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I found the definition of a scheme to be slightly unclear/insufficiently precise at one point, so I have tweaked things slightly, and added more details. Indeed, it is quite common to find a formulation similar to ’every point has an open neighbourhood isomorphic to an affine scheme’, whereas I think it important to be clear that one does not have the freedom to choose the sheaf of rings on the local neighbourhood, it must be the restriction of the structure sheaf on $X$.
Right, so the open neighbourhood has to be isomorphic to an affine scheme as locally ringed spaces, not as topological spaces. (Or as ringed spaces – a morphism of ringed spaces which happen to be locally ringed is an isomorphism of ringed spaces if and only if it is an isomorphism of locally ringed spaces. But leaving the category of locally ringed spaces doesn’t seem right to me, since it is that category as opposed to the category of ringed spaces which has geometric relevance.)
I feel slightly uneasy with the new formulation “for every point, there is an open subset”: I’d prefer “there is an open covering”. This is because the latter generalizes better to schemes construed as locally ringed locales, while the former is only meaningful for locally ringed locales which happen to have enough points.
I propose either changing this part of the definition, or adding a new subsection “As locally ringed locales” detailing these concerns. What do you think?
Thanks for the thoughts!
I feel slightly uneasy with the new formulation “for every point, there is an open subset”: I’d prefer “there is an open covering”.
In fact, it did say “with an open covering (as locally ringed spaces)” before. But, even if I might prefer philosophically the “open covering” definition, I find that I’d like to be able to just look up a fully spelled out classical definition as well, just for ease of comparison with the literature.
I think all of the material that you add to the nLab is great. So definitely I’d be very happy for you to add a discussion of this to the entry, this would be a great addition. But I’d suggest to follow your second proposal, of separating out the “open covering” definition to a second definition or subsection, and adding a few words about why it’s preferable constructively.
Right, so the open neighbourhood has to be isomorphic to an affine scheme as locally ringed spaces, not as topological spaces.
Indeed. The point I really wished to emphasise is that ’open neighbourhood’ has to be interpreted in terms of locally ringed spaces, so that one does not have a choice about the sheaf. At least to me, it is possible to interpret the quote above as just saying that the isomorphism is as locally ringed spaces, leaving a bit ambiguous what sheaf one has on the open neighbourhood (if the latter is interpreted in topological spaces).
Okay! I’ll add an appropriate section. I totally agree that it’s important to specify the choice of sheaf, and am happy that this is now spelled out in more detail. :-)
Great, thanks!
I fixed a few typos, but I think there are more and I don’t understand the material sufficiently well. I’m specifically interested in the definition of open and closed subfunctors in terms of V(E) and D(E) for a set of “functions” on a functor. Is there a textbook where I can find more details? I’m a bit surprised that open subfunctors are defined in terms of V(E), since those look like zero loci to me.
Jonas Frey
@Jonas have you seen Zhen Lin Low’s thesis? This seems one of the likely modern places this is discussed. The only other one that comes to mind is Demazure and Gabriel’s book Groupes algebriques, but I can’t say for sure the general theory is covered much there.
@David: Thanks, I’ll have a look! Mathieu Anel also recommended these notes by Toen.
I felt it was unclear what the structure on $O(X)$ was making it into a $k$-ring and established this in some detail. The existing definition did not clarify in what sense one could carry out componentwise addition and multiplication, or with regards to what components (for each $k$-ring $R$?)
Patrick N.
Is there an approach to scheme theory in which schemes are viewed as algebraic objects instead of geometric ones?
Here’s what I mean: From the functorial perspective, a scheme is a certain kind of functor $F: \text{CRing}\to\text{Sets}$. These functors are geometric objects because they are generalized objects of $\text{CRing}^{\text{op}}$. What if I instead want to consider functors $G: \text{CRing}^{\text{op}}\to \text{Sets}$. These functors would be algebraic objects because they are generalized objects of $\text{CRing}$. Is there a natural way to define the opposite category of schemes in terms of certain functors $G: \text{CRing}^{\text{op}}\to \text{Sets}$? What topology on $\text{CRing}$ could these opposite-schemes be sheaves with respect to?
Perhaps take a look at Isbell duality. In the table there in Section 6 are schemes opposite to finitely generated commutative algebras.
Added:
The category of schemes admits small coproducts.
It does not admit coequalizers: https://mathoverflow.net/questions/9961/colimits-of-schemes/23966#23966
The category of schemes admits finite limits.
It does not admit infinite products: https://mathoverflow.net/questions/9134/arbitrary-products-of-schemes-dont-exist-do-they/65534#65534
There is some use of the terminology “algebraic scheme” in this page (as well as a few others), but I find it a bit confusing. For instance, an EGA prescheme is just a scheme today, not an algrebraic scheme (== scheme whose struct5ure map is of finite type). Right?
Seems a bit odd that this page devotes the introduction to the locally ringed space approach. Then when providing definitions via this approach and then via sheaves on $CRing^{op}$, it ends the subsection on the former with a link to functor of points, where we learn that it’s the latter that’s favoured by Grothendieck.
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