Synopsis of material from EGA Chapter II, 4
4. Projective bundles. Ample sheaves
4.1. Definition of projective bundles.
Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent OY -module.
The projective bundle over Y defined by E is the
Synopsis of material from EGA Chapter II, 3
3. Homogeneous spectrum of a sheaf of graded algebras
3.1. Homogeneous spectrum of a graded quasi-coherent OY algebra.
(3.1.1). Let Y be a prescheme. A sheaf of graded OY algebras S = n Sn is quasi-coherent
Synopsis of material from EGA Chapter II, 1
1. Affine morphisms
1.1. S-preschemes and OS -algebras.
(1.1.1). Given an S-prescheme f : X S, A(X) denotes the sheaf of OS algebras f OX .
Given a sheaf of OX modules (or OX algebras) F, A(F) denotes the sheaf
Synopsis of material from EGA Chapter IV, 1.11.7
1. Relative finiteness condtions. Constructible subsets of preschemes.
Some of the concepts to follow were introduced in Chapter I, 6, but are given a more
compete treatment here.
1.1. Quasi-compact morphis
Synopsis of material from EGA Chapter 0 (Vol. I) 14
1. Rings of fractions
1.0. Rings and algebras.
(1.0.1). All rings have a unit element 1. If we dont specify otherwise, rings are commutative and modules over non-commutative rings are left modules.
Synopsis of material from EGA Chapter II, 5
5. Quasi-affine, quasi-projective, proper and projective morphisms
5.1. Quasi-affine morphisms.
Definition (5.1.1). A scheme is quasi-affine if it is isomorphic to a quasi-compact open
subscheme of an affine sch
Synopsis of material from EGA Chapter II, 2.12.4
2. Homogeneous prime spectra
2.1. Generalities on graded rings and modules.
(2.1.1). Notation. Let S beL
an non-negatively graded ring. Its degree n component is
denoted Sn . The subset S+ = n>0 Sn is a gra
Synopsis of material from EGA Chapter II, 2.52.9
2.5. Sheaf associated to a graded module.
(2.5.1). If M is a graded S module, then M(f ) is an S(f ) module, giving a quasi-coherent
f(f ) on Spec(S(f ) ) = D+ (f ) Proj(S) (I, 1.3.4).