# Schemes

The topics we will discuss are: Sheaves on topological spaces, ringed spaces, sheaves of modules, injective sheaves, cohomology of sheaves, locally ringed spaces, schemes, morphisms of schemes, properties of morphisms of schemes, closed immersions, open immersions, immersions, separated morphisms, morphisms of finite type, morphisms of finite presentation, finite morphisms, proper morphisms, invertible sheaves, Weil divisors, effective Cartier divisors, Cartier divisors, projective morphisms, projective spaces. Hopefully we will be able to say something about cohomology of projective space, duality, and prove the Riemann-Roch theorem for curves in the ``correct generality''.

I will be using the book by Robin Hartshorne, Algebraic Geometry.

It is strongly encouraged to go to the lectures, which are on Tuesday and Thursday 2:40-3:55 in Mathematics 520.

Problem sets will be announced in lecture on Thursdays and on this web page. They are due in lecture on the next Thursday. Please write out all arguments completely.

The TA for the course is Matt Deland (I think?). His email address is deland@math.columbia.edu and his office is Mathematics 408. He will have office hours on Monday probably between 10-11AM. Details TBA.

Grades are computed by a weighted average between the scores on problem sets and final. The weights are 2/3 and 1/3 respectively.

The final will be a written exam.

Here are the weekly problem sets. Please hit the refresh button on your browser to make sure you have to latest list.

• First set due Tuesday 23 January: all problems of set-1 below.
• Second set due Thursday Feb 1: all of set-2 below.
• Third set due Thursday Feb 8: all of set-3 below.
• Fourth set due Thursday Feb 15: all of set-4 below. (Last modified 9:23 AM Friday Feb 9, 2007.)
• Fifth set due Thursday Feb 22: Please catch up and hand in any exercises you haven't handed in yet.
• Sixth set due Thursday March 1: Set-5 below.
• Seventh set due Thursday March 8: Set-6 below. Added on March 15 a description of the boundary map H^0 to H^1 in Cech cohomology.
• Eigth set due Thursday March 22: As much as you can of Set-7 below.
• Nineth set due Thursday April 5: Set-8 below.