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. Now that we're getting into scheme theory, we'll use Eisenbud and Harris' The Geometry of Schemes. 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.

    HW due 9/28

    HW due 10/5

    HW due 10/26 (or the following Tuesday, for undergrads who'd rather be studying for the GRE): exercises I-9, 10, 15, 17, 20 from Eisenbud and Harris.

    HW due 11/9:

  • 1. Let X be a scheme and R = Gamma(X; O_X) the global functions. Show in excruciating detail that there is a map of schemes from X to Spec R.
  • 2. Let R be a commutative O_X-algebra, i.e. a sheaf of algebras over X with a compatible O_X-module structure. Show in detail how to define the scheme Spec R in terms of the Specs of the stalks, and its map to X. In particular if R = O_X you'd better get the identity X -> X.
  • 3. Exercise I-43 of [EH]
  • 4. Exercise I-46 of [EH]
  • 5a. Compute the fibers of Spec of the map RR[y] -> RR[x,y] / < y-x^2 > over the points < y-r > for r in RR (here RR = the reals).
  • 5b. Compute the fibers of Spec of the map ZZ -> ZZ[x] / < x^2+1 > over the points < p > for p in Spec ZZ (here ZZ = the integers).