(S) \) as follows:

The underlying set consists of all homogeneous prime ideals \( \mathfrak{p} \subseteq S \) that do not contain \( S_+ \).
The topology: closed sets are \( V_+(\mathfrak{a}) = \{ \mathfrak{p} \in \mathrm{Proj}(S) \mid \mathfrak{a} \subseteq \mathfrak{p} \} \) for homogeneous ideals \( \mathfrak{a} \).
The structure sheaf: on the basic open set \( D_+(f) = \{ \mathfrak{p} \mid f \notin \mathfrak{p} \} \) for homogeneous \( f \in S_+ \), one sets \( \mathcal{O}(D_+(f)) = (S_f)_0 \), the degree-zero part of the localization \( S_f = S[f^{-1}] \).

The key point is that \( D_+(f) \cong \mathrm{Spec}((S_f)_0) \) as ringed spaces, making \( \mathrm{Proj}(S) \) a scheme. If \( S = A[x_0, \ldots, x_n] \) with \( A \) a ring and all \( x_i \) of degree 1, then \( \mathrm{Proj}(S) = \mathbb{P}^n_A \), projective \( n \)-space over \( A \).

4.2 Projective \( n \)-Space

Definition. Let \( k \) be a field. Projective \( n \)-space over \( k \) is \[ \mathbb{P}^n_k = \mathrm{Proj}(k[x_0, x_1, \ldots, x_n]), \]

where all generators have degree 1. It is covered by \( n+1 \) affine open sets \( U_i = D_+(x_i) \cong \mathrm{Spec}(k[x_0/x_i, \ldots, \hat{x_i}/x_i, \ldots, x_n/x_i]) \cong \mathbb{A}^n_k \).

Example. \( \mathbb{P}^2_k \) has three affine charts \( U_0, U_1, U_2 \). On \( U_0 \), the coordinates are \( u = x_1/x_0, v = x_2/x_0 \). On \( U_1 \), the coordinates are \( x_0/x_1, x_2/x_1 \). On the overlap \( U_0 \cap U_1 \), we have \( x_0/x_1 = 1/u \), and the transition functions are as expected from classical projective geometry.

4.3 Closed Subschemes and Homogeneous Ideals

For a projective scheme \( \mathbb{P}^n_k \), closed subschemes correspond to homogeneous ideals.

Definition. A projective variety over \( k \) is a closed integral subscheme of \( \mathbb{P}^n_k \) for some \( n \). It corresponds to a homogeneous prime ideal \( \mathfrak{p} \subseteq k[x_0, \ldots, x_n] \) with \( \mathfrak{p} \neq S_+ \).

Theorem (Projective Nullstellensatz). Let \( k \) be algebraically closed, and let \( I \subseteq k[x_0, \ldots, x_n] \) be a homogeneous ideal. The closed subscheme \( V_+(I) \subseteq \mathbb{P}^n_k \) is empty if and only if \( \sqrt{I} \supseteq (x_0, \ldots, x_n) = S_+ \). Moreover, there is an order-reversing bijection between radical homogeneous ideals \( I \not\supseteq S_+ \) and closed subvarieties of \( \mathbb{P}^n_k \).

Example (Plane Curve). The zero set of a homogeneous polynomial \( F(x_0, x_1, x_2) \in k[x_0, x_1, x_2] \) of degree \( d \) defines a hypersurface \( V_+(F) \subseteq \mathbb{P}^2_k \), a projective plane curve of degree \( d \). The curve \( y^2 z = x^3 - xz^2 \) (homogenized Weierstrass form) defines an elliptic curve.

Chapter 5: Morphisms Between Schemes

5.1 Types of Morphisms

Many geometric properties of maps between spaces are encoded in the structure of corresponding morphisms of schemes. We survey the most important classes.

Definition. A morphism \( f : X \to Y \) of schemes is:

Locally of finite type if there exists an affine open cover \( Y = \bigcup V_i \) with \( V_i = \mathrm{Spec}(B_i) \) and for each \( i \), a cover of \( f^{-1}(V_i) \) by affines \( \mathrm{Spec}(A_{ij}) \) where each \( A_{ij} \) is a finitely generated \( B_i \)-algebra.
Of finite type if additionally each \( f^{-1}(V_i) \) can be covered by finitely many such affines.
Finite if there is an affine cover \( Y = \bigcup V_i \), \( V_i = \mathrm{Spec}(B_i) \), with \( f^{-1}(V_i) = \mathrm{Spec}(A_i) \) and \( A_i \) a finite \( B_i \)-module.
Affine if the preimage of every affine open in \( Y \) is affine in \( X \).

Finite morphisms are affine, and affine morphisms are separated (defined below).

5.2 Separated and Proper Morphisms

Separatedness is the scheme-theoretic analogue of the Hausdorff condition; properness corresponds to compactness.

Definition (Separated Morphism). A morphism \( f : X \to Y \) is separated if the diagonal morphism \( \Delta : X \to X \times_Y X \) is a closed immersion. A scheme \( X \) is separated if \( X \to \mathrm{Spec}(\mathbb{Z}) \) is separated.

Theorem (Affine Communication Lemma). \( f : X \to Y \) is separated if and only if for every pair of affine opens \( U = \mathrm{Spec}(A) \subseteq X \) and \( V = \mathrm{Spec}(B) \subseteq X \) with the same image in \( Y \), the intersection \( U \cap V \) is affine and the natural map \( A \otimes_\mathcal{O} B \to \mathcal{O}(U \cap V) \) is surjective.

