INFO FOR THE FINAL EXAM
MATH 538 FALL 2011
The nal exam will be oral, individual and in a seminar room. The exam will be 20-30 minutes. You can
schedule your exam for any time after thanksgiving including during nals week. Send me an email saying when
you
HOMEWORK # 6
DUE FRIDAY NOVEMBER 18TH
MATH 538 FALL 2011
1. Let A be a ring and suppose that a is an ideal. Dene a ring Ga (A) = an /an+1 where a0 := A. This is
n=0
a graded ring with multiplication induced by multiplication on the Rees-algebra. If A is N
HOMEWORK # 7
DUE FRIDAY DECEMBER 9TH
MATH 538 FALL 2011
1. Suppose that A is a ring and that M and N are A-modules. A module L together with a short exact sequence
0 M L N 0 is called an extension of M and N . For example, M N is an extension of M and N
w
A FACT ABOUT REGULAR LOCAL RINGS
MATH 538 FALL 2011
1. Prime avoidance and regular local rings are domains
We will prove that regular rings are integral domains. Before continuing however, I need
a stronger form of prime avoidance.
Lemma 1.1 (Prime avoida
WORKSHEET # 1
MATH 538 FALL 2011
In this worksheet, well go through the proof of primary decomposition of ideals in a Noetherian ring. In particular,
we will prove the following
Theorem. In a Noetherian ring A, every ideal I has a primary decomposition. T
WORKSHEET # 2
MATH 538 FALL 2011
In this worksheet, well explore normalization of an integral domain.
Denition 0.1. Suppose R is an integral domain with eld of fractions K (R). We dene the normalization of R to
be the integral closure of R inside K (R). W
WORKSHEET # 3
MATH 538 FALL 2011
In this worksheet, well explore valuation rings. The content of this worksheet is taken largely from AtiyahMacdonald.
Denition 0.1. Suppose that B is an integral domain and K = K (B ) is its eld of fractions. We say that B
WORKSHEET # 4 SOLUTIONS
MATH 538 FALL 2011
Our goal in this worksheet is to play with faithful atness. Recall the following denition:
Denition 0.1. If R is a ring and M is an R-module, we say that M is faithfully at if M is at and if for any two
R-modules
HOMEWORK # 1
DUE FRIDAY SEPTEMBER 2ND
MATH 538 FALL 2011
1. Suppose that : R S is a ring homomorphism and I is an ideal of R. If a is an ideal of S , prove that 1 (a)
is an ideal of R. More generally, if b is an ideal of R and a = (b) S is the ideal of S
HOMEWORK # 3
DUE MONDAY OCTOBER 3RD
MATH 538 FALL 2011
1. Suppose that k is an algebraically closed eld, and that R and S are two nite generated k algebras (in other
words, R = k [x1 , . . . , xm ]/I and S = k [y1 , . . . , yn ]/J . Prove that there is a
HOMEWORK # 5
DUE FRIDAY NOVEMBER 4TH
MATH 538 FALL 2011
1. Use Nakaymas lemma and results from a worksheet to show that if (A, m) is a Noetherian local ring, then the
maximal ideal m is principal if and only if m/m2 is 1-dimensional over k = R/m.
Solution
HOMEWORK # 4
DUE WEDNESDAY OCTOBER 19TH
MATH 538 FALL 2011
1. Is the following true or false. If it is true, prove it. If it is false, give a counter-example. If R is a ring, M is an
R-module and M1 and M2 are submodules of M such that M = M1 + M2 , then
HOMEWORK # 3
DUE MONDAY OCTOBER 3RD
MATH 538 FALL 2011
1. Suppose that k is an algebraically closed eld, and that R and S are two nite generated k algebras (in other
words, R = k [x1 , . . . , xm ]/I and S = k [y1 , . . . , yn ]/J . Prove that there is a
SOLUTIONS TO HOMEWORK # 1
DUE FRIDAY SEPTEMBER 2ND
MATH 538 FALL 2011
1. Suppose that : R S is a ring homomorphism and I is an ideal of R. If a is an ideal of S , prove that 1 (a)
is an ideal of R. More generally, if b is an ideal of R and a = (b) S is th
HOMEWORK # 2
SOLUTIONS
MATH 538 FALL 2011
1. Suppose that R := Z/ m and that S 1 cfw_1, n, n2 , n3 , . . . is a multiplicative system. Determine S 1 R.
Hint: Note S 1 R (S 1 Z)/(S 1 m ) (we will prove this in class soon, or see the book).
=
Solution:
Not
HOMEWORK # 3
DUE MONDAY OCTOBER 3RD
MATH 538 FALL 2011
1. Suppose that k is an algebraically closed eld, and that R and S are two nite generated k algebras (in other
words, R = k [x1 , . . . , xm ]/I and S = k [y1 , . . . , yn ]/J . Prove that there is a
HOMEWORK # 4
DUE WEDNESDAY OCTOBER 19TH
MATH 538 FALL 2011
1. Is the following true or false. If it is true, prove it. If it is false, give a counter-example. If R is a ring, M is an
R-module and M1 and M2 are submodules of M such that M = M1 + M2 , then
HOMEWORK # 5
DUE FRIDAY NOVEMBER 4TH
MATH 538 FALL 2011
1. Use Nakaymas lemma and results from a worksheet to show that if (A, m) is a Noetherian local ring, then the
maximal ideal m is principal if and only if m/m2 is 1-dimensional over k = R/m.
Solution
HOMEWORK # 6
DUE FRIDAY NOVEMBER 18TH
MATH 538 FALL 2011
1. Let A be a ring and suppose that a is an ideal. Dene a ring Ga (A) = an /an+1 where a0 := A. This is
n=0
a graded ring with multiplication induced by multiplication on the Rees-algebra. If A is N
HOMEWORK # 7
DUE FRIDAY DECEMBER 9TH
MATH 538 FALL 2011
1. Suppose that A is a ring and that M and N are A-modules. A module L together with a short exact sequence
0 M L N 0 is called an extension of M and N . For example, M N is an extension of M and N
w
INFO FOR THE FINAL EXAM
MATH 538 FALL 2011
The nal exam will be oral, individual and in a seminar room. The exam will be 20-30 minutes. You can
schedule your exam for any time after thanksgiving including during nals week. Send me an email saying when
you
NOTES ON HOMOLOGICAL ALGEBRA
MATH 538 FALL 2011
1. Tor
Fix a ring A and a module M . A free resolution of M is a set of free modules Fi = Rni , i Z0 and
maps Fi Fi1 (set F1 = 0) as well as a map : F0 M such that
F3 F2 F1 F0 M 0
is exact. The data:
F3 F2
HOMEWORK # 2
DUE FRIDAY SEPTEMBER 16TH
MATH 538 FALL 2011
1. Suppose that R := Z/ m and that S 1 cfw_1, n, n2 , n3 , . . . is a multiplicative system. Determine S 1 R.
Hint: Note S 1 R (S 1 Z)/(S 1 m ) (we will prove this in class soon, or see the book).