# Opinions about the book “Lectures on Algebraic Geometry 1: Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces” [closed]

I'm not sure whether it is appropriate to ask this question here. I apologize if it isn't. I am deciding whether to buy the book "Lectures on Algebraic Geometry 1: Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces" by Gunter Harder. I like the content and the few pages I could find on Google. However, the editing seems very sloppy as I could spot several obvious typos and spelling errors. At one place the words "vfill" appear! I'm quite alarmed by the editing and wonder whether there might be serious mistakes in the mathematics too.

So I would like to know if anyone has read this book and could say something about it. I want to overlook the annoying typos if the mathematical content is great. Also, Gunter Harder seems like a serious mathematician in an important field.

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## closed as off topic by Loop Space, François G. Dorais♦, S. Carnahan♦, Kevin H. Lin, Anton GeraschenkoMay 4 '10 at 18:35

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What do you want to learn from the book? What is your background? That information is rather helpful in deciding whether to buy a book, and probably anyone trying to answer your question will need it... Now I gotta run to google books and search for «vfill»! – Mariano Suárez-Alvarez May 3 '10 at 22:47
My background is in number theory and algebraic geometry. I've studied Harshorne's and other algebraic geometry books. I just want to read Harder's to get more perspectives from (perhaps) an important mathematician. The book also discusses some topics I haven't learned like Hodge Theory. – Anon May 3 '10 at 22:55
This question now has a meta thread - tea.mathoverflow.net/discussion/383/opinions-about-books – François G. Dorais May 4 '10 at 5:47
There is a review of this book on MathSciNet ams.org/mathscinet-getitem?mr=2382668 – François G. Dorais May 4 '10 at 6:33
To the two commentators at that thread: I didn't mean to discuss things like how many typos were "too many". To me the book just seems like it's been very sloppily edited and it made me worry about the math content. Besides, I couldn't find any review of this book, so I asked for more info about it here. – Anon May 4 '10 at 6:38

Here is the MathSciNet review by Andrei D. Halanay:

The book under review is the first of a two-volume introduction to algebraic geometry. This first volume deals mostly with prerequisites, namely homological algebra and sheaf theory. The last chapter, occupying about one third of the book, is devoted to the classical theory of Riemann surfaces and abelian varieties. In this context the methods and results of the previous chapters are applied. The exposition is very pedagogical, every notion being first motivated by an example. For instance, derived functors are introduced after the reader is presented with the functors of invariants and co-invariants for a $G$-module ($G$ is an abelian group). These functors are not right exact (respectively left exact), thus prompting the need to consider the derived functors.

The book opens with a short chapter on category theory of a more abstract character. In this chapter the notions of product and of inductive and projective limit are considered in full generality. The second chapter is dedicated to homological algebra with an emphasis on cohomology and homology of groups. The ${\rm Ext}$ and ${\rm Tor}$ functors are also discussed. About half of the book is occupied by the third and fourth chapters, which are about sheaf theory and cohomology of sheaves. The fourth chapter goes much deeper than just ordinary'' sheaf cohomology and can be considered as a short course in algebraic topology. Thus spectral sequences and the cohomology of manifolds, including the theorems of de Rham and Dolbeault, are considered here. The chapter ends with a treatment of Hodge theory. The last chapter treats the subject of Riemann surfaces and abelian varieties. The exposition uses the results from previous chapters, but also classical complex analytic results. In this chapter the Riemann-Roch theorem, the algebraicity of Riemann surfaces (the reconstruction of a surface from its function field) and Serre duality are proved. Abelian varieties are first introduced as Jacobians of the surfaces and then studied per se. Thus the Kodaira embedding theorem is proved in this context. Also, a first discussion about moduli spaces takes place with the introduction of the Poincaré bundle, and line bundles on abelian varieties are investigated. The chapter (and this first volume) ends with a short introduction to the algebraic methods which can be used over arbitrary ground fields, methods to be fully developed in the next volume.

The book is very well written and the author has found the right measure between proving everything and leaving some aspects as exercises. Thus the Riemann-Roch theorem, Serre duality and many other results are almost fully proved (some hard-analytic aspects being just quoted, as is to be expected) while some others are left as exercises, for instance the link between the homotopy and homology groups of manifolds or some cohomological computations. It is worth mentioning that the author introduces the derived category of the category of sheaves as the natural setting for derived functors. It is a remarkable fact that he has managed to put together in a harmonious manner very abstract notions and constructions with classical ones from the time of Abel, Jacobi and Riemann. The book can be very valuable as a course text, not only in algebraic geometry, but also in sheaf theory, modern algebraic topology or classical theory of Riemann surfaces and their Jacobians due to its pedagogical nature.

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@Clark: thank you very much for posting this review. It is very helpful. – Anon May 4 '10 at 16:21

Personally I would be very sceptical about a book that at even a first glance has that many typographical and editing errors. I would, as you do, think more than once before spending serious money on something like that. I'm very sensitive to aesthetics (a bit silly perhaps).

However, as you point out, Harder is an extremely respectable and fine mathematician so I doubt that there are any serious mathematical errors in the book. So if you can overlook (and I know this is hard) the packaging and focus on the contents (which you seem to be more in favour of), I think you should buy it.

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I second the comment on Harder's reputation. There's a lot to learn from him, even if the edition of this book is sloppy (it probably is - I've seen versions of it floating around the web about 10 years ago). – Franz Lemmermeyer May 4 '10 at 5:45
@Daniel Larsson and @Franz Lemmermeyer: thank you for your opinions. I wasn't sure about prof. Harder and your opinions of him confirm my view and I will buy the book just because he's a fine mathematician and I really like the topics is his books. – Anon May 4 '10 at 6:33