## Advanced Algebraic Geometry

See also the mastermath page for this course.### Lecturers

Robin de Jong (Leiden) and Lenny Taelman (UvA).### Prerequisites

This is a WONDER graduate-level course. We expect students to be familiar (and comfortable) with algebraic geometry at the level of the mastermath*Algebraic Geometry*course. Prerequisites for

*Advanced Algebraic Geometry*are basic algebra, the language of modules and categories (see also the intensive course offered in mastermath), and basic algebraic geometry corresponding roughly to sections I.1-I.6 (varieties) and II.1 (sheaves) of Hartshorne's book Algebraic Geometry.

### Literature

Robin Hartshorne,*Algebraic Geometry*. Atiyah and MacDonald,

*Introduction to Commutative Algebra*. We will also regularly refer to sections of the online

*Stacks Project*.

### Examination

There will be two take-home assignments (one in October, and one in December), and a final oral exam on January 8 or 9. The take-home assignments will be given a preliminary grade, the final grade for the course will only be determined at the oral exam.

**Oral exam.** At the oral exam, the take-home assignments (and the material related to these assignments) will be discussed,
and definitions, (counter)examples and basic constructions may be asked for. There is no need to learn
long proofs by heart. The exam lasts for 30 minutes. Please bring the marked copies of your
assignments to the exam.

**Schedule**. If you want to change your time slot, please find a student willing to
swap. Please be present 20 minutes in advance of your exam. All exams are in Utrecht.

Time |
Thu, 8 Jan (BBG 005) |
Fri, 9 Jan (BBG 069) |

10.00 | Tom Wennink | Alexander Gietelink |

10.40 | Ties Laarakker | Wibrich Drost |

11.20 | Emma Brakkee | Wessel Bindt |

13.00 | Stefan van der Lugt | Jeroen Hanselman |

13.40 | Mohamed Hashi | Maarten Zwaan |

14.20 | Freek Witteveen | Arend de Jonge |

15.00 | Reinier Kramer | Nguyen Tu Thinh |

16.00 | Michiel Jespers | Nandi Schoots |

16.40 | Peter Koymans | |

17.20 | David Kok |

**Assignment 2**. The second take-home assignment is due on December 11
(by email to both instructors.)

**Assignment 1**. The first take-home assignment is due on November 6
(by email to both instructors.)

### Program

**Week 14** (dec 18). Serre duality and Riemann-Roch for curves over a field.

**Week 13** (dec 11). Grothendieck vanishing, Cech cohomology of quasi-coherent
sheaves, cohomology of projective space, higher direct images.

*Literature*. Hartshorne III.3, III.4, III.5. Stacks project:
Grothendieck vanishing,
Cech resolution of a sheaf,
Alternating and ordered Cech complexes,
Cech cohomology of quasi-coherent sheaves,
Cohomology of projective space,
Higher direct images of coherent sheaves.

*Exercises*. Hartshorne: III.4.1, III.4.3, III.4.7, III.5.1, III.5.2.

**Week 12** (dec 4). Additive functors, left exact functors, right derived functors,
sheaf cohomology, acyclic resolutions, flasque sheaves, Cech complex.

*Literature*. Hartshorne III.1, III.2. Stacks project:
Abelian categories,
Additive functors,
Right derived functors and injective resolutions,
Cohomology of sheaves,
Flasque sheaves.

*Exercises*. See this file.

**Week 11** (nov 27). Formally smooth and formally étale morphisms,
cochain complexes, homotopy category of cochain complexes, injective resolutions,
right derived functors.

*Literature*. Hartshorne: III.1.
Stacks project: Formally smooth morphisms, Formally étale morphisms, Cochain complexes and homotopies, Homotopy category, Injective objects, Injective resolutions.

*Exercises*. See this file.

**Week 10** (nov 20). Sheaves of differentials, flat morphisms, smooth morphisms,
étale morphisms, formally smooth and formally étale morphisms.

*Literature*. Hartshorne II.8, III.10 (warning: we used a different definition of
smoothness, which works in greater generality). Stacks project:
Module of differentials,
Sheaf of relative differentials,
Flat morphisms,
Smooth morphisms,
Etale morphisms.

*Exercises*. HAG II.8.1, III.10.1, III.10.3.

**Week 9** (nov 13). Weil divisors. Class group. Relation with Pic.
Examples: **P**^{n}, finite covers of **P**^{1}.

*Literature*. Hartshorne II.6. Stacks project:
Divisors,
Effective Cartier divisors.

*Exercises*. HAG II.6.2, 6.6, 7.1.

