18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
More properties of morphisms (updated 5 Mar 09)
Note that nite presentation is not discussed in EGA 1; see EGA 4.1 instead.
1
More about separated morphisms
Lemma. The composition of closed immer
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Problem Set 5 (due Friday, March 13, in class)
Please submit exactly eleven of the following exercises, including all exercises marked
Required. In case it comes up, you may use Hartshornes more
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Morphisms of schemes (updated 20 Feb 09)
We next introduce morphisms of locally ringed spaces and schemes. Same references as
the previous handout.
Missing remark from last time: when EGA was wri
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Projective morphisms, part 1 (updated 3 Mar 08)
We now describe projective morphisms, starting over an ane base.
1
Proj of a graded ring
The construction of Proj of a graded ring was assigned as
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Problem Set 1 (due Friday, February 13, in class)
Since I didnt do so earlier, let me take space here to enunciate the collaboration policy
for this class. You may and should collaborate freely w
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Pro jective morphisms, part 2 (updated 7 Mar 09)
I particularly recommend Eisenbud-Harris for the material in this section; they give a
complete description of the relationship b etween the two d
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Schemes
We next introduce locally ringed spaces, ane schemes, and general schemes. References:
Hartshorne II.2, Eisenbud-Harris I.1, EGA 1.1.
1
Ringed and locally ringed spaces
A ringed space is
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Introduction
In this lecture, Ill give a bit of an overview of what we will be doing this semester, and
in particular how it will dier from 18.725. We will start in earnest (with the rudiments of
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Sheaves of modules (updated 27 Feb 09)
Having discussed sheaves of sets, abelian groups, and rings, we now consider sheaves of
modules over a locally ringed space, with an emphasis on the situati
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Category theory (updated 8 Feb 09)
Were going to use the language of category theory freely. Fortunately, its easy to learn
b ecause it corresponds naturally to the way you (hopefully) already th
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Sheaves (updated 12 Feb 09)
We are now ready to introduce the basic building blo ck in the theory of schemes, the
notion of a sheaf. See also: Hartshorne II.1, EGA 1 0.3. (The latter means: look
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Problem Set 2 (due Friday, February 20, in class)
Please submit exactly twelve of the following exercises, including all exercises marked
Required.
1. (Required) Prove that Set and its opposite c
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Problem Set 3 (due Friday, February 27, in class)
Please submit exactly thirteen of the following exercises, including all exercises marked
Required.
1. (Required) Use the fact that the structure
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
More on abelian sheaves
We now specialize the discussion of sheaves to the situation where the target category
consists of abelian groups. At the end, Ill explain how to generalize to the case of
18.726: Algebraic Geometry (K.S. Kedlaya, MIT, Spring 2009)
Problem Set 4 (due Friday, March 6, in class)
Please submit exactly twelve of the following exercises, including all exercises marked
Required.
1. Eisenbud-Harris II-11 and II-20. (If you need m