Algebraic Geometry: Part I: Schemes. With Examples and Exercises (Advanced Lectures in Mathematics) ペーパーバック – 2010/6/25
Kindle 端末は必要ありません。無料 Kindle アプリのいずれかをダウンロードすると、スマートフォン、タブレットPCで Kindle 本をお読みいただけます。
This comprehensive introduction to schemes is complemented by many exercises that serve to check the comprehension of the text, treat further examples and give an outlook on further results. Includes details from commutative algebra in an appendix.
Prof. Dr. Ulrich Görtz, Institut für Experimentelle Mathematik, Universität Duisburg-Essen.Essen.
Prof. Dr. Torsten Wedhorn, Institut für Mathematik, Universität Paderborn.
Prof. Dr. Ulrich Görtz, Institute of Experimental Mathematics, University Duisburg-Essen.
Prof. Dr. Torsten Wedhorn, Department of Mathematics, University of Paderborn.
Some of the topics treated thoroughly in this text which are not treated in as much detail (or at all!) in the other standard texts include relative $\Spec$ and relative $\Proj$, (quasi-)projectivity (in the sense of EGA), base change and direct images, and non-Noetherian schemes. Generality is particularly important for arithmetic geometry, and this means working with schemes which aren't necessarily Noetherian. There is an entire chapter (Chapter 10) devoted to a systematic treatment of approximating non-Noetherian schemes and their morphisms by Noetherian ones, i.e., reduction to the Noetherian case, beginning with morphisms locally of finite presentation. The only reference works that cover more material are EGA and de Jong's Stacks Project. I haven't read much of EGA, but I've read quite a bit of the Stacks Project, and it is wonderful, but it is absolutely massive, and written in such a way that, to get to interesting results, one sometimes needs to read hundreds of pages of preliminary material. This is not a deficiency, in my opinion, because Stacks is meant to be a comprehensive reference on which to build the theory of more general objects. But it's not the most practical place to learn the basics. This book strikes an excellent balance between coverage and readability. I have studied it fairly completely, already knowing a fair amount of algebraic geometry, and feel that I've learned more from it than from Liu or Hartshorne (which is not to fault those books, which, again, have lots of good qualities of their own). The book also contains several very useful appendices. There is one devoted to category theory, one on the necessary results in commutative algebra (not many proofs but precise references), one on permanence for properties of morphisms, and one on relations between properties of morphisms. As with most algebraic geometry textbooks, this one has plenty of exercises at the end of each chapter. However I would say that the ones in this textbook are more valuable than the infamous Hartshorne exercises. Being the first printing of a new textbook, there are a fair number of typos. But there is an errata for the book at the website http://www.algebraic-geometry.de/.
My only complaint with the book, and it is minor, is that it is a very thick paperback. This means that, when reading certain sections (i.e. those not in the middle of the book), it has trouble staying open. Of course this also means the book is cheaper than something comparable in hardcover. But I would really like to have it in hardcover, just because hardcover texts tend to stay open and hold up better with time. But I'll take this one any way I can get it.
In summary, I highly recommend this to graduate students interested in learning and using modern algebraic geometry in their work, as well as to working mathematicians seeking a fairly comprehensive (without being gargantuan) reference for scheme theory. For graduate students, this book paves the way to more advanced reference\research works, such as Katz-Mazur Arithmetic Moduli of Elliptic Curves. (AM-108) (Annals of Mathematics Studies), Neron Models Neron Models (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics), and Cornell-Stevens Arithmetic Geometry. Lastly, I'll just say that this is the first book of which I've purchased two copies: one for home and one for the office. I just like it that much.
This new textbook (by Goertz and Wedhorn) seems the find the good balance: the text is crystal clear and very well organized, and the authors give a lot of motivation and examples. In particular there is a very nice chapter with interesting examples at the end of the book (cubic surfaces, abelian varieties, determinantal varieties, weighted projective spaces, ...). The material is presented in great generality (the authors are arithmetic geometers after all!) - fields are almost never supposed to be algebraically closed, hypotheses on morphisms are always minimal ("locally of finite presentation", "quasi-compact quasi-separated", ...). All this feels very natural - to my own surprise. There are more details than in the books by Hartshorne in Liu, and yet the text makes you think a lot. Each chapter has a nice exercise set. The appendices are fantastic.
You might think that the book - with its 600 pages - is too long; after all this is just the first of two volumes (and the second volume hasn't appeared yet). Curiously enough doesn't bother me, because the text is such a pleasure to read. The length of the book can be seen as a disadvantage, but I prefer looking at it as an advantage. I can't wait for the second volume!