5.10 For a point P on a variety X, let m be the maximal ideal of the local ring OP . We dene
the Zariski tangent space TP (X) of X at P to be the dual k-vector space of m/m2 .
a) For any point P X, dim TP (X) dim X, with equality if and only if
a) Let f be the rational function on P2 given by f = x1 /x0 . Find the set of points where
f is dened and describe the corresponding regular function.
b) Now think of this function as a rational map from P2 to A1 . Embed
2.8 If X is a classical variety, and U X open, then f : U k is regular in the above sense if
and only if it is regular in the sense we dened for classical varieties.
Solution: This problem is actually straightforward once one
3.2 Show that if X is a variety, and for some P, Q X we have OP,X OQ,X K(X), then
P = Q.
Solution: Let U and V be ane open neighborhoods of P, Q, respectively. Then A(U ) OP,X
and A(V ) OQ,X . As in Exercise I.4.7 of Hartshor
a) Let Y be the plane curve y = x2 . Show that A(Y ) is isomorphic to a polynomial ring
in one variable over k.
b) Let Z be the plane curve xy = 1. Show that A(Z) is not isomorphic to a polynomial
ring in one variable ove
2.9 If Y An is an ane variety, we identify An with an open set U0 Pn by the homeomorphism 0 . Then we can speak of Y , the closure of Y in Pn , which is called the projective
closure of Y.
a) Show that I(Y ) is the ideal gene
2. Week 2
Unless otherwise noted, the letters R and A denote commutative rings with unity, the letters I and J denote
ideals, the letter k denotes a eld, and letters like Xi denote variables.
Denition 2.1. Let : Rn R be a li
1. Week 1
Unless noted otherwise, the letters R and A denote commutative rings with identity. The letters I and J
Exercise 1.1. Let R be a ring (not necessarily commutative, not necessarily with identity). For
3. Week 3
Exercise 3.1. Let f and g be monomials in R = k[x1 , . . . , xn ].
Say f = xa1 xan and g = xb1 xbn . Show f | g if and only if for all 1 i n, we have ai bi .
Prove that if f (g)R, then deg(f ) d
4. Week 4
For the following exercise set, the Newton polytope (unbounded) of I is the UNBOUNDED convex hull
of all the monomials in I. It is what Hbl denotes K(I).
Exercise 4.1. Show that if I and J are monomial ideals, th
3.2 A morphism whose underlying map on the topological spaces is a homeomorphism need
not be an isomorphism.
a) For example, let : A1 A2 be dened by t (t2 , t3 ). Show that denes a
bijective bicontinuous morphism of A1 onto t
5. Week 5
Denition 1. Let R be a ring, and S be another ring with R S. An element of S is integral over R if
it satises a monic polynomial with coecients in R.
Show (1 + 5)/2 is integral over Z.
(2) Show (
The rst two exercises are from the Chevalleys theorem and complete varieties lecture
2.13 Chows lemma. It is clear that every complete variety is birational to some projective
variety. However, a much stronger statement is true: given a
Problem Set 6 Math 221 Fall 2007 This problem set will cover some long overdue homological algebra. Let A be a ring. Modules will be modules over A unless otherwise specified. Remark: In many places "Noetherian ring and finitely generated module" can be r
Mandatory Assignment MAT4200 Solutions
Let k be any eld and let be any element in k. Show that following identitiy between ideals in the polynomial ring
k[X, Y ] holds:
(X 2 , XY ) = (X) (X 2 , Y + X).
Show that (X 2 , Y + X) = (X 2 , Y + X) if
MAT 4200, eksamen 12.12.13, solutions
La p1 , p2 , p3 vre forskjellige primtall, og la n = p1 p2 p3 .
1a. Hva er de maksimale idealene i Z/nZ? Hva er nilradikalet? Gi et eksempel p et
primrt ideal i Z/nZ som ikke er et primideal. (2pt)
Homework 1 (due: 10-7-11).
All rings are commutative with identity!
(1) [4pts] Let R be a nite ring. Show that R = NZD(R).
(2) [8pts] Suppose that R is a subring of a ring S and that R is a direct summand
of S as an R-module (i.e. S = R N as an R-module).
Solutions Ark 1
From the book: Number 8, 9, 10 and 12 on page 11.
Number 8 : Show that the set of prime ideals of A has a minimal element with respect
Solution: By Zorns lemma it sucies to show that any descending chain of prime
From the book: Number 10, 11 and 12 on page 32.
Number 10 : Let A be a ring, a an ideal contained in the Jacobson radical of A;
let M be an A-module and N a nitely generated A-module, and let u : M N be
homomorphism. If the induced homomorp
From the book: Number 1,2 and 4 on page 78.
Number 1: Let M be A-module and u : M M a module homomorphism.
i) If M is Noetherian and u is surjective, then u is an isomprphism.
ii) If M is Artinian and u is injective, then u is an isomorphis
From the book: Number 2, 4, 5, 10, and 11 on page 55 and 56.
Number 2: If a = a, then a has no embedded prime ideals.
Solution: Let a = q1 qr be a minimal primary decomposition end let pi = qi
be the assoicated p
arime ideals of a. As takin
From the book: Number 1, 2, 3, 5 and 6 on page 43 and 44.
Number 1: Let S be a multiplicatively closed subset of a ring A, and let M be a
nitely generated A-module. Then S 1 M = 0 if and only if there exists an element
s S such that sM = 0.
Problem Set 11 Math 221 Fall 2007 In this problem set we will study the module of relative dierentials. You may skip any one problem of your choice. Derivations and dierentials. Let A B be a ring homomorphism making B an A-algebra, and let M be a B-module
Homework 3 due: 11-18-11.
(1) [6pts] For a polynomial P (t) Q show that the following conditions are equivalent:
(a) P (n) Z for all integers n Z.
(b) P (n) Z for all but nitely many integers n Z.
(c) P (t) = i=0 ai t with ai Z and n N suitable.
Homework 4 (due: 12-9-11).
(1) [10pts] Let R be a semilocal Noetherian ring and I R an ideal of R. Show
that the following conditions are equivalent:
I is an ideal of denition of R.
I Jrad(R) and R/I is an Artinian ring.
I Jrad(R) and R/I
Homework 2 (due: 10-28-11).
(1) [4pts] Let R be a ring and Q R an ideal with radQ = P where P R is a
prime ideal. Show that Q is P -primary if and only if for all a, b R with ab Q
and a P we have that b Q.
(2) [10pts] Let R be a Noetherian ring, P R a p
Homework 4 (due: 12-10-07).
(1) Prove the Five Lemma: Consider a commutative diagram with exact rows: A1 A2 A3 A4 A5 t t t t t
1 2 3 4
B1 B2 B3 B4 B5 and prove: (a) If t2 and t4 are surjective and t5 is injective, then t3 is surjective. (b)
6. Supplemental Exercises
Exercise 6.1. This exercise goes back to review the denition of a ring, and explore an object that is
almost a ring. It should be signicantly more elementary than most of the other exercises in the