1971
DOI: 10.1007/bfb0059750
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Algebraic Spaces

Abstract: This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by with the publisher.

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Cited by 252 publications
(185 citation statements)
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“…Aside from using a couple of self-contained results (Lemma 3.1.4 and Proposition A.1.3), our proof uses Corollary 3.1.12, which rests on Theorem 1.2.2, whose proof in turn relies on almost everything in §3.1 and §A.3. However, the noetherian case of Corollary 3.1.12 is an old result of Knutson [K,III,Thm. 3.3].…”
Section: Appendix a Some Foundational Facts For Algebraic Spacesmentioning
confidence: 99%
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“…Aside from using a couple of self-contained results (Lemma 3.1.4 and Proposition A.1.3), our proof uses Corollary 3.1.12, which rests on Theorem 1.2.2, whose proof in turn relies on almost everything in §3.1 and §A.3. However, the noetherian case of Corollary 3.1.12 is an old result of Knutson [K,III,Thm. 3.3].…”
Section: Appendix a Some Foundational Facts For Algebraic Spacesmentioning
confidence: 99%
“…Let S be any algebraic space. In [K,II,§6] the concept of points of S and the associated topological space |S| is defined in general but is only developed in a substantial manner when S is quasi-separated (i.e., ∆ S/ Spec Z is quasi-compact). The key to the theory is the fact [K,II,6.2] that if S is quasi-separated and s : Spec k → S is a morphism with k a field then there is a unique subfield k 0 ⊆ k such that s factors through a (necessarily unique) monomorphism s 0 : Spec k 0 → S. This can fail when S is not quasi-separated, as the following example shows, so the theory of points of S needs to be modified in general.…”
Section: Appendix a Some Foundational Facts For Algebraic Spacesmentioning
confidence: 99%
“…The representability of the moduli functor of ordinary curves of genus g with a level-n structure means that there is a moduli scheme M and a universal curve XM -+ M. Remark 3.6. For every trivializing covering p : Us ---> Ms, the fibred product R = Us x Ms Us defines an &ale equivalence relation (Pl, P2) : Us Xm, Us=~Us, whose quotient (in the category of locally ringed spaces [ 16]) is Ms. Then, the functor of spin curves 8spin ~ Ms e is the quotient of equivalence relation R" _~ Us" x m," Us" :::¢ Us" in the category of sheaves of sets.…”
Section: The Moduli Scheme Of Spin Curves Ms Is Quasi-projective Of Dmentioning
confidence: 99%
“…But even in the category of schemes, 6tale equivalence relations may fail to have a categorical quotient [16]. This problem is solved with the introduction of Artin's algebraic spaces [1,16], which are natural outgrows of schemes. In this larger category any &ale equivalence relation has a categorical quotient [ 16].…”
Section: The Representability Theorem For Susy Curvesmentioning
confidence: 99%
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