Definition (Proper Morphism). A morphism \( f : X \to Y \) is proper if it is separated, of finite type, and universally closed: for every \( Y' \to Y \), the base change \( f' : X \times_Y Y' \to Y' \) is a closed map.

Projective morphisms are proper. The converse is not true in general (there exist proper non-projective varieties), but for many purposes one works with projective schemes.

5.3 The Valuative Criteria

Separatedness and properness admit elegant reformulations using discrete valuation rings (DVRs).

Theorem (Valuative Criterion for Separatedness). Assume \( f : X \to Y \) is of finite type. Then \( f \) is separated if and only if for every DVR \( R \) with fraction field \( K \), every commutative square \[ \mathrm{Spec}(K) \to X, \quad \mathrm{Spec}(R) \to Y \]

has at most one lift \( \mathrm{Spec}(R) \to X \) making the diagram commute.

Theorem (Valuative Criterion for Properness). Under the same hypotheses, \( f \) is proper if and only if such a lift exists and is unique.

These criteria capture the geometric intuition: separatedness says limits of arcs are unique when they exist; properness says limits always exist and are unique (the “valuative criterion for compactness”).

5.4 Immersions and Base Change

Definition. An open immersion is a morphism \( j : U \to X \) that is an isomorphism onto an open subscheme. A closed immersion is a morphism \( i : Z \to X \) such that \( i \) is a homeomorphism onto a closed set and \( i^\# : \mathcal{O}_X \to i_* \mathcal{O}_Z \) is surjective (equivalently, locally of the form \( \mathrm{Spec}(A/\mathfrak{a}) \hookrightarrow \mathrm{Spec}(A) \)).

Definition (Base Change). Given morphisms \( f : X \to S \) and \( g : T \to S \), the base change of \( f \) along \( g \) is the morphism \( f_T : X_T = X \times_S T \to T \). Many properties of morphisms are preserved under base change: open and closed immersions, separated, proper, finite, of finite type.

Chapter 6: Fibered Products

6.1 Existence of Fibered Products

Theorem. The category of schemes has all fibered products (pullbacks). That is, given morphisms \( f : X \to S \) and \( g : Y \to S \), the fibered product \( X \times_S Y \) exists as a scheme.

Proof sketch. For affine schemes \( X = \mathrm{Spec}(A) \), \( Y = \mathrm{Spec}(B) \), \( S = \mathrm{Spec}(R) \), the fibered product is \( \mathrm{Spec}(A \otimes_R B) \). For general schemes, cover by affines and glue. The tensor product is the coproduct in the category of \( R \)-algebras, which corresponds to the product in affine schemes over \( S \).

6.2 Fiber Products of Affine Schemes

The affine case is central. Suppose \( A \) and \( B \) are \( R \)-algebras. Then:

\[ \mathrm{Spec}(A) \times_{\mathrm{Spec}(R)} \mathrm{Spec}(B) = \mathrm{Spec}(A \otimes_R B). \]

Example. Let \( k \) be a field, \( X = \mathrm{Spec}(k[x]/(f(x))) \) and \( Y = \mathrm{Spec}(k[y]/(g(y))) \), both over \( S = \mathrm{Spec}(k) \). Then \[ X \times_S Y = \mathrm{Spec}\!\left(\frac{k[x,y]}{(f(x), g(y))}\right). \]

This is the product variety (scheme) cut out by \( f(x) = 0 \) and \( g(y) = 0 \) in \( \mathbb{A}^2_k \).

Example (Intersection via fibered product). If \( Z_1, Z_2 \hookrightarrow X \) are closed subschemes, their scheme-theoretic intersection \( Z_1 \cap Z_2 \) is the fibered product \( Z_1 \times_X Z_2 \), which is the closed subscheme of \( X \) corresponding to the sum of the two ideals. Unlike the topological intersection, this remembers multiplicities.

6.3 Geometric Fibers

Given \( f : X \to S \) and a point \( s \in S \), the fiber over \( s \) is

\[ X_s = X \times_S \mathrm{Spec}(k(s)), \]

where \( k(s) = \mathcal{O}_{S,s} / \mathfrak{m}_s \) is the residue field at \( s \). For \( S = \mathrm{Spec}(\mathbb{Z}) \) and \( X = \mathrm{Spec}(\mathbb{Z}[x, y]/(y^2 - x^3 + x)) \), the fiber over \( (p) \in \mathrm{Spec}(\mathbb{Z}) \) is the reduction \( X_{(p)} = \mathrm{Spec}(\mathbb{F}_p[x,y]/(y^2 - x^3 + x)) \), a curve over \( \mathbb{F}_p \).

The geometric fiber over a geometric point \( \bar{s} : \mathrm{Spec}(\bar{k}) \to S \) (where \( \bar{k} \) is an algebraic closure of \( k(s) \)) is \( X_{\bar{s}} = X \times_S \mathrm{Spec}(\bar{k}) \). Many properties of morphisms (e.g., smoothness, irreducibility of fibers) are best checked on geometric fibers.

6.4 Flatness (Brief Introduction)

Definition (Flat Morphism). A morphism \( f : X \to S \) is flat if for every \( x \in X \), the local ring map \( \mathcal{O}_{S, f(x)} \to \mathcal{O}_{X, x} \) makes \( \mathcal{O}_{X, x} \) a flat \( \mathcal{O}_{S, f(x)} \)-module.

