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nLab > Latest Changes: scheme

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  1.  
    • CommentRowNumber1.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 29th 2018
    • (edited Apr 29th 2018)

    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 XX.

    diff, v30, current

  2. 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?

    • CommentRowNumber3.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 1st 2018
    • (edited May 1st 2018)

    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).

  4. 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. :-)

  5. Great, thanks!

  6. 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

    diff, v32, current

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 19th 2020

    @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.

    • CommentRowNumber8.
    • CommentAuthorJonasFrey
    • CommentTimeJul 19th 2020

    @David: Thanks, I’ll have a look! Mathieu Anel also recommended these notes by Toen.

  9. I felt it was unclear what the structure on O(X)O(X) was making it into a kk-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 kk-ring RR?)

    Patrick N.

    diff, v33, current

    • CommentRowNumber10.
    • CommentAuthorGuest
    • CommentTimeSep 10th 2020
    To elaborate, the previous draft was ambiguous in the sense that the reader might think that given two natural transformations $f, g: X \to O_k$, one should add them by defining, for a $k$-ring $R$, $f+g$ at $R$ to act on some $x\in X(R)$ and return $(f+g)_R(x) \in Hom(k[t], R)$ defined by $(f+g)_R(x)(p(t)) = f_R(x)(p(t)) + g_R(x)(p(t))$. But under this definition $(f+g)_R(x)$ will not be a ring homomorphism, as it will send the constant polynomial $1$ to $2$ in $R$.

  11. I reverted the edits I made as they ended up making things more confusing. Sorry.

    Patrick N.

    diff, v35, current

    • CommentRowNumber12.
    • CommentAuthorn.mertes
    • CommentTimeMar 27th 2021

    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:CRing→SetsF: \text{CRing}\to\text{Sets}. These functors are geometric objects because they are generalized objects of CRing op\text{CRing}^{\text{op}}. What if I instead want to consider functors G:CRing op→SetsG: \text{CRing}^{\text{op}}\to \text{Sets}. These functors would be algebraic objects because they are generalized objects of CRing\text{CRing}. Is there a natural way to define the opposite category of schemes in terms of certain functors G:CRing op→SetsG: \text{CRing}^{\text{op}}\to \text{Sets}? What topology on CRing\text{CRing} could these opposite-schemes be sheaves with respect to?

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 27th 2021

    Perhaps take a look at Isbell duality. In the table there in Section 6 are schemes opposite to finitely generated commutative algebras.

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 24th 2021
    • CommentRowNumber15.
    • CommentAuthorjonsterling
    • CommentTimeAug 10th 2022

    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?

  16. Change one instance of “algebraic scheme” to “scheme”.

    Anonymous

    diff, v37, current

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 13th 2022

    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 opCRing^{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.

  18. fix typo (\to vs \mapsto)

    diff, v41, current

  19. link to “Zariski site” seems appropriate for “Zariski Grothendieck topology”

    diff, v41, current

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 8th 2024

    Can we add precise references establishing an equivalence between the locally ringed space definition and the functor of points definition?

    There is something fishy about the currently stated definition using the functor of points, since it talks about sheaves on a large site, with no attempt to remedy the size problems, e.g., by passing to small presheaves or restricting the cardinalities of algebras forming affine schemes, making the site small.

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