By Roger A. Johnson

This vintage textual content explores the geometry of the triangle and the circle, focusing on extensions of Euclidean concept, and studying intimately many fairly contemporary theorems. numerous hundred theorems and corollaries are formulated and proved thoroughly; a number of others stay unproved, for use by way of scholars as workouts. 1929 version.

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Push forwards and flat pull backs commute with base field extensions. Proof. For flat pull backs, this follows from the general fact that fiber products commute with base field extensions. 2). By linearity, it is sufficient to consider the case when f : Vk -. Wk is a morphism between irreducible and reduced schemes over k. The assertion is trivial if f is not generically finite. Let (Rj, mj) be the local rings of the generic points of W K and (R ij , mij) the local rings of the generic points of VK.

1). Set K = ker[Hom(J, k) -+ Ext1(lz , Oz)] and let qi : J -+ k i ~ k be a basis of K. Let Since Ext1(lz,oz) ® Jm ~ LExt1(lz,Oz) ® ki' we conclude that [EB"" (ZA')] = O. Conversely, if J -+ J' is a quotient such that [EB' (ZA)] = 0 and p : J -+ J' -+ k is any quotient, then also [EBI'(ZA)] = 0 where BP = B/kerp. This implies that every quotient of J' is a quotient of J m , hence J' itself is a quotient of Jm. Finally, by construction, dim ker q = dimJ - dimK ~ dimObs(Z). 3). 2 Infinitesimal Study of the Hilbert Scheme 31 As an application, first consider the Hilbert scheme of a scheme over a field.

1). 5). 5 Proposition. Let Y = yk be a scheme over a field k and Z c Y a closed subscheme defined over k with ideal sheaf 1= I z . Let yB be a scheme fiat over B such that Y B X Spec B Spec k ~ y. Let Y A = Y B X Spec B Spec A and ZA C yA a closed subscheme, fiat over A extending Z. 1) The obstruction [EB(ZA)] to extend ZA to a subscheme ZB c yB fiat over B lies in Ext} (Iz, 0 z ® J). 2) The set of extensions is either empty or they form a principal homogeneous space over Homy(lz, Oz ® J). Proof.