Flat morphisms are the “continuous families” of algebraic geometry: the fibers vary in families without jumping. A key example is that a morphism \( \mathrm{Spec}(B) \to \mathrm{Spec}(A) \) is flat iff \( B \) is a flat \( A \)-module. Projections, open immersions, and étale morphisms are flat. The generic fiber of a flat morphism over an integral scheme determines the fibers at closed points in many respects.

Chapter 7: Dimension and Regularity

7.1 Krull Dimension

Definition (Krull Dimension). The Krull dimension of a ring \( A \), denoted \( \dim A \), is the supremum of the lengths of chains of prime ideals \[ \mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n. \]

The Krull dimension of a scheme \( X \) is the supremum of the lengths of chains of irreducible closed subsets \( Z_0 \subsetneq Z_1 \subsetneq \cdots \subsetneq Z_n \subseteq X \).

For a topological space, the dimension defined by chains of irreducible closed subsets agrees with Krull dimension when the space is the spectrum of a ring. Basic examples: \( \dim \mathbb{A}^n_k = n \), \( \dim \mathbb{P}^n_k = n \), \( \dim \mathrm{Spec}(\mathbb{Z}) = 1 \). For varieties over a field, Krull dimension coincides with the transcendence degree of the function field over \( k \).

Theorem. If \( A \) is a finitely generated \( k \)-algebra that is an integral domain, then \( \dim A = \mathrm{tr.deg}_k \, \mathrm{Frac}(A) \).

7.2 Codimension

Definition. The codimension of an irreducible closed subvariety \( Z \subseteq X \) is \[ \mathrm{codim}(Z, X) = \dim \mathcal{O}_{X, \eta_Z}, \]

where \( \eta_Z \) is the generic point of \( Z \) and \( \mathcal{O}_{X, \eta_Z} \) is its local ring in \( X \). For well-behaved (e.g., equidimensional) schemes, \( \dim Z + \mathrm{codim}(Z, X) = \dim X \).

7.3 Regular Local Rings and Smooth Varieties

Definition (Regular Local Ring). A Noetherian local ring \( (R, \mathfrak{m}, k) \) is regular if \( \dim R = \dim_k (\mathfrak{m}/\mathfrak{m}^2) \). The integer \( \dim_k(\mathfrak{m}/\mathfrak{m}^2) \) is the embedding dimension, and the inequality \( \dim R \leq \dim_k(\mathfrak{m}/\mathfrak{m}^2) \) always holds (Nakayama's lemma); regularity demands equality.

Definition (Smooth Variety). A variety \( X \) of dimension \( n \) over a field \( k \) is smooth at a point \( x \) if the local ring \( \mathcal{O}_{X,x} \) is a regular local ring of dimension \( n \). It is smooth (or non-singular) if it is smooth at every point.

Theorem (Jacobian Criterion). Let \( X = V(f_1, \ldots, f_r) \subseteq \mathbb{A}^n_k \) (or \( \mathbb{P}^n_k \)) with \( k = \bar{k} \). Then \( X \) is smooth at a closed point \( x \in X \) if and only if the Jacobian matrix \[ J = \left(\frac{\partial f_i}{\partial x_j}(x)\right)_{1 \le i \le r, 1 \le j \le n} \]

has rank \( n - \dim X \) at \( x \).

Example. The curve \( C : y^2 = x^3 \) in \( \mathbb{A}^2_k \) (char \( k \neq 2, 3 \)) has Jacobian \( (-3x^2, 2y) \). At the origin \( (0, 0) \), this vanishes, so \( C \) is singular there (it has a cusp). The curve \( y^2 = x^3 - x \) has Jacobian \( (-3x^2 + 1, 2y) \), which is non-zero at every point of the curve (over \( \bar{k} \) with char \( k \neq 2, 3 \)), so this elliptic curve is smooth.

7.4 Normal Varieties

Definition (Normal Scheme). A scheme \( X \) is normal if every local ring \( \mathcal{O}_{X,x} \) is an integrally closed domain. A variety is normal if and only if it satisfies Serre's conditions \( R_1 \) (regular in codimension 1) and \( S_2 \) (depth \( \geq 2 \) at points of codimension \( \geq 2 \)).

Regular implies normal (regular local rings are integrally closed). Normal varieties have good divisor theory: in particular, the Weil divisor class group is well-defined, and on a smooth variety it coincides with the Cartier divisor class group (the Picard group).

Chapter 8: Line Bundles and Divisors

8.1 Invertible Sheaves and the Picard Group

Definition (Invertible Sheaf / Line Bundle). An invertible sheaf (or line bundle) on a scheme \( X \) is a sheaf \( \mathcal{L} \) of \( \mathcal{O}_X \)-modules that is locally free of rank 1: there exists an open cover \( X = \bigcup U_i \) with \( \mathcal{L}|_{U_i} \cong \mathcal{O}_{U_i} \) for each \( i \).

The tensor product \( \mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{M} \) of two invertible sheaves is again invertible. The inverse of \( \mathcal{L} \) is \( \mathcal{L}^{-1} = \mathcal{H}om_{\mathcal{O}_X}(\mathcal{L}, \mathcal{O}_X) \). Under tensor product, the isomorphism classes of invertible sheaves form a group:

Definition (Picard Group). The Picard group \( \mathrm{Pic}(X) \) is the group of isomorphism classes of invertible sheaves on \( X \), with group operation given by tensor product.

