Algebraic Geometry: Part I: Schemes. With Examples and Exercises (Advanced Lectures in Mathematics) 2010th Edition by Ulrich Görtz (PDF)

Description

This book introduces the reader to modern algebraic geometry. It presents Grothendieck’s technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get startet, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐In the interest of full disclosure, I should note at the outset that I do arithmetic geometry, so the amount of modern algebraic geometry I consider to be foundational is considerable (I’ve been learning this stuff for over 4 years and there is still a ton that I don’t know). Having said that, this is absolutely my (current) favorite introductory textbook on the subject. It does not contain everything that someone working in arithmetic geometry needs to know, but the most important missing topics, namely cohomology and smooth, unramified, and etale morphisms, are supposed to be covered in the sequel (Part II). (Besides, if the book had given a treatment of cohomology at the same level of generality as it treats other topics, it would have been over 1000 pages long.) As it is, the book covers more important material than any of the other introductory texts on the subject, including Liu and Hartshorne, and in greater depth and generality. While I have personally found it very valuable to learn from all of the major texts, as each has its own strengths and weaknesses, if I were only able to have one textbook on scheme theory, I would choose Goertz-Wedhorn. 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 referenceresearch works, such as Katz-Mazur

⭐, Neron Models

⭐, and Cornell-Stevens

⭐. 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.

⭐Overwhelmingly my favorite text on algebraic geometry. I would rather not say much beside mentioning it is comprehensive, so long as the sequel is published (for reasons already covered above). I especially enjoy the text because it covers non-noetherian topics, as well some valuable insights into torsors for those learning the subject. This text together with FGA Explained are, in my opinion, is the most direct route to research level material.

⭐This is an amazing book for students of algebraic geometry. It complements Hartshorne well, and can be used as both a reference and a learning source.

⭐Very helpful companion to chapter two of Hartshorne. Would recommend to anyone learning sheaves and schemes for first time. Looking forward to their second volume (hope it comes out soon!).

⭐I am very excited about this new textbook on scheme theory. The canonical references for scheme theory are “Algebraic geometry” by Hartshorne, “Algebraic geometry and arithmetic curves” by Liu and “The red book of varieties and schemes” by Mumford. Each of these books has its problems: Hartshorne is sometimes a little bit too brief, leaves too many important things as exercises and does geometry over algebraically closed fields – so his treatment is often not sufficiently general for arithmetic geometers. Liu is much better for arithmetic geometry and the last few chapters are fantastic, but is a bit chaotic and lacking motivation and intuition, and his treatment of sheaf cohomology is disappointing (but it is still a good book!). We all know that Mumford has a terrific style and his “red book” is a gem, but it is not really a comprehensive introduction to scheme theory – too many topics are missing, but it is great to gain some intuition.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!

⭐I found the writing style, the amount of explanation, examples and exercises all very good. I would recommend to anyone learning algebraic geometry to a masters or starting PhD level, and certainly if you want to get a good grounding in the subject.

⭐Good introduction into the Field . Recommended on a masterlevel

⭐A wonderful book. It helps you to make your first steps in the world of algebraic geometry providing clear proofs and a lot of examples. At the end of every chapter there are a lot of instructive exercises.

⭐

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