Timeline for Clarifying an interpretation of algebraic spaces
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 18, 2018 at 23:14 | comment | added | Marc Hoyois | To elaborate on Denis' comment, a Deligne-Mumford stack can be defined as a ringed topos locally equivalent to the etale topos of an affine scheme. I think this definition is due to Grothendieck, who called such topoi "multiplicités schématiques", but it is equivalent to the more standard definition. An algebraic space is precisely a 0-truncated DM stack (i.e. such that the groupoid of maps from any other DM stack is discrete). | |
| Apr 12, 2018 at 17:26 | vote | accept | Wenzhe | ||
| Feb 27, 2018 at 15:33 | history | edited | Qfwfq | CC BY-SA 3.0 |
added 11 characters in body
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| Feb 12, 2018 at 8:55 | comment | added | Denis Nardin | @nfdc23 Well, it is a ringed topos though... | |
| Feb 12, 2018 at 8:33 | answer | added | Niels | timeline score: 3 | |
| Feb 11, 2018 at 21:44 | comment | added | nfdc23 | It is only a vague heuristic, demystified by the real definition: an algebraic space is a functor $F$ on a certain category of schemes (it is not a ringed space!) such that it satisfies (i) the sheaf axiom for the etale topology, (ii) a "relative representability condition" for its diagonal, and (iii) admits an "etale cover" by the functor $h_X$ of points of a scheme $X$. So for an affine open cover $\{U_i\}$ of $X$, the functors $h_{U_i}$ constitute an "etale cover" of $F$ by (iii) and informally $F$ is a "gluing" of the $U_i$'s along the fiber products $U_i \times_F U_j$ (schemes by (ii)). | |
| Feb 11, 2018 at 21:06 | history | asked | Wenzhe | CC BY-SA 3.0 |