Example. For \( X = \mathbb{P}^n_k \), the Picard group is \( \mathrm{Pic}(\mathbb{P}^n_k) \cong \mathbb{Z} \), generated by \( \mathcal{O}(1) \), the tautological line bundle. Its sections over \( \mathbb{P}^n_k \) are the homogeneous polynomials of degree 1, and \( \mathcal{O}(d) = \mathcal{O}(1)^{\otimes d} \) has sections that are homogeneous of degree \( d \).

8.2 Weil Divisors and Cartier Divisors

On a normal variety, one can describe line bundles geometrically via divisors.

Definition (Weil Divisor). A Weil divisor on an integral scheme \( X \) of dimension \( n \) is a formal \( \mathbb{Z} \)-linear combination \( D = \sum n_i Z_i \) of irreducible codimension-1 subvarieties \( Z_i \). The Weil divisor class group \( \mathrm{Cl}(X) \) is the quotient of the group of Weil divisors by the subgroup of principal divisors \( \mathrm{div}(f) = \sum_Z v_Z(f) Z \), where \( v_Z(f) \) is the order of vanishing of a rational function \( f \) along \( Z \).

Definition (Cartier Divisor). A Cartier divisor on \( X \) is a global section of the sheaf \( \mathcal{K}_X^*/\mathcal{O}_X^* \), where \( \mathcal{K}_X \) is the sheaf of total quotient rings. Equivalently, it is given by an open cover \( \{U_i\} \) of \( X \) and rational functions \( f_i \in \mathcal{K}^*(U_i) \) with \( f_i/f_j \in \mathcal{O}^*(U_i \cap U_j) \). The Cartier divisor class group (or Picard group) is the group \( \mathrm{CaCl}(X) = \mathrm{Pic}(X) \).

On a smooth variety over a field, every Weil divisor is Cartier and \( \mathrm{Cl}(X) = \mathrm{Pic}(X) \). The correspondence between Cartier divisors and line bundles: a Cartier divisor \( \{(U_i, f_i)\} \) determines a line bundle \( \mathcal{O}_X(D) \) by taking \( \mathcal{O}(D)|_{U_i} = f_i^{-1} \mathcal{O}_{U_i} \subseteq \mathcal{K}_{U_i} \).

8.3 Linear Systems and Maps to Projective Space

Definition (Linear System). Given a line bundle \( \mathcal{L} \) on \( X \), a linear system is a projective subspace of the complete linear system \( |L| = \mathbb{P}(H^0(X, \mathcal{L})) \), the projective space of global sections of \( \mathcal{L} \) modulo scalars. Each effective divisor \( D \) in the linear system corresponds to a section \( s \in H^0(X, \mathcal{L}) \).

Theorem. A line bundle \( \mathcal{L} \) on a variety \( X \) over \( k \) determines a morphism \( \phi_\mathcal{L} : X \to \mathbb{P}^n_k \) (for \( n + 1 = \dim H^0(X, \mathcal{L}) \)) if and only if \( \mathcal{L} \) is globally generated (i.e., the map \( H^0(X, \mathcal{L}) \otimes \mathcal{O}_X \to \mathcal{L} \) is surjective). The pullback \( \phi_\mathcal{L}^* \mathcal{O}(1) \cong \mathcal{L} \).

Definition (Ample / Very Ample). A line bundle \( \mathcal{L} \) is very ample if \( \phi_\mathcal{L} \) is a closed immersion into projective space. It is ample if some power \( \mathcal{L}^{\otimes m} \) is very ample. By Serre's theorem, \( \mathcal{L} \) is ample iff for every coherent sheaf \( \mathcal{F} \), the sheaves \( \mathcal{F} \otimes \mathcal{L}^{\otimes m} \) are globally generated for \( m \gg 0 \).

8.4 Cohomology of Sheaves

Sheaf cohomology measures the obstruction to extending local sections globally.

Definition. For a sheaf \( \mathcal{F} \) on a scheme \( X \), the cohomology groups \( H^i(X, \mathcal{F}) \) are the right derived functors of the global sections functor \( \Gamma(X, -) \). Concretely, one takes an injective resolution \( 0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots \) in the category of sheaves, applies \( \Gamma \), and takes cohomology.

For practical computation on projective schemes, Čech cohomology is very useful. Given an open affine cover \( \mathfrak{U} = \{U_i\}_{i=0}^n \) of \( X \), the Čech complex is

\[ 0 \to \prod_i \mathcal{F}(U_i) \to \prod_{i < j} \mathcal{F}(U_i \cap U_j) \to \cdots \to \mathcal{F}(U_0 \cap \cdots \cap U_n) \to 0, \]

and the cohomology of this complex computes \( H^*(X, \mathcal{F}) \) when the cover consists of open affines and \( \mathcal{F} \) is quasicoherent (by a theorem of Cartan).

