Math 6670, Fall 2017

Allen Knutson

Tues/Thurs 11:40-12:55

Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.
—Sir Michael Atiyah, 2002

The book we'll use for reference is at the bottom here. Topics:

  • Varieties and their dimension theory
  • Subvarieties of Cn and the Nullstellensatz
  • Operations on ideals
  • Subvarieties of CPn
  • Nilpotents
  • Bezout's theorem
  • Hilbert functions
  • Hilbert dimension vs. Krull dimension
  • Heading toward schemes
  • Specm and Spec
  • The structure sheaf on Spec R [ch 4]
  • Sheaves, presheaves, sheafification, stalks, sheaf operations [ch 2]
  • Local rings and DVRs
  • Schemes
  • The equivalence of categories between affine schemes and commutative rings [ch 6]
  • ...
  • Initial notes here. Next notes.

    If you're getting a grade in this class, turn in HW. Due 8/31:

  • Ex 1.1, 1.2 from those notes
  • HW due 9/7:
  • Give the analogue of Taylor's theorem when expanding a function NN->ZZ as f(d) = \sum_n c_n (d+n choose n).
  • In particular, show that a polynomial f is integer-valued iff these c_n are all integer.
  • What does your analogue give for the non-polynomial function f(d)=2^d?
  • Prove the Hilbert syzygy theorem for the case of monomial ideals in x,y.
  • Let I = < xy-z > and J = < z-t > be ideals in C[x,y,z], where t is a number. What's the prime decomposition of I+J?
  • Let I be a homogeneous radical ideal, the intersection of some minimal prime ideals {P}. Find a formula for the degree of I (the leading coefficient of the Hilbert polynomial) in terms of the {P}.
  • HW due 9/21. Personally, I find the easiest way to use Macaulay2 to be on my machine from within emacs, but it is possible to use it online.