Talk:Proj construction

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Great Work!T3kcit 21:50, 30 October 2006 (UTC)

Glad you like it. Ryan Reich 22:20, 30 October 2006 (UTC)

[edit] The twisting sheaf of Serre

minor problem here. Take k to be a field, and S=k[x] where x has degree 2. Then Proj(S)=Spec(k) is a single point, and yet the twisting sheaf as defined here is O(1). Let's see...the global sections of this are the degree one elements of k[x,x^{-1}] but all elements have even degree. But there is no nonzero coherent sheaf on Spec(k) without nonzero global sections....so somewhere you need to start assuming S is generated by elements of degree 1 in this section.89.242.8.39 23:05, 20 March 2007 (UTC)

Well, we do assume that in the Global Proj section, so it seems reasonable. However, I am not an expert on these nuances and I do actually make the note that it is erroneous that the ring can be reconstructed from the various powers of the twisting sheaf. I don't have my copy of Hartshorne with me and I don't remember this being in there, but then again, he might simply have stated at the beginning of the chapter that all his rings are generated in degree 1. This is exactly the sort of issue that could turn out to be "not a bug, but a feature" of the general theory, and I'm certainly not going to go around creating suspect mathematics by imposing a condition like this, even if it seems reasonable. I'll check later, and so should you. Ryan Reich 14:06, 21 March 2007 (UTC)

Ok I will check. By the way do you think the first half of Hartshorne is essentially a copy of Elements de Geometrie Algebrique by Grothendieck? (same person writing from different isp)137.205.232.153 08:43, 30 March 2007 (UTC)

I had a look myself and it does seem that he specifically mentions "generated in degree 1" in one of his big theorems. I don't really mention specific results by name here; I just wrote a sort of motivational spiel about what O(1) means, and that spiel does actually have the disclaimer that things don't always work out nicely. My one actual example is a polynomial ring, which is generated in degree 1. I guess: if you're sufficiently bothered by the distinction you should add some material that uses this assumption crucially and point it out. As for EGA, chapter 2 is most definitely a clone of (the table of contents of) EGA1, which is, I think, what was intended. At the time (1977) EGA was the only real reference for schemes (Mumford's Red Book was popular but not even officially published and also, not very complete), and it was in French and written like Wittgenstein's Tractatus Logico-Philosophicus, which is to say, with nearly unintelligible rigor. Hartshorne transformed it into a useful text for people trying to understand the (by then not-so-) new ideas, albeit incompletely. As for chapter 3...I'm just glad I don't have to read EGA3 to learn how one defines sheaf cohomology. BTW, why don't you open a Wikipedia account? It's easy and you get a name. Ryan Reich 13:21, 30 March 2007 (UTC)