1971
DOI: 10.1007/bfb0059753
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Algebraic spaces

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Cited by 37 publications
(64 citation statements)
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“…Since Z is of finite length, Z ~ will be free over Z, and we chose a basis. Next, the descent data Zi~X ~ is given canonically by our choice of Z ~ It will lift to some affines U i over X ~ [12,II,6.4]. These choices determine a lifting of ( to one of the above schemes V. Now given any infinitesimal extension Z cZ~, the etale affine Y over Z x rX ~ extends canonically to u over Z 1 XF X~ [12,III,3.4], and so on.…”
Section: Reduction Of a Flat Groupoid To An Algebraic Stackmentioning
confidence: 95%
“…Since Z is of finite length, Z ~ will be free over Z, and we chose a basis. Next, the descent data Zi~X ~ is given canonically by our choice of Z ~ It will lift to some affines U i over X ~ [12,II,6.4]. These choices determine a lifting of ( to one of the above schemes V. Now given any infinitesimal extension Z cZ~, the etale affine Y over Z x rX ~ extends canonically to u over Z 1 XF X~ [12,III,3.4], and so on.…”
Section: Reduction Of a Flat Groupoid To An Algebraic Stackmentioning
confidence: 95%
“…(Even though such an X is initially built only as a separated algebraic space, it is a scheme. This can be seen in a couple of ways, perhaps the most concrete being that the jmap from X to P 1 ZOE1=N is quasi-finite, and any algebraic space that is separated and quasi-finite over a Noetherian scheme is a scheme [K,II,Section 6.16].) Remark A.1 For the reader who is interested in schemes being projective rather than just proper, we make some side remarks now (not to be used in what follows).…”
Section: Brian Conradmentioning
confidence: 99%
“…An algebraic space is separated if the equivalence relation defining it is a closed embedding ( [30], Def. II.1.6) and this is the case by Lemma 2.27.…”
Section: Lemma 229mentioning
confidence: 99%
“…II.1.6) and this is the case by Lemma 2.27. Thus in addition if S is separated, so is the morphism f : X 0 (B, P, s) → S ( [30], Prop. II.3.10).…”
Section: Lemma 229mentioning
confidence: 99%