MATH 210A SOLUTIONS TO THE FINAL EXAM
2
n
1. (1) Only need to check that g (x)p R if g (x) R. In fact (a0 x + a1 xp + a2 xp + + an xp )p =
n+1
ap xp + + ap xp . (2) Only nontrivial part is f (g (x) + h(x) = f (g (x) + f (h(x), which follows from
n
0
n
p
p
MATH 210A HOMEWORK 7: SOLUTIONS
Note: I only give sketches of the nontrivial part of the exercises.
1. Theorems
2. Exercises
2.1. Each element in M is a linear map M A. Dene a multilinear : M1 Mn (M1 A
A Mn ) by sending linear maps (fi ) (wherefi : Mi A)
MATH 210A HOMEWORK 8: SOLUTIONS
Note: I only give sketches of the nontrivial part of the exercises.
1. Theorems
2. Exercises
2.1.
2.2. Since F is exact, F (ker(fi ) = ker(F (fi ) and F (Im(fi ) = Im(F (fi ). We have an exact sequence
0 Im(fi1 ) ker(fi ) H
MATH 210A HOMEWORK 9: SOLUTIONS
Note: I only give sketches of the nontrivial part of the exercises.
1. Theorems
2. Exercises
2.1. Pick a free resolution F of M . Then
TorA (lim Ni , M ) = H n (lim Ni ) A F ) H n (lim(Ni A F ) lim H n (Ni A F ) lim TorA (N
MATH 210A HOMEWORK 6: SOLUTIONS
1. Theorems
Theorem 1.
B A C is the direct sum of B and C in the category of commutative A-algebras means
(1) There are algebra maps i : B B A C and j : C B A C .
(2) For any A-algebra X with algebra maps f : B X and g : C
MATH 210A HOMEWORK 5: SOLUTIONS
Note: I only give sketches of the nontrivial part of the exercises.
1. Theorems
2. Exercises
2.1. (2) We need to show that m1 (x+In ) is still open for any x A. In fact, m1 (In ) = a,bA,abx+In (a+
In ) (b + In ), which is a
MATH 210A HOMEWORK 2: SOLUTIONS
Note: I only give sketches of the nontrivial part of the exercises.
1. Theorems
2. Exercises
2.1. (3) No. Consider Z Q.
2.2.
2.3. (1)(2) Decompose into a sum M = Ar (M [p]) (where p runs over a set of prime elements and
M [
MATH 210A HOMEWORK 3: SOLUTIONS
Note: I only give sketches of the nontrivial part of the exercises.
1. Theorems
2. Exercises
2.1. Initial: ; nal: singleton.
2.2. Direct sum: disjoint union of Xi then identify all the base points xi to one point, which is
MATH 210A HOMEWORK 4: SOLUTIONS
Note: I only give sketches of the nontrivial part of the exercises.
1. Theorems
2. Exercises
2.1. (1) Take A = B C , the product of two rings. Then (1, 0), (0, 1) D but their sum is the unit
element. (2) Let I D be an ideal
MATH 210A HOMEWORK 1: SOLUTIONS
Note: I only give sketches of the nontrivial part of the exercises.
1. Theorems
2. Exercises
2.1.
2.2.
2.3. No. R = k [x, y ] = M and N = (x, y ).
2.4. Take A = k [x, y ], n = 1 and N = (x, y ).
2.5. (3) Take any non-comm R