 
Summary: 1 Various remarks and comments at the begin
ning of the lecture regarding previous lectures
Remark 1.1. Restriction maps need not be injective.
Germs of sections of vector bundles form a sheaf (See Hartshorne Ex II.1.?)
Regular functions on a variety form a sheaf with the additional condition
that F() = {0}.
2 Discussion about topics for the courses over
the next two semesters
Material omitted here. (It will probably appear in the syllabus when the next
course is taught.)
3 Proj
Recall: If Ais a ring, we define SpecA which is the analog of an affine variety.
What is the analog of a projective variety in the theory of schemes?
(One answer is to use the functor t: varieties schemes.)
Let S be a graded ring, i.e. there exists a decomposition
S =
d0
Sd
as a direct sum of abelian groups such that d, e 0 SdSe Sd+e.
Example 3.1. S = k[x0, . . . , xn] Sd = k linear combinations of elements of S
