MATH 614 LECTURE NOTES, FALL, 2013
by Mel Hochster
Lecture of September 4
We assume familiarity with the notions of ring, ideal, module, and with the polynomial
ring in one or nitely many variables over a commutative ring, as well as with homomorphisms of
Math 614, Fall 2013
Due: Wednesday, October 9
Problem Set #2
1. If R is a ring, f R and I R, I :R f denotes the ideal cfw_r R : rf I.
(a) Prove that for each prime P of R, the image of f in RP is not in IRP i P I :R f .
(b) Let K be a eld, let R = K[x1 ,
Math 614, Fall 2013
Problem Set #1: Solutions
1. Since D is not a eld, it has a nonzero maximal ideal m. Let P , Q be the inverse images
in R of (0) and m. Since the map is surjective, P = Q: in fact, Q/P m. Since P Q,
=
Q is in the closure of P . Hence,
Math 614, Fall 2013
Problem Set #2: Solutions
1. (a) The image of f is in IRP if and only if the image of f in RP /IRP (R/I)P is 0,
=
and that is the case if and only if some w R P kills the image of f in R/I, i.e., wf I
or w I :R f . Thus, the image of f
Math 614, Fall 2013
Problem Set #3: Solutions
1. T K[x1 , . . . , xn , z]/(x1 xn z 1). If we let yi = xi z ki , the equation becomes
=
(y1 + z k1 ) (yn + z kn )z which is monic in z provided that all the ki 1. Thus, we may
choose all the ki to be 1 and ta
Math 614, Fall 2013
Due: Monday, September 23
Problem Set #1
1. Let R be a commutative ring and suppose that there is a surjective ring homomorphism
of R onto an integral domain D that is not a eld. Prove that Spec (R) is not a Hausdor
space.
2. Let C[x]
Math 614, Fall 2013
Due: Wednesday, October 30
Problem Set #3
1. Let T = K[x1 , . . . , xn ] be a polynomial ring and let R = Tf , where f = x1 xn .
Determine an explicit K-subalgebra A R that is generated over K by algebraically
independent elements and
Math 614, Fall 2013
Due: Friday, November 22
Problem Set #4
1. Let x and y be relatively prime elements in a UFD R such that I = (x, y)R is a proper
ideal. You may assume that the surjection R2
I that sends (r, s) sx ry has kernel
Rv spanned by v = (x, y)
Math 614, Fall 2013
Due: Wednesday, December 11
Problem Set #5
1. Let R = K[w, x, y, z], a polynomial ring. Find an irredundant primary decomposition
for (x3 , xy 5 z, x2 yw, z 7 w7 )R consisting of ideals generated by monomials. Which ideals are
unique i
Math 614, Fall 2013
Problem Set #4: Solutions
1. We verify that the surjection R2
(x, y) sending (r, s) sx ry has kernel spanned
by v, although this was not required. If (r, s) 0 then sx = ry. Since x and y have no
common factor and x divides ry, it follo