5/3/20: I have graded your HW 7. Here are some stats: mean = 5.8,
median = 6. I have used Adobe sharing to share the annotated pdf files
with you. Let me know if there is any problem.
5/3/20: I have graded your HW 7. Here are some stats: mean = 5.8,
median = 6. I have used Adobe sharing to share the annotated pdf files
with you. Let me know if there is any problem.
5/3/20: I have posted a few last-minute hints to HW 7 on the homework
page.
5/2/20: Some urgent things have come up, and I have not been able to
grade your HW 6 yet. I will do that on Sunday. I apologize.
5/2/20: Sean has drawn my attention to an inconsistency in Figure
20.2. Those typos may indeed be very annoying! Sorry about that. It
should be R3, rather than R2 in
both cases in the figure. The diagonal arrows denote multiplication of
row vectors by the given column vectors on the right. (x,y,z) in the
wording of the problem is meant to be (x1, x2,
x3).
4/24/20: I have replaced Exercise 17.2 with 20.16 on HW7 to support
our studying of Chaper 20. Sorry about doing this this late. But you
still have two weekends before the HW is due!
4/22/20: The last homework, #7, due Monday, May 4, is posted.
4/15/20: Homework 5 is graded. Some stats: mean = 4, median = 4, max =
6. I have shared your graded papers with you via the Adobe Acrobat
cloud. If you cannot access that or see my comments in the pdf file,
please let me know.
4/7/20: Homework 6, due Monday, April 20, is posted.
4/6/20: Homework 4 is graded. Some stats: mean = 4.8, median = 4, max
= 6.
3/24/20: Homework 5, due Monday, April 6, is posted.
3/22/20: Zane has pointed out another error in the wording of homework
Exercise E. The ring in Exercise E must be local, as we talked about
systems of parameters only for local rings. However, the ring in
Exercise E, as it was given to you originally, is not local: all
points (x,y,z) on X ⊂ A3 correspond to
maximal ideals. I was supposed to localize the ring at the origin
(0,0,0). I apologize and have corrected this on the online homework.
Here is also a little clarification for Exercise 10.8: the ring R is the coordinate ring A(X) = k[x,y,z]/(xz,yz). The module over that ring is meant to be the very ring A(X), as a module over itself. And the principal ideal by which you mod out A(X) is meant to be something like (z - 1).
3/19/20: Zane has pointed out to an error in the wording of homework
problem 11.3 from the text. The problem should be
reworded: Exercise 11.3: Show that a valuation ring is Noetherian
iff it is a DVR or a field. By the way, there is another error in
the textbook in the very definition of a valuation ring. Here is a
correct definition: A valuation ring is a domain R with a
valuation ν: K(R)* → G such that R = { x ∈
K(R)* | ν(x) ≥ 0} ∪ {0}.
3/19/20: So far, I have seen only one person at my Zoom office hours,
and that was yesterday. I hope everyone got a Google Calendar
invitation for them. Please break my solitude! ;-)
3/18/20: Oof! We have made through the first online class! And that
one had possibly the toughest proof in this course... I am open to
suggestions on how to improve online teaching. For instance, I can get
an iPad from my department and try doing lectures more similar to what
we had in the classroom, using the whiteboard feature of Zoom.
3/17/20: Folks, Homework 4 due date is postponed to Monday, March 23,
because of the delayed first instruction day after the COVID-19
extended Spring Break. Let us all stay at home, healthy and safe, and
Zoom!
3/9/20: Sorry for getting stuck in class on Friday before the break
when discussing algebraic Hartogs's theorem. As usual, things are
simpler than they looked back then. Here is how it works. If R is the
affine ring of a normal affine algebraic variety X (i.e., R is a
normal Noetherian domain over a field), and f is a rational function
on X (meaning f is in the quotient field K(R)), and q is a prime ideal
of R, then f ∉ Rq means f has the denominator in q,
that is, the denominator vanishing on the subvariety Z(q) ⊂ X,
whose codimension is codim q, by definition. Let us agree to say that
if f ∉ Rq, then f has a pole at Z(q), and if f ∈
Rq, then f does not have a pole at Z(q). Corollary 11.4
says that our ring R is the intersection of localizations
Rq over all primes q of codim 1. Geometrically, it means
that if a rational function f does not have a pole at any codim-1
subvariety Z(q), then f is regular, i.e., in R. Thus, if a rational
function f on X has a pole only at a subvariety Y ⊂ X of
codimension at least two, then f does not have a pole at any
subvariety Z(q) of codimension 1 and hence must be regular.
3/6/20: I have reduced the reading part of Homework 4 to adjust to
what will be covered in class through Monday after the Spring Break.
3/5/20: I have to move today's (Thursday, March 5) office to the union
of two time slots tomorrow (Friday, March 6), 11:15-12:00 and
1:00-1:30. Sorry about a late notice. I forgot that we were going to
have a faculty meeting today at 2:30.
3/2/20: My correction in class of K(R)p to K(Rp)
was wrong. It must be K(R)p. At the moment, I thought
K(R)p did not make sense, but what actually does not make
sense is the image of x from K(R) in K(Rp), whereas the
image of x from K(R) in K(R)p makes perfect sense.
3/2/20: Homework 4, due March 20, is posted.
2/28/20: I will be out of town Tuesday through Wednesday next week
(March 3-4), giving a colloquium talk at UC Riverside. This
means: no class meeting or office hours on Wednesday, March 4.
2/19/20: I have to cancel my office hours 3:35-4:25 p.m. tomorrow,
Thursday, February 20, and move them to 1:15-2;15 p.m., Friday,
February 21, just this week. I apologize for the inconvenience. My
office hours is the time you can be sure to find me in my office. I
will be happy to see you any other time --- make an appointment to
make sure I will be there in my office!
2/15/20: Homework 3, due February 28, is posted.
2/13/20: All homework is assigned based on Eisenbud's third printing (1999 or
later). If you are using an older printing, make sure to check the
correction lists on our class home page against the assigned
exercises.
2/11/20: I have added the comments which I have previously sent you by
email to Homework 2.
2/4/20: Homework 2, due February 14, is posted.
2/3/20: I have to cancel my regular office hours 3:35-4:25 p.m. today,
Monday, February 3. However, I will be available from 11:15 a.m. till
1:15 p.m. Stop by to break my solitude, especially if you have
questions!
1/29/20: I have to move my office hours for this Thursday, January 30,
to start one our later than usual, that is to say, start at 3:35
p.m. Since that will be one day before homework is due, to make it
convenient for all of you, I will be there in my office till 5:15 p.m.
1/22/20: I will check with you at the first class meeting whether my
office hours are okay with you. However, the announced office hours
for this Thursday, January 23, are not okay for me. I am moving them
to 1:25-2:15 p.m. for the time being, unless we agree upon something
else during the first class meeting.
1/16/20: I recommend the following way to study for this class. Attend
each class, take notes, participate in class actively. After each
class review your notes and study the corresponding part of the
text. You can find out which part of the text at the Class Outlines
page. Then do the assigned homework problems pertinent to that
material. I also encourage you to work in study groups on the
homework, as most of you did in the first term. Here is a guide
on How
to Form a Successful Study Group.
1/16/209: If you need to register for this course, please, send a
message to Stephanie Lawson at ugrad@math.umn.edu and ask for a permission number. If
permission is granted, go to OneStop and register.
Last modified: (2021-09-07 21:12:17 CDT)