Theorem (Cohomology of \( \mathbb{P}^n \)). For \( \mathcal{O}(d) \) on \( \mathbb{P}^n_k \):

\( H^0(\mathbb{P}^n, \mathcal{O}(d)) \) is the space of homogeneous polynomials of degree \( d \) (zero for \( d < 0 \)).
\( H^n(\mathbb{P}^n, \mathcal{O}(d)) \cong H^0(\mathbb{P}^n, \mathcal{O}(-d - n - 1))^\vee \) (Serre duality).
\( H^i(\mathbb{P}^n, \mathcal{O}(d)) = 0 \) for \( 0 < i < n \) and all \( d \).

Chapter 9: Curves and the Riemann-Roch Theorem

9.1 Smooth Projective Curves

Definition. A smooth projective curve over a field \( k \) is a scheme \( C \) that is integral, smooth, projective, and of dimension 1 over \( k \).

The theory of smooth projective curves over \( k \) is one of the most beautiful chapters of algebraic geometry. Every smooth projective curve \( C \) over an algebraically closed field \( k \) is determined up to isomorphism by its function field \( K = k(C) \): the closed points of \( C \) correspond bijectively to discrete valuations of \( K/k \) (the places of \( K \)).

9.2 Divisors on Curves

Since \( \dim C = 1 \), the only irreducible closed subsets of dimension 0 are points. Thus:

Weil divisors on \( C \) are formal \( \mathbb{Z} \)-linear combinations of closed points: \( D = \sum_{P \in C} n_P \cdot P \), with \( n_P = 0 \) for all but finitely many \( P \).
The degree of \( D \) is \( \deg D = \sum n_P [k(P) : k] \).
A rational function \( f \in K^* \) has divisor \( \mathrm{div}(f) = \sum_P v_P(f) \cdot P \), where \( v_P \) is the valuation at \( P \).
Principal divisors have degree 0 (by the product formula).

Definition. For a divisor \( D \) on \( C \), the associated line bundle is \( \mathcal{O}_C(D) \). Its global sections are \[ H^0(C, \mathcal{O}(D)) = \{ f \in K^* \mid \mathrm{div}(f) + D \geq 0 \} \cup \{0\} =: L(D), \]

the Riemann-Roch space of \( D \). Set \( \ell(D) = \dim_k L(D) \).

9.3 Genus

Definition (Geometric Genus). The genus of a smooth projective curve \( C \) over \( k \) is \[ g = g(C) = \dim_k H^1(C, \mathcal{O}_C). \]

Equivalently, for \( k = \mathbb{C} \), the genus is the number of handles of the topological surface \( C(\mathbb{C}) \).

Alternative definitions: if \( C \) is a smooth plane curve of degree \( d \), then \( g = (d-1)(d-2)/2 \). For an elliptic curve (smooth genus-1 curve with a marked point), \( g = 1 \). The projective line \( \mathbb{P}^1_k \) has \( g = 0 \).

9.4 The Riemann-Roch Theorem

The Riemann-Roch theorem is the central result in the theory of curves. It computes \( \ell(D) \) and relates it to the geometry of the canonical divisor.

Definition (Canonical Divisor). The canonical divisor class \( K_C \) of a smooth curve \( C \) is the divisor class corresponding to the cotangent sheaf \( \Omega_{C/k} \cong \omega_C \) (the dualizing sheaf). It has degree \( \deg K_C = 2g - 2 \).

Theorem (Riemann-Roch for Curves). Let \( C \) be a smooth projective curve of genus \( g \) over an algebraically closed field \( k \), and let \( D \) be a divisor on \( C \). Then \[ \ell(D) - \ell(K_C - D) = \deg D - g + 1. \]

Proof sketch. We use coherent sheaf cohomology. From the short exact sequence defining \( \mathcal{O}(D) \) and the Euler characteristic \( \chi(C, \mathcal{O}(D)) = h^0 - h^1 \), we get: \[ \ell(D) - h^1(C, \mathcal{O}(D)) = \deg D - g + 1, \]

by comparing with the case \( D = 0 \) (where \( \ell(0) = 1 \) and \( h^1 = g \)) and using additivity of Euler characteristic when adding or removing points. Serre duality then gives \( h^1(C, \mathcal{O}(D)) = h^0(C, \Omega_C(-D)) = \ell(K_C - D) \).

Remark. When \( \deg D > 2g - 2 \), we have \( \ell(K_C - D) = 0 \) (since \( \deg(K_C - D) < 0 \) means \( K_C - D \) has no effective representative), so the theorem reduces to \( \ell(D) = \deg D - g + 1 \). This is the "non-special" range.

9.5 Applications of Riemann-Roch

Curves of low genus.

\( g = 0 \): Every smooth curve with a rational point is isomorphic to \( \mathbb{P}^1_k \). (Take any point \( P \); then \( \ell(P) = 2 \) gives a morphism to \( \mathbb{P}^1 \) of degree 1.)
\( g = 1 \): Elliptic curves. For a point \( P \) on an elliptic curve, Riemann-Roch gives \( \ell(3P) = 3 \), so \( 3P \) embeds \( C \) in \( \mathbb{P}^2 \) as a cubic, and \( C \) has a Weierstrass equation \( y^2 = x^3 + ax + b \).
\( g = 2 \): The canonical map \( \phi_{K_C} : C \to \mathbb{P}^1 \) is a degree-2 map; \( C \) is a hyperelliptic curve. The map is given by \( \ell(K_C) = g = 2 \).

Embeddings of curves.

