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I'm currently trying to learn algebraic geometry from Hartshorne's Algebraic Geometry. I've often heard it said, both on MathOverflow and in my department, that Hartshorne's treatment of certain topics is objectionable (especially in the foundational chapters II and III). Here I'm not talking about the mechanics of the presentation -- being too terse, leaving important theorems to the exercises, and so on. Rather, I take these comments to mean that he develops certain parts of the theory in ways that are philosophically questionable, or technically "hacky." In particular, his treatment of sheaves and cohomology is often singled out, though this is far from the only thing I've heard people grumble about.

What parts of the theory does Hartshorne do in a way that might be considered morally "wrong," and what would a "correct" treatment look like?

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    $\begingroup$ This is too opinion based to get anywhere good. $\endgroup$
    – Pedro
    Oct 30, 2015 at 17:57
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    $\begingroup$ Agreed. Hartshorne himself owns up to having elided some details about the development of sheaf cohomology, but he certainly didn't want to dedicate 50 or 100 pages to spelling out this theory. I'd be interested to know what other objections there are, especially from Chapter II. Considering he was in such a tricky place -- trying to bridge Atiyah-MacDonald (or maybe Matsumura) to EGA -- I still regard the book as a remarkable achievement. $\endgroup$ Oct 30, 2015 at 18:18
  • $\begingroup$ A good number of the big-list questions are almost entirely opinion-based. I'm not saying this justifies anything here. $\endgroup$
    – Hoot
    Oct 30, 2015 at 21:33
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    $\begingroup$ I think that the reason that Hartshorne's book is so widely used is that, despite any flaws it may have (and there are quite a few), we don't have anything [objectively/unquestionably] better. It deals with a large number of topics into a staggering depth (this is why the book always seems so intimidating to the novice). The high content-to-size ratio makes it a very good reference work; much more manageable than, say, EGA, the stacks project, or Vakil's notes (each of which, of course, is also invaluable). $\endgroup$
    – Remy
    Nov 1, 2015 at 2:01
  • $\begingroup$ @Remy Could you elaborate on what those flaws might be? I get the impression there's some quasi-canonical list of philosophical gripes. If there's not, perhaps this is a poor question. $\endgroup$
    – Potato
    Nov 1, 2015 at 2:03

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It seems that Hartshorne gets the definition of immersion "wrong." On page 120, section II.5, he defines an immersion to be an open immersion followed by a closed immersion. One might argue that a better definition is that an immersion is a closed immersion followed by an open immersion.

Hartshorne's definition creates problems, because according to his definition, a composition of immersions may not be an immersion. In particular, this makes exercise II.5.12 very awkward. See this answer for further discussion.

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    $\begingroup$ An example of an immersion which is not a composition of an open immersion followed by a closed immersion is given here, Example 28.3.4. $\endgroup$
    – Watson
    Feb 20, 2018 at 17:41

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