The Geometry of Schemes

Graduate Texts in Mathematics #197

by David Eisenbud, Joe Harris

Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.

Genres: Mathematics, Geometry, Textbooks

304 pages, Paperback

First published January 1, 2000

Reviews

[Jon Paprocki · Rating 4 out of 5]

I haven't read the whole thing yet, but the textbook that one reads cover-to-cover is extremely rare, so I'm reviewing it with approximately 60% of the material digested.

I've mostly been studying schemes on my own, which given extreme abstraction and difficulty typically associated with the topic, has been slow-going. I've tried using a variety of textbooks (included the (in)famous Hartshorne), but this is the first one I've tried where I can actually feel material sticking to the walls of my brain.

For somebody with a more geometric than algebraic brain, it is a godsend. Despite the fact that schemes are notoriously difficult to visualize, the book comes with an incredible number of pictures, and establishes geometric intuition with many references to the less abstract settings of varieties and manifold. It also includes lots and lots of examples and doesn't leave far too many important facts to be proven by the reader like Hartshorne tends to do.

No review on an introductory book on schemes would be complete without addressing the difficulty of the codependence between commutative algebra and schemes. Commutative algebra by itself seems strange and unmotivated (or that was my experience learning it last year), full of lots of definitions and theorems that don't seem to be leading anywhere in particular. The real intuition for commutative algebra comes from scheme theory, but unfortunately one must already know commutative algebra in order to understand schemes! It is a bit of a catch 22, but looking back on it, I do believe that the correct way to tackle this problem is to just keep your head down and learn some commutative algebra first (say, from Atiyah and MacDonald's Introduction to Commutative Algebra), and only then move towards schemes. I tried reading this book at the same time I was learning commutative algebra last year, and it was an exercise in frustration, and extremely slow. But after letting the commutative algebra settle into my brainstem for a year, I found that I could digest enough scheme theory in a day what previously took me a week.

The book concludes with the functor of points, which is an excellent jumping-off point into another entirely different viewpoint on schemes, and which also leads higher into the strata of algebraic spaces and stacks, where the air starts to become quite thin. I found it fun to pick out random results (mostly just the low-lying fruit) and then translate them into the functor of points language. For someone who finds the Yoneda lemma a warm and comfortable place, it is just what the doctor ordered after weeks of hard slogging in the algebraic trenches.

[Geoffrey Lee · Rating 4 out of 5]

To me, this book would seem a lot more effective if Eisenbud provided answers for some of the exercises in the first chapter.

[Lulu]

Algebraic Geometry