Theorem. If \( \deg D \geq 2g + 1 \), the line bundle \( \mathcal{O}(D) \) is very ample, and \( D \) embeds \( C \) into \( \mathbb{P}^{d-g} \) where \( d = \deg D \). For \( \deg D \geq 2g \), \( \mathcal{O}(D) \) is globally generated and non-special.

Clifford’s Theorem.

Theorem (Clifford). Let \( D \) be a special divisor on \( C \) (meaning \( \ell(K_C - D) > 0 \), so \( 0 \leq \deg D \leq 2g - 2 \)). Then \[ \ell(D) \leq \frac{\deg D}{2} + 1. \]

Equality holds only for \( D = 0 \), \( D = K_C \), or when \( C \) is hyperelliptic and \( D \) is a multiple of the hyperelliptic class.

Clifford’s theorem controls how many independent sections a special divisor can have, and is a key tool in Brill-Noether theory.

Chapter 10: Serre Duality

10.1 The Dualizing Sheaf

Serre duality is the algebraic geometry analogue of Poincaré duality on compact oriented manifolds. To state it, we need the dualizing sheaf.

Definition (Dualizing Sheaf). Let \( X \) be a projective variety of dimension \( n \) over a field \( k \). A dualizing sheaf \( \omega_X \) is a coherent sheaf on \( X \) together with a trace map \[ t : H^n(X, \omega_X) \to k, \]

such that for every coherent sheaf \( \mathcal{F} \) on \( X \), the natural pairing

\[ H^i(X, \mathcal{F}) \times H^{n-i}(X, \mathcal{H}om(\mathcal{F}, \omega_X)) \to H^n(X, \omega_X) \xrightarrow{t} k \]

is a perfect pairing of \( k \)-vector spaces.

Theorem (Existence and Uniqueness of Dualizing Sheaf). For a smooth projective variety \( X \) of dimension \( n \) over \( k \), the dualizing sheaf exists and is given by \[ \omega_X = \bigwedge^n \Omega_{X/k}, \]

the top exterior power of the sheaf of Kähler differentials. It is an invertible sheaf (a line bundle) and is uniquely characterized by Serre duality.

For a smooth projective curve \( C \), \( \omega_C = \Omega_{C/k} \) is just the sheaf of 1-forms, and its degree is \( 2g - 2 \) (as mentioned above).

For projective space \( \mathbb{P}^n_k \), we have \( \omega_{\mathbb{P}^n} = \mathcal{O}(-n-1) \). This can be computed from the Euler sequence and the formula for the canonical class of a hypersurface.

10.2 Statement of Serre Duality

Theorem (Serre Duality). Let \( X \) be a smooth projective variety of dimension \( n \) over a field \( k \), and let \( \mathcal{F} \) be a locally free sheaf on \( X \). Then there are natural isomorphisms of \( k \)-vector spaces \[ H^i(X, \mathcal{F}) \cong H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)^\vee \]

for all \( 0 \leq i \leq n \), where \( \mathcal{F}^\vee = \mathcal{H}om(\mathcal{F}, \mathcal{O}_X) \) is the dual sheaf. In particular, \( h^i(X, \mathcal{F}) = h^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X) \).

Proof sketch (for projective space). On \( \mathbb{P}^n_k \), any locally free sheaf is a direct sum of twists \( \mathcal{O}(d) \). By the explicit Čech cohomology computation, \( H^i(\mathbb{P}^n, \mathcal{O}(d)) \) is nonzero only for \( i = 0 \) (\( d \geq 0 \)) and \( i = n \) (\( d \leq -n-1 \)), and the duality \( H^n(\mathbb{P}^n, \mathcal{O}(d)) \cong H^0(\mathbb{P}^n, \mathcal{O}(-d-n-1))^\vee \) can be verified directly. For general smooth projective varieties, the result follows by embedding into projective space and using adjunction or abstract machinery (injective resolutions).

10.3 Serre Duality for Curves

For a smooth projective curve \( C \) of genus \( g \), \( n = 1 \) and \( \omega_C = \Omega_{C/k} \). Serre duality reads: for any line bundle \( \mathcal{L} \) on \( C \),

\[ H^1(C, \mathcal{L}) \cong H^0(C, \mathcal{L}^{-1} \otimes \omega_C)^\vee = H^0(C, K_C - D)^\vee, \]

where \( D \) is the divisor corresponding to \( \mathcal{L} \). This is precisely the identification \( h^1(C, \mathcal{O}(D)) = \ell(K_C - D) \) used in the proof of Riemann-Roch.

Example. For \( C = \mathbb{P}^1_k \), \( g = 0 \) and \( \omega_{\mathbb{P}^1} = \mathcal{O}(-2) \). Serre duality gives \( H^1(\mathbb{P}^1, \mathcal{O}(d)) \cong H^0(\mathbb{P}^1, \mathcal{O}(-d-2))^\vee \). For \( d \geq 0 \), both sides vanish (since \( -d-2 < 0 \)); for \( d \leq -2 \), \( H^1(\mathbb{P}^1, \mathcal{O}(d)) \cong H^0(\mathbb{P}^1, \mathcal{O}(-d-2)) \), which has dimension \( -d-1 \).

10.4 Applications: Vanishing and Bounds

Serre duality, combined with Riemann-Roch and other tools, yields powerful vanishing results for cohomology.

