### Course Meeting Times

Lecture: 3 sessions / week, 1 hour / session

### Prerequisites

#### Required

18.705 Commutative Algebra (or equivalent)

18.725 Algebraic Geometry (or equivalent)

A suitable equivalent for 18.725 is familiarity with one of:

Shafarevich, Igor R. *Basic Algebraic Geometry 1: Varieties in Projective Space*. Translated by Miles Reid. New York, NY: Springer-Verlag, 1994. ISBN: 9783540548126.

or

Mumford, David. *The Red Book of Varieties and Schemes*. New York, NY: Springer, 2009. ISBN: 9783540632931.

#### Recommended

Some exposure to homological algebra and rudimentary category theory (e.g. in 18.905 Algebraic Topology). I’ll introduce what I need as I go along, but possibly in a pretty loose way with a lot of assertions left as “exercises for the listener.”

### Overview

It is easiest to explain what 18.726 is about by contrasting it with 18.725. In 18.725, one studies algebraic geometry using powerful techniques but with a “classical” frame of reference set up in the early 20th century. In the 1960s, a school of mostly French mathematicians led by Grothendieck developed a new language and toolkit for dealing with algebro-geometric objects, centered around the notion of a scheme. The construction of a scheme is the algebraic geometry analogue of the construction of a manifold (or differentiable manifold, or…) by glueing together small easily understood pieces; working with schemes makes it possible to distinguish “local” and “global” phenomena as is typical in other types of modern geometry. The definition of an “abstract algebraic variety” from 18.725 approximates this, but the notion of a scheme is much more general and much more flexible; it correctly accounts for nilpotents (a big help to intersection theory), deals well with base fields which are not algebraically closed, and easily accommodates objects of number-theoretic interest (like the ring of rational integers).

In some ways, this course is a language class: we will be learning how to read, write and speak the language of schemes. Along the way, we will absorb some of Grothendieck’s insights into this language: the importance of working locally, the relevance of relative properties of morphisms, base change, and how to think algebraically about the cohomology of algebro-geometric objects.

### Textbook

Hartshorne, Robin. *Algebraic Geometry*. New York, NY: Springer, 1997. ISBN: 9780387902449.

This is required mostly because I will draw many of the homework problems from it.

### Homework

Problem sets will be assigned each week, to be turned in the following week. Each problem set will involve doing a certain number of problems chosen from a larger set of offerings; students should choose what to submit based on their own background and interests. You should plan to spend a lot of time on the homework, since much of the challenge of this material is getting comfortable with the basic notions, and this is best achieved by grappling with them at close range.

### Exams

There are no exams in this course.

### Grading

The course grade will be based solely on the weekly problem sets.

### Note for Undergraduates

This class is not particularly intended for undergraduates, and is not appropriate as a first course in algebraic geometry (remember that 18.725 and 18.705 are both prerequisites). Also, the time required to complete the homework in this class may seem large even compared to other graduate courses. However, undergraduates with adequate preparation can take this class. Grading criteria will not distinguish between undergraduate and graduate students.

### Calendar

LEC # | TOPICS |
---|---|

1 | Introduction and overview |

2 | Basics of category theory |

3-5 | Sheaves |

5 | Abelian sheaves |

6-7 | Schemes |

7-9 | Morphisms of schemes |

9-10 | Sheaves of modules |

11-12 | More properties of morphisms |

12-13 | Projective morphisms, part 1 |

13-14 | Projective morphisms, part 2 |

15 | More properties of schemes |

16-17 | Flat morphisms and descent |

17-18 | Differentials |

18-19 | Divisors |

19-20 | Divisors on curves |

21-23 | Homological algebra |

24-26 | Sheaf cohomology |

27 | Cohomology of quasicoherent sheaves |

27-29 | Cohomology of projective spaces |

29-30 | Hilbert polynomials |

30-33 | GAGA |

33-34 | Serre duality for projective space |

35-36 | Dualizing sheaves and Riemann-Roch |

36-37 | Cohen-Macaulay schemes and Serre duality |

38 | Higher Riemann-Roch |

39 | Étale cohomology |