# Errata to Principles of Algebraic Geometry

The Principles of Algebraic Geometry is a great book with, IMHO, many typos and mistakes. Why don't we collaborate to write a full list of all of its typos, mistakes etc? My suggestions:

Page 10 top, definition of $\mathcal{O}_{n,z}$ is wrong (or at least written in a confusing way)

Page 15, change of coordinates given for the projective spaces only work when $i < j$. It states that the given transitions also work in the case when $j< i$.

Page 27, need to put a bar on the second entry of the $h_ij(z)$ operator defined. Also, shouldn't the title of this section be geometry of complex manifolds, instead of calculus on complex manifolds?

Page 35, definition of what is a sheaf is wrong. The gluing condition should be for any family of open sets, not just for pairs of open sets! (I've seem PhD students presenting this definition of sheaf on pg seminars...)

Page 74, writes $D(\psi \wedge e)$, but $\psi$ and $e$ are in two different vector spaces, and one cannot wedge vectors in different vector spaces... I guess they mean tensor product.

Page 130, definition of divisor: it says it's a linear combination of codim 1 of irreducible subvarieties. By linear it means over $\mathbb{Z}$ not over the complex numbers (better should say, like Hartshorne, that $Div$ is the free abelian group generated by the irreducible subvarieties).

Page 180, equation (*) has target a direct sum of line bundles, not tensor.

Page 366, when it says "supported smooth functions over $\mathbb{R}^n$, are these complex valued or real valued functions?

Page 440 top equation. Is it really correct?

Page 445 Second phrase of hypercohomology section; it says sheaves of abelian sheaves. Probably means set of abelian sheaves.

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@SpecR, I'd encourage you to get in touch with a moderator (e.g. me, scott@tqft.net) -- you didn't leave an email address. Errata requests are difficult and dangerous projects to attempt on mathoverflow, and I'd like to make sure this is going to work out. –  Scott Morrison Jan 26 '10 at 1:26
In particular, unless you've read and digested this discussion: tea.mathoverflow.net/discussion/154/… about a previous errata question, I would be inclined to discourage this. –  Scott Morrison Jan 26 '10 at 1:28
Scott, I've read the above thread, but still don't get why people are so worried about errata questions. These are mathematically meaningful and can render a service to the community, especially when it comes to books that many people use, like Griffiths-Harris. Moreover, spotting a mistake can be non-trivial, so I don't understand the rationale for making it community wiki either. Of course, this may mean that the questions would pop up from time to time, but I personally don't mind and, judging by the number of upvotes this post and Kevin's have got, there are others who won't mind either. –  algori Jan 26 '10 at 6:35
The greatest error of the book is the lack of exercises :-) –  Kevin Lin Jan 26 '10 at 7:32
For what it's worth, here's some comments to the OP about what I learnt from the Cassels-Froehlich errata thread: (1) state which edition of the book you're talking about. (2) don't just ask here, ask in other places on the internet. (3) Be prepared to put in a lot of work collating responses. I could also add that in the Cassels-Froehlich case there was no chance of getting the authors to do the dirty work (too many authors of the articles, and too old a book). Here another approach might be to push the authors to help you out. –  Kevin Buzzard Jan 26 '10 at 12:34