Theorem (Kodaira Vanishing, in characteristic 0). Let \( X \) be a smooth projective variety over a field of characteristic 0, and let \( \mathcal{L} \) be an ample line bundle on \( X \). Then \[ H^i(X, \omega_X \otimes \mathcal{L}) = 0 \quad \text{for all } i > 0. \]

By Serre duality, this is equivalent to \( H^{n-i}(X, \mathcal{L}^{-1}) = 0 \) for \( i > 0 \), i.e., \( H^j(X, \mathcal{L}^{-1}) = 0 \) for \( j < n \).

Theorem (Riemann-Hurwitz Formula). Let \( f : C \to D \) be a non-constant morphism of smooth projective curves of degrees \( n \) and genera \( g(C) \), \( g(D) \) respectively. Then \[ 2g(C) - 2 = n(2g(D) - 2) + \deg R, \]

where \( R = \sum_{P \in C} (e_P - 1) P \) is the ramification divisor, and \( e_P \) is the ramification index at \( P \). (Assumes separable \( f \) and that the characteristic does not divide any \( e_P \).) This follows from the fact that \( f^* \omega_D \) is a subsheaf of \( \omega_C \), with quotient supported on the ramification locus.

Example. A degree-2 map \( f : C \to \mathbb{P}^1 \) (a hyperelliptic curve) has \( g(D) = 0 \), so \( 2g(C) - 2 = -4 + \deg R \), giving \( \deg R = 2g(C) + 2 \). Since each ramification point has \( e_P = 2 \) (in characteristic \( \neq 2 \)), there are \( 2g + 2 \) branch points in \( \mathbb{P}^1 \). For \( g = 1 \) (elliptic curves), this gives 4 branch points, consistent with the Weierstrass form \( y^2 = (x - a_1)(x-a_2)(x-a_3)(x-a_4) \).

Chapter 11: Supplementary Topics

11.1 Coherent Sheaves and Quasi-Coherent Sheaves

Definition. A sheaf \( \mathcal{F} \) of \( \mathcal{O}_X \)-modules on a scheme \( X \) is quasi-coherent if locally (on an affine cover \( \{U_i = \mathrm{Spec}(A_i)\} \)), \( \mathcal{F}|_{U_i} \cong \widetilde{M_i} \) for some \( A_i \)-module \( M_i \). Here \( \widetilde{M} \) denotes the quasi-coherent sheaf associated to an \( A \)-module \( M \), with \( \widetilde{M}(D(f)) = M_f \). It is coherent if additionally each \( M_i \) can be taken to be finitely presented.

On a Noetherian scheme, coherent sheaves are the central objects. The global sections functor, pushforward, pullback, tensor product, and sheaf Hom all preserve (quasi-)coherence under mild hypotheses. Coherent sheaves on \( \mathrm{Spec}(A) \) correspond exactly to finitely generated \( A \)-modules.

Theorem (Serre's Theorem — Finiteness). Let \( X \) be a projective scheme over a Noetherian ring \( A \), and let \( \mathcal{F} \) be a coherent sheaf on \( X \). Then:

\( H^i(X, \mathcal{F}) \) is a finitely generated \( A \)-module for all \( i \geq 0 \).
\( H^i(X, \mathcal{F} \otimes \mathcal{O}(m)) = 0 \) for all \( i > 0 \) and all \( m \gg 0 \).

11.2 The Euler Characteristic

Definition. For a coherent sheaf \( \mathcal{F} \) on a projective scheme \( X \) over a field \( k \), the Euler characteristic is \[ \chi(X, \mathcal{F}) = \sum_{i \geq 0} (-1)^i h^i(X, \mathcal{F}), \]

where \( h^i = \dim_k H^i(X, \mathcal{F}) \). This is an integer, finite by Serre’s theorem.

The Euler characteristic is additive in short exact sequences: if \( 0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0 \) is exact, then \( \chi(\mathcal{F}) = \chi(\mathcal{F}') + \chi(\mathcal{F}'') \). It is also more accessible to computation than individual cohomology groups, and often determines them.

Theorem (Hirzebruch-Riemann-Roch). For a smooth projective variety \( X \) of dimension \( n \) over \( \mathbb{C} \) and a vector bundle \( \mathcal{E} \) on \( X \), \[ \chi(X, \mathcal{E}) = \int_X \mathrm{ch}(\mathcal{E}) \cdot \mathrm{td}(T_X), \]

where \( \mathrm{ch}(\mathcal{E}) \) is the Chern character of \( \mathcal{E} \) and \( \mathrm{td}(T_X) \) is the Todd class of the tangent bundle. For a curve (\( n = 1 \)) and a line bundle \( \mathcal{L} = \mathcal{O}(D) \), this reduces to \( \chi = \deg D - g + 1 \), recovering Riemann-Roch.

11.3 The Relative Canonical Sheaf and Adjunction

When studying a smooth hypersurface \( X = V(F) \subseteq \mathbb{P}^n \) of degree \( d \), the adjunction formula computes the dualizing sheaf:

Theorem (Adjunction Formula). Let \( X \subseteq Y \) be a smooth hypersurface in a smooth variety \( Y \). Then \[ \omega_X = (\omega_Y \otimes \mathcal{O}_Y(X))|_X. \]

For \( X \subseteq \mathbb{P}^n \) a smooth hypersurface of degree \( d \), since \( \omega_{\mathbb{P}^n} = \mathcal{O}(-n-1) \) and \( \mathcal{O}(X) = \mathcal{O}(d) \), we get

\[ \omega_X = \mathcal{O}(d - n - 1)|_X. \]

Example. A smooth plane curve \( C \subseteq \mathbb{P}^2 \) of degree \( d \) has \( \omega_C = \mathcal{O}(d-3)|_C \), so \( \deg \omega_C = d(d-3) \). The formula \( \deg \omega_C = 2g - 2 \) then gives \( g = (d-1)(d-2)/2 \), consistent with what we stated earlier. For \( d = 3 \) (cubic plane curve), \( g = 1 \), confirming that smooth cubics are elliptic curves.

