Introduction

This lecture course is a continuation of the course Algebraic Geometry 1 which covered the definition of schemes, and some basic notions about schemes and scheme morphisms, such as immersions and subschemes, fiber products of schemes and separated morphisms.

The main object of study of this term’s course will be the notion of \({\mathscr O}_X\)-module, a natural analogue of the notion of module over a ring in the context of sheaves of rings. As we will see, the \({\mathscr O}_X\)-modules on a scheme \(X\) contain a lot of information about the geometry of this scheme, and we will study them using a variety of methods. In particular, we will introduce the notion of cohomology groups, a powerful algebraic tool that makes its appearance in many areas of algebra and geometry.

These notes are based on the notes for similar courses which I taught in 2023 and 2019, and will be updated or modified to reflect the content of the current class while the term proceeds.

References. The books [ GW1 ] , [ GW2 ] by Wedhorn and myself, by Hartshorne [ H ] , and by Mumford [ Mu ] . More precise references are given in most of the individual sections. Mumford still uses the ancient terminology and calls a prescheme what we call a scheme, and a scheme what we call a separated scheme. Further references: [ Stacks ] , [ EGA ] .

We give a brief overview of some of the topics we will discuss in this class.

Proper schemes and morphisms. (Section 1.5) Similarly as separatedness is the algebro-geometric version of the Hausdorff property, properness is the algebro-geometric version of compactness; we will see that - as expected, cf. the Riemann sphere - projective space and all of its closed subschemes are proper. (The converse is, interestingly, not true: There exist proper schemes (over a field, say) which cannot be embedded into any projective space.))

Divisors and line bundles. (Chapter 3) Assume we have a closed subscheme \(X \subseteq \mathbb P^n_k\). We can intersect \(X\) with the "coordinate hyperplanes" \(V_+(X_i)\) to obtain subschemes of \(X\) of a very special form, so-called divisors. We will discuss several perspectives on the notion of divisor. It is an interesting question whether, given a divisor, it arises in the way described above.

\(\mathscr O_X\)-Modules. (Chapter 2) Similarly to the notion of module over a ring, there is a notion of sheaves of modules over a sheaf of rings which we can apply in particular to the structure sheaf of a scheme or of any ringed space. Such \(\mathscr O_X\)-modules contain a lot of interesting information about the underlying space \(X\). There is a close connection to the notion of divisor which we will discuss in the course. (And for this reason, in the course we will first talk about \({\mathscr O}_X\)-modules, and then about divisors.)

Cohomology of sheaves. (Chapter 6) We know that a surjective sheaf morphism might not induce surjections on the sets of sections over a fixed open of the underlying space (for simplicity take the whole space \(X\)). So the functor \(\Gamma (X, -)\) taking global sections (on the category of sheaves of abelian groups say, where we have a notion of exact sequences) is not exact. Studying in which way exactness fails leads, by the general formalism of "derived functors", to the notion of cohomology groups \(H^i(X, \mathscr F)\) on \(X\) with coefficients in a sheaf \(\mathscr F\) of abelian groups. (Often \(\mathscr F\) will be a \(\mathscr O_X\)-module). This is an extremely powerful tool to "algebraize" geometric information. After this "controled loss of information" often things are easier to work with.

The Theorem of Riemann-Roch. (Theorem 3.13, Section 6.17) This famous theorem is a specific example where cohomology of \(\mathscr O_X\)-modules attached to divisors on a smooth projective curve can be used very profitably in order to study the geometry of the curve. Among many other things it yields a very clean description of the group structure on an elliptic curve, giving us the associativity (which we had to leave open in Part 1) basically for free.