Summary:
The goal of this thesis is to present some tools that are fundamental for
modern Algebraic Geometry. More specifically, we will present the notions
of quasi-coherent and coherent sheaves, their sheaf cohomology, and also
étale cohomology.
In the first chapter we define the kernels, images and cokernels of sheaf
morphisms. This means that we can also define what it means to have an
exact sequence of sheaves. We prove basic properties for all these
notions. Then we define O_X-modules, and quasi-coherent and coherent
sheaves (which are O_X-modules of a certain kind). After that we study
direct and inverse images of sheaves, at the third section of the first
chapter. At the end of the chapter, we see some applications of
quasi-coherent sheaves at the definition of closed subschemes and at the
characterization of affine morphisms over a given scheme.
The reason why quasi-coherent and coherent sheaves are important is that
their cohomology theory is a “good” cohomology theory over schemes. We
present this theory at the second chapter. First we introduce the more
abstract theory of sheaf cohomology which is defined using flabby
resolutions of sheaves. Afterwards we define a second sheaf cohomology
theory, Čech cohomology, which is easier to compute. The important
assumption that we have to make in order to have the same cohomology
groups from both theories is that the sheaves we’re using are
quasi-coherent or coherent.
In the second part of the thesis we introduce étale cohomology. In order
to do that we have to introduce a lot of new notions and techniques. First
we define what is an étale map, explaining the geometric idea behind the
definition. In most parts we omit any proof that is purely in the domain
of Commutative Algebra (which is something analogous to Calculus for
Algebraic Geometry). Then we talk about the groundbreaking idea of
Grothendieck: In his search for a suitable cohomology theory which would
give him the right point of view to look at the Weil conjectures, he
thought of defining a new (and more general) notion of topology (!)
instead of looking for some combinatorial definition of a new cohomology
theory, or perhaps a new algebraic notion. The indications towards that
idea are related to the limitations of the usual Zariski topology that we
consider over varieties or schemes (e.g. the Inverse Function Theorem
doesn’t hold and we also don’t have an interesting cohomology theory with
constant coefficients over irreducible varieties).
This new notion of topology is expressed using the language of sites. We
focus our study on the étale site on a variety. Now we’re able to develop
a new sheaf theory over this site, which will have many common
characteristic as the usual sheaf theory over the Zariski site, but will
be much more powerful (we pay the price for this power because the proofs
for similar results are considerably harder to get). With this new sheaf
theory, comes a new sheaf cohomology. We repeat the constructions of the
derived functors cohomology groups and Čech cohomology groups and point
out the differences with the usual cases (e.g. in order to get the same
groups we don’t need quasi-coherence or coherence).
In the last chapter, we state and sketch (most of) the proof of the Weil
conjectures. To be precise we don’t sketch the proof for the last (and far
more difficult) conjecture – the Riemann hypothesis over finite fields. We
define l-adic cohomology using étale cohomology and list a few of its
properties which are needed for the proof of the conjectures. l-adic
cohomology was the theory that gave the “correct” cohomology groups with
constant coefficients, analogous to what singular cohomology does in
Algebraic Topology. These two theories have very similar properties, but
the proofs for l-adic cohomology are very “deep” theorems, as expected.
Finally, we try to show how one can derive the proofs of the conjectures
as exercises, once the theory of l-adic cohomology is well-established.
Keywords:
algebraic geometry, sheaf cohomology, etale cohomology, Weil conjectures
