11

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?

Potato
  • 41,411
  • 5
    This is too opinion based to get anywhere good. – Pedro Oct 30 '15 at 17:57
  • 4
    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. – John Brevik Oct 30 '15 at 18:18
  • A good number of the [tag:big-list] questions are almost entirely opinion-based. I'm not saying this justifies anything here. – Hoot Oct 30 '15 at 21:33
  • 2
    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). – Remy Nov 01 '15 at 02:01
  • @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. – Potato Nov 01 '15 at 02:03
  • It's certainly not the case that algebraic geometers go around passing down the list of objections to Hartshorne ;). I could think about what my objections are, but I'm not completely confident I am qualified to speak about this. And at any rate, I would need to think about it for a while. – Remy Nov 01 '15 at 02:14
  • I'm curious to know what the whispered objections to the sheaf and cohomology bits are. – Hoot Nov 02 '15 at 13:03
  • @Hoot I forget what I was thinking when I wrote that. I just noticed that Mark Haiman on his 256AB course website complains that EGA doesn't use derived categories, which I suppose is a complaint that could also be directed at Hartshorne. Whether that's a good complaint about something that wants to get to the geometry with a minimum of fuss is ... debatable ... – Potato Dec 09 '15 at 04:41
  • « Hartshorne's definition [of coherent sheaf] is incorrect », from https://math.stackexchange.com/questions/52856/is-noetherian-condition-always-needed-when-speaking-of-a-coherent-sheaf – Watson Feb 20 '18 at 17:39

1 Answers1

8

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.

Community
  • 1
Potato
  • 41,411
  • 2
    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. – Watson Feb 20 '18 at 17:41