Postgraduate Thesis uoadl:3245422 181 Read counter

Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2022-11-07

2022

Lygdas Orestis

Κοντογεώργης Αριστείδης Καθηγητής Τμήμα Μαθηματικών ΕΚΠΑ

Συνομολογία συναφών δεματιών & Etale συνομολογία

Greek

Coherent sheaf cohomology & Etale cohomology

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.

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.

Science

algebraic geometry, sheaf cohomology, etale cohomology, Weil conjectures

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