Talk:Algebraic space

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At some point, the alternate definition of an algebraic space in terms of the functor of points for the étale topology probably should be added. I don't have a copy of Knutson handy, though. -- Spireguy (talk) 03:50, 7 March 2008 (UTC)

In addition, the bijective correspondence between smooth, proper algebraic spaces / C and Moishezon spaces should be added. This makes it possible to give an example (the classical one by Hironaka) of an algebraic space / C which is not a scheme. All of this obviously once there is enough in the article to define smooth and proper. Stca74 (talk) 19:30, 10 August 2008 (UTC)

The sentence ¨algebraic curves are schemes¨ reads a bit odd. Most people who dont know about algebraic spaces would define an algebraic curve to be a scheme! I guess it should read ¨algebraic spaces of dimension one are schemes¨. Similarly for smooth surfaces. —Preceding unsigned comment added by 130.237.48.109 (talk) 13:16, 3 November 2008 (UTC)

Trivial equivalence relation?

The meaning of

3. R is the trivial equivalence relation over each connected component of U

is not clear to me. I can think of a number of possible interpretations:

• The equivalence classes of R are identical with the connected components of U.
• Each connected component of U is contained in an equivalence class of R.
• For all x, y belonging to the same connected component of U, we have xRy iff x=y.
• For all x, y in U we have xRy iff x=y.

The third interpretation seems to allow for the connected components of U to be glued together in lots of weird ways, so I tend towards the fourth interpretation, which however doesn't involve the connected components of U at all.

Further, it is unclear to me whether the sentence

Thus, an algebraic space allows a single connected component of U to cover X with many "sheets".

applies to all algebraic spaces, or only to the ones satisfying the property 3 above. AxelBoldt (talk) 00:27, 31 December 2009 (UTC)

I believe the third interpretation is actually correct. Since the equivalence relation is etale, given the third condition, the gluing will be just the normal kind of gluing already allowed for schemes. As to the second sentence ask about, I clarified it in the article as well. -- Spireguy (talk) 21:05, 31 December 2009 (UTC)

Undefined terms

In the section #Algebraic_spaces_as_sheaves, the notation $h_X$ is never defined. I assume it means the functor $Hom(-,X)$ but not sure enough to put that in. Also it talks about morphisms between sheaves as though they were morphisms of schemes, without explanation. What does it mean for a morphism of sheaves (functors on $(Sch/S)_{et}$) to be surjective and etal? or representable? DSZ~enwiki (talk) 17:56, 25 January 2024 (UTC)