Algebraic spaces

Versal deformations and algebraic stacks

“…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.…”

Generalized Heegner cycles and p-adic Rankin L-series

“…(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).…”

Mirror Symmetry via Logarithmic Degeneration Data I

“…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.…”

“…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).…”