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1. • CommentRowNumber1.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 22nd 2012
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   Author: Stephan A Spahn
   Format: MarkdownItexI edited [[group scheme]] and [[scheme]] a bit.

   I edited group scheme and scheme a bit.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 22nd 2012
   • (edited May 22nd 2012)
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   Author: zskoda
   Format: MarkdownItexForget, I wrote some nonsense

   Forget, I wrote some nonsense

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 8th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexWith some of the discussion taking place [here](http://nforum.mathforge.org/discussion/5149/extensive-category/) in mind, I added some remarks regarding set-theoretic issues at [[scheme]]. I hope I got it right; please check.

   With some of the discussion taking place here in mind, I added some remarks regarding set-theoretic issues at scheme. I hope I got it right; please check.

4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeAug 8th 2013
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   Author: Zhen Lin
   Format: MarkdownItexIt looks correct. I fixed some funny typesetting of primes. The second axiom in the sheaf-style definition can probably be replaced by "there is an epimorphism $\coprod_i U_i \to X$ in the category of sheaves", but I don't immediately see how to show that the two conditions are equivalent without knowing that both definitions lead to schemes in the locally ringed space sense.

   It looks correct. I fixed some funny typesetting of primes. The second axiom in the sheaf-style definition can probably be replaced by “there is an epimorphism $\coprod_i U_i \to X$ in the category of sheaves”, but I don’t immediately see how to show that the two conditions are equivalent without knowing that both definitions lead to schemes in the locally ringed space sense.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 1st 2014
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   Author: David_Corfield
   Format: MarkdownItexNot very helpful comment, but [[scheme]] doesn't strike me as particularly clear. Couldn't the functor-of-points approach be given greater prominence? This could be related to 'Examples and similar dualities' of [[Isbell duality]], no? As a more concrete point, this >the category $Aff:=Psh(CRing^{op})$ of presheaves on the opposite category of commutative rings should presumably be >the category $Psh(Aff):=Psh(CRing^{op})$ of presheaves on the opposite category of commutative rings. Or the defn $Aff :=CRing^{op}$ happens earlier. The page [[generalized schemes]] seems friendlier. But while on that page >To recall the equivalence between the two points of view, every scheme $X$ gives rise to a representable [[presheaf]] on the formal dual of commutative rings >$$ X(-) : CRing \to Set $$ >$$ A \mapsto Hom_{CRing}(Spec A, X) $$ is taking Hom space in the wrong category, no?

   Not very helpful comment, but scheme doesn’t strike me as particularly clear. Couldn’t the functor-of-points approach be given greater prominence? This could be related to ’Examples and similar dualities’ of Isbell duality, no?

   As a more concrete point, this

   the category $Aff:=Psh(CRing^{op})$ of presheaves on the opposite category of commutative rings

   should presumably be

   the category $Psh(Aff):=Psh(CRing^{op})$ of presheaves on the opposite category of commutative rings.

   Or the defn $Aff :=CRing^{op}$ happens earlier.

   The page generalized schemes seems friendlier. But while on that page

   To recall the equivalence between the two points of view, every scheme $X$ gives rise to a representable presheaf on the formal dual of commutative rings

   $$X(-) : CRing \to Set$$ $$A \mapsto Hom_{CRing}(Spec A, X)$$

   is taking Hom space in the wrong category, no?

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 1st 2014
   • (edited Aug 1st 2014)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex[[affine scheme]] isn't so clear either. E.g., in the term $Spec(\Gamma_Y \mathcal{O}_Y)$, we can find the $\mathcal{O}_Y$ from the linked page [[ringed space]], but I can't see $\Gamma_Y$ defined anywhere.

   affine scheme isn’t so clear either. E.g., in the term $Spec(\Gamma_Y \mathcal{O}_Y)$, we can find the $\mathcal{O}_Y$ from the linked page ringed space, but I can’t see $\Gamma_Y$ defined anywhere.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 1st 2014
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   Author: Todd_Trimble
   Format: MarkdownItex> is taking Hom space in the wrong category, no? I fixed it.

   is taking Hom space in the wrong category, no?

   I fixed it.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 1st 2014
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   Author: David_Corfield
   Format: MarkdownItexThanks. I tried to liven up [[functor of points]] with an illustration. Is there a standard example of a non-representable functor of points?

   Thanks.

   I tried to liven up functor of points with an illustration. Is there a standard example of a non-representable functor of points?

9. • CommentRowNumber9.
   • CommentAuthorZhen Lin
   • CommentTimeAug 1st 2014
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   Author: Zhen Lin
   Format: MarkdownItexThe functor of points of a non-affine scheme is not representable by an affine scheme (of course). For instance, $\mathbb{P}^n$. Or do you mean a (pre)sheaf that isn't representable by any scheme?

   The functor of points of a non-affine scheme is not representable by an affine scheme (of course). For instance, $\mathbb{P}^n$. Or do you mean a (pre)sheaf that isn’t representable by any scheme?

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 1st 2014
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    Author: David_Corfield
    Format: MarkdownItexRight. I guess I was just asking for a simple non-affine scheme.

    Right. I guess I was just asking for a simple non-affine scheme.