18.726 (Algebraic Geometry): Syllabus

Note: any changes or additions to this syllabus will be posted on the main course web page: math.mit.edu/18.726.

About the class

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.


Kiran Kedlaya, 2-165, x3-2946, kedlaya[at]mit[dot]edu (web site, more contact info). There may be a homework grader assigned based on class size and availability, but I haven't been promised one yet.


Tuesdays and Thursdays, 11-12:30, room 4-261 (campus map). Note: MIT protocol dictates that lectures start 5 minutes after the posted start time and end 5 minutes before the posted end time.


Required: Algebraic Geometry, by Robin Hartshorne. (Warning: although this book is extremely useful, and you will learn a lot from it, it is also a book that many love to hate, for different and sometimes contradictory reasons!)

Recommended: download Éléments de Géométrie Algébrique, by Grothendieck and Dieudonné, from NUMDAM, and keep it on hand in case you need to look up one of the odd references to it I will be sprinkling throughout the course. (Most of those references will be to EGA1, which is also available in English translation.) However, do not attempt to read EGA directly; this has been likened to reading a dictionary, with similarly unfruitful results. On the other end of the intuitive-to-formal spectrum, you may also find helpful The Geometry of Schemes, by Eisenbud and Harris; it provides some intuition lacking in Hartshorne (e.g., the functor of points).

On top of all that, I expect to supplement Hartshorne with some typed lecture notes. These will be distributed in class and also via this web site.


18.725 or equivalent; since 18.725 includes 18.705 (commutative algebra) as a corequisite, that means 18.705 is an implicit prerequisite here. More explicitly, a course in classical algebraic geometry at the level of, say, Shafarevich's Basic Algebraic Geometry I, or Part I of Mumford's The Red Book of Varieties and Schemes. Some familiarity with (or at least lack of fear of) homological algebra may come in handy later in the course.

If you did not take 18.725 (either last term or in a prior academic year), please come talk to me at your earliest convenience, so I can verify the "equivalence" of your background. One blanket warning: 18.725 covers sheaves pretty thoroughly, so I will not be spending much time on them at the beginning of the course; if you are not yet comfortable with them, you may start to have trouble in the course very soon!


Problem sets will be assigned each Thursday to be turned in the following Thursday; late homeworks will be accepted only by prior arrangement (i.e., sometime before the due date). I tend to be pretty reasonable about late homeworks; just don't push it. However, the material in this course (especially the basic facility with manipulating schemes that is crucial to further work in the subject) is largely delivered through homework, so getting behind on the homework is an extremely risky proposition!

The format for homeworks will be about the same as last term: on each set you will be required to do about 10 problems out of a pool of somewhat more. Unlike last term, some of the problems will not be given explicitly, but only by reference into Hartshorne.

Note that if you intend to submit homeworks for grading, you must be enrolled in the class (unless that is impossible, e.g., if you are not enrolled at MIT or a cross-registration school; in that case, see me for guidance). If you do not intend to submit homeworks for grading, you are still encouraged to register (or cross-register) as a listener. If nothing else, that will make it more likely that the class is assigned a grader.


There are no exams in this course.


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

Syllabus and lecture calendar

The syllabus and (proposed) lecture calendar are here.