**Week 8** (nov 6). Coherent sheaves on a locally noetherian scheme. Functor of points of
(closed subschemes of) **P**^{n}. Group schemes. Examples.

*Literature*. Hartshorne II.5 and II.7.1. Stacks project:
Coherent sheaves on a locally noetherian scheme,
Functor of points of projective space,
Group schemes,
Examples of group schemes.

*Exercises*. HAG II.5.16, and these.

**Week 7** (oct 30). Locally free sheaves. Tautological sheaf on projective space.
Tilde-construction on Spec and Proj. Quasi-coherent sheaves of modules.

*Literature*. Hartshorne II.5 up to Proposition 5.12. Stacks project: for more on
the tilde construction
see the sections
Affine schemes and
Projective schemes. See also
Quasi-coherent sheaves on Proj,
Invertible sheaves on Proj,
Quasi-coherent modules,
Quasi-coherent sheaves on affines.

*Exercises*. HAG II.5.2 (add that the homomorphism \rho should be K-linear), 5.3, and
these exercises.

**Week 6** (oct 23). Gluing. Proj. Projective space is proper over **Z**. Sheaves of modules
on a ringed space. Locally free sheaves.

*Literature*. Hartshorne II.2 (but we gave a different construction of Proj),
II.4.9 (with a different proof), II.3 (up to the adjunction between push-forward and pull-back).
Stacks project:
The category of *O*-modules,
Tensor product,
Sheaf hom,
Pushforward and pull-back,
Locally free sheaves.

*Exercises*. HAG II.2.14, II.3.12, II.5.1. Try also these exercises.

**Week 6** (oct 16). Separated morphisms. Proper morphisms. Graded rings. Proj and projective space.

*Literature*. Hartshorne II.4 but skipping the valuative criteria.
An alternative construction of Proj is in Hartshorne II.2. Stacks project:
Separation axioms,
Proper morphisms,
Graded rings,
Proj of a graded ring.

*Exercises*. See this file.

**Week 4** (oct 9). Fiber products. Cartesian diagrams. Base change. Fibers of a morphism. Separated morphisms.

*Literature*. Hartshorne: II.3, II.4 up to Cor. 4.6 but skipping the valuative criterion of separatedness. Stacks project:
Existence of fiber products of schemes,
Fiber products of schemes,
Base change in algebraic geometry,
Separation axioms.

*Exercises*. See this file.

**Week 3** (oct 2). Open immersions. Closed immersions. Integral schemes. Finiteness properties of schemes and morphisms. Yoneda and the functor of points of a scheme.

*Literature*. Hartshorne: II.3 up to 3.2.3. Some extra references from the stacks project:
Open immersions of LRS,
Integral schemes,
Noetherian schemes,
Morphisms of finite type,
Morphisms of finite presentation.

*Exercises*. See this file. Also: Hartshorne II.2.3, II.2.8, II.2.18 (if you have not done it yet), II.3.2, II.3.6, II.3.14.

**Week 2** (sept 25). Spec(*R*). Examples of spectra. Nullstellensatz for rings. Functoriality of Spec for ring maps. Examples. Spec (on Rings) and global sections (on LRS) are adjoint. Affine schemes, schemes. Schemes over a base scheme *S*. *k*-Varieties and their associated *k*-schemes.

*Literature*. Hartshorne: II.2. Stacks project:
Locally ringed spaces,
Affine schemes,
The category of affine schemes,
Local rings,
Irreducible components of spectra,
Schemes,
Zariski topology of schemes.

*Exercices*. See this file. Also: Hartshorne II.2.13, II.2.16, II.2.17, II.2.18, II.2.19.

**Week 1** (sept 18). Differentiable manifolds and algebraic varieties using sheaves of functions. Presheaves and sheaves on a topological space. Sheaves of groups, rings, etc. Sheafification. Pushforward and pullback of sheaves. f-maps between sheaves. Ringed spaces (RS), locally ringed spaces (LRS). Tangent space of locally ringed space at a point. Localization of (modules over) commutative rings. Definition of affine schemes (Aff) and schemes (Sch).

*Literature*. Atiyah-Macdonald: see chapter 3 for localization. Hartshorne: II.1 and II.2 up to 2.3.4. A lot more detail can be found in the Stacks project:
Localization,
Sheaves and continuous maps,
Ringed spaces,
Locally ringed spaces,
Spectrum of a ring,
Affine schemes.

*Exercises*. Hartshorne II.1.2-II.1.8, II.2.3, II.2.5, II.2.7, II.2.9, II.2.10. (Try to do a few exercises from both II.1 and II.2).