**Preface**

**1. Schemes and sheaves: definitions **

1.1. \(\mathrm{Spec}(R)\)

1.2. \(\tilde{M}\)

1.3. Schemes

1.4. Products

1.5. Quasi-coherent sheaves

1.6. The functor of points

1.7. Relativization

1.8. Defining schemes as functors

Appendix: Theory of sheaves

**2. Exploring the world of schemes**

2.1. Classical varieties as schemes

2.2. The properties: reduced, irreducible and finite type

2.3. Closed subschemes and primary decompositions

2.4. Separated schemes

2.5. \(\mathrm{Proj} R\)

2.6. Proper morphisms

**3. Elementary global study of \(\mathrm{Proj} R\)**

3.1. Invertible sheaves and twists

3.2. The functor of \(\mathrm{Proj} R\)

3.3. Blow ups

3.4. Quasi-coherent sheaves on \(\mathrm{Proj} R\)

3.5. Ample invertible sheaves

3.6. Invertible sheaves via cocycles, divisors, line bundles

**4. Ground fields and base rings**

4.1. Kronecker’s big picture

4.2. Galois theory and schemes

4.3. The Frobenius morphism

4.4. Flatness and specialization

4.5. Dimension of fibres of a morphism

4.6. Hensel’s lemma

**5. Singular vs. non-singular**

5.1. Regularity

5.2. Kähler differential

5.3. Smooth morphisms

5.4. Criteria for smoothness

5.5. Normality

5.6. Zariski’s Main Theorem

5.7. Multiplicities following Weil

**6. Group schemes and applications**

6.1. Group schemes

6.2. Lang’s theorems over finite fields

**7. The cohomology of coherent sheaves**

7.1. Basic Čech cohomology

7.2. The case of schemes: Serre’s theorem

7.3. Higher direct images and Leray’s spectral sequence

7.4. Computing cohomology (1): Push \(\mathcal{F}\) into a huge acyclic sheaf

7.5. Computing cohomology (2): Directly via the Čech complex

7.6. Computing cohomology (3): Generate \(\mathcal{F}\) by “known” sheaves

7.7. Computing cohomology (4): Push \(\mathcal{F}\) into a coherent acyclic sheaf

7.8. Serre’s criterion for ampleness

7.9. Functorial properties of ampleness

7.10. The Euler characteristic

7.11. Intersection numbers

7.12. The criterion of Nakai-Moishezon

7.13. Seshadri constants

**8. Applications of cohomology **

8.1. The Riemann-Roch theorem

Appendix: Residues of differentials on curves

8.2. Comparison of algebraic with analytic cohomology

8.3. De Rham cohomology

8.4. Characteristic p phenomena

8.5. Deformation theory

**9. Two deeper results**

9.1. Mori’s existence theorem of rational curves

9.2. Belyi’s three point theorem

**References**

**Index**