11.4 Moduli Spaces and Functorial Perspective

One of the goals of modern algebraic geometry is to construct moduli spaces parametrizing geometric objects.

Definition (Moduli Problem). A moduli problem consists of a functor \( \mathcal{M} : \mathbf{Sch}^{\mathrm{op}} \to \mathbf{Set} \) where \( \mathcal{M}(T) \) is the set of isomorphism classes of "families of objects over \( T \)." A scheme \( M \) represents the moduli problem if \( \mathcal{M} \cong h_M \) (the functor of points of \( M \)). A coarse moduli space is a scheme \( M \) with a natural transformation \( \mathcal{M} \to h_M \) that is a bijection on algebraically closed field-valued points and is initial among all such natural transformations.

The moduli space \( \mathcal{M}_g \) of smooth projective curves of genus \( g \) is a fundamental object. For \( g \geq 2 \), \( \mathcal{M}_g \) exists as a coarse moduli space of dimension \( 3g - 3 \). Its construction uses geometric invariant theory (GIT) or the theory of Hilbert schemes, both natural extensions of the material in this course.

Chapter 12: Connections and Outlook

12.1 Étale Morphisms and the Étale Site

A morphism \( f : X \to Y \) is étale if it is flat, unramified, and locally of finite type. Étale morphisms are the algebraic geometry analogue of local homeomorphisms in topology. They form the basis of the étale topology, a Grothendieck topology on the category of schemes, and the étale cohomology \( H^i_{\text{ét}}(X, \mathbb{Z}/\ell\mathbb{Z}) \) — constructed by Grothendieck to prove the Weil conjectures. The comparison theorem of Artin identifies étale cohomology with singular cohomology over \( \mathbb{C} \).

12.2 Intersection Theory

On a smooth projective surface \( X \) (a variety of dimension 2), the intersection product of two divisors \( D_1, D_2 \) is defined as

\[ D_1 \cdot D_2 = \chi(\mathcal{O}_X) - \chi(\mathcal{O}(-D_1)) - \chi(\mathcal{O}(-D_2)) + \chi(\mathcal{O}(-D_1 - D_2)). \]

For curves on a surface, this gives a symmetric bilinear form on \( \mathrm{Pic}(X) \), and the Hodge Index Theorem asserts that this form has signature \( (1, \rho - 1) \) where \( \rho = \mathrm{rank}\,\mathrm{Pic}(X) \). Intersection theory is extended to higher-dimensional varieties by Chow groups and Chern classes.

12.3 Deformation Theory

Infinitesimal deformations of a scheme \( X \) — i.e., liftings of \( X \) from \( \mathrm{Spec}(k) \) to \( \mathrm{Spec}(k[\epsilon]/(\epsilon^2)) \) — are classified by \( H^1(X, T_X) \), the first cohomology of the tangent sheaf. Obstructions to deformation live in \( H^2(X, T_X) \). For curves of genus \( g \geq 2 \), the deformation space has dimension \( h^1(C, T_C) = h^1(C, \Omega_C^{-1}) = 3g - 3 \) (by Serre duality and Riemann-Roch), explaining the dimension of \( \mathcal{M}_g \).

12.4 Grothendieck’s Relative Point of View

Much of modern algebraic geometry takes place not over a fixed base field but “relatively”: given a morphism \( f : X \to S \), one studies \( X \) as a family over \( S \). The relative perspective — introduced and systematized by Grothendieck in EGA and SGA — has been enormously productive:

Relative differentials \( \Omega_{X/S} \) and relative dualizing sheaf \( \omega_{X/S} \).
Relative cohomology and the Leray spectral sequence \( H^p(S, R^q f_* \mathcal{F}) \Rightarrow H^{p+q}(X, \mathcal{F}) \).
Base change theorems: when the formation of \( R^i f_* \mathcal{F} \) commutes with base change \( S' \to S \).

The schemes encountered in number theory — integral models of varieties over \( \mathbb{Q} \), Shimura varieties, the arithmetic surface attached to a number field — are naturally objects over \( \mathrm{Spec}(\mathbb{Z}) \) or a ring of integers, and the relative perspective is indispensable.

Remark (Further Directions). This course provides the foundation for many advanced topics in modern research:

Abelian varieties and the Weil conjectures: the cohomological formalism of \( \ell \)-adic cohomology and Frobenius.
Algebraic \( K \)-theory: the relationship between sheaves, vector bundles, and \( K \)-groups of rings and schemes.
Derived algebraic geometry: replacing rings by differential graded algebras and ∞-categories; deformation theory in its natural habitat.
The Langlands program: geometric Langlands replaces automorphic forms with \( \mathcal{D} \)-modules or perverse sheaves on moduli stacks of bundles.
Mirror symmetry and derived categories: Fourier-Mukai transforms, derived categories of coherent sheaves, and their role in symplectic geometry.