Uli Walther
Math 746
walther@math.purdue.edu
494-1959 (work)
teaching Math 598E Spring 2008
MWF: 12:30-1:20 Math 211
Textbook: Iyengar, Leuschke, Leykin, Miller, Miller, Singh and
Walther: "24 h of local cohomology", Grad Studies in Math 87 (AMS).
Homework questions:
9.18, A4, A5, 1.6, A.11, 2.10, 2.15, 19.28, 21.18,
17.6, 20.8, 20.24,
Computer project:
Type "M2" into any math department computer, this will start
the computer algebra system Macaulay2. See
www.math.uiuc.edu/~Macaulay2 for stuff on the program, specifically
look at the documentation
http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.1/share/doc/Macaulay2/Macaulay2Doc/html/
where I find particularly the index useful.
Here are some things you should try: Define a ring R over a field KK of
your choice with 6 variables. Let I=(f1,f2,f3) be the ideal of the 2x2
minors discussed in class. Let J=(f1,f2,f3*f3). Compute minimal free
resolutions for both, and also compute all non-vanishing Ext's of R/I
and R/J into R. For the adventurous: determine the image of the map
Ext^3(R/J,R) -> Ext^3(R/Frob(J),R) where KK is characteristic p and
Frob is the Frobenius functor.
Now use Macaulay2 to to compute H^3_I(R) if KK=QQ. (Hint: "load
"D-modules.m2" " and then "help localCohom"). Compute the
characteristic variety "charVar" of this module and discuss what this module
looks like.
Can you find an (interesting) ideal I in ZZ[x_1,...,x_n] such that its
reduction mod p has a different projective dimension than its
reduction mod p' ? (I am excluding the uninteresting case I=(p)...)
Reading material for the week after Spring break:
1. The attached article.
2. In THE BOOK, read section 22.1 on Segre products of graded rings
and a particularly weird family of varieties related to number theory.
3. If I am not back on Friday, read sections 20.1, 20.2 and 20.3 on
semigroup rings.
Comments to
walther@math.purdue.edu