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M ATH 894
Take Home Exam
April 2016
I NSTRUCTIONS : The exam has five sets of questions labeled I through V. You should do all five
questions.
You may freely use any notes or statements from class or the book. It is also permissible to use
outside sources but if you do this be sure to acknowledge these references in your write up of those
questions. (You can also come and ask me questions if you get stuck).
The exam is due at Friday, April 15, 2016 at 13:00.
(Of course you may hand it in earlier if you wish.)
Questions:
I. — A DJOINT FUNCTORS .
II. — I NTEGRAL CLOSURES .
III. — R IGHT EXACTNESS .
IV. — I NTEGRALITY OF C HARACTERS .
V. — Q UESTIONS ABOUT Tor.
I. — A DJOINT F UNCTORS
Let C and D be categories, and F : C −→ D a functor which has a left adjoint.
Prove that F preserves fibre products. This means that, given any three objects X, Y , and Z of C,
with morphisms π1 : X −→ Z and π2 : Y −→ Z, prove that if X ×Z Y exists in C then F (X)×F (Z) F (Y )
exists in D and that F (X) ×F (Z) F (Y ) = F (X ×Z Y ).
II. — I NTEGRAL C LOSURES
Let A be a ring. Let a0 , . . . , an−1 be elements of A, and let p(x) = xn + an−1 xn−1 + · · · + a1 x + a0 . Set
M = A[x]/(p(x)); this is a free A-module of rank n.
(a) Let ϕx : M −→ M be the A-module homomorphism which is “multiplication by x”, i.e., by the
image of x under the quotient map A[x] −→ M. Find the characteristic polynomial of ϕx .
Now suppose that A is a domain, with quotient field K, and L any extension field of K. An element
α of L is called integral over A if there is a monic polynomial p(x) = xn + an−1 xn−1 + · · · + a1 x + a0 in
A[x] with α as a root. (Monic means that the leading coefficient of the polynomial, the coefficient of
xn , is 1.)
(b) Prove that the following are equivalent:
(b) The element α ∈ L is integral over A.
(b) There is a free A-module M and an A-module map ϕ : M −→ M such that α is an eigenvalue of ϕ ⊗ IdL : M ⊗A L −→ M ⊗A L. (N OTE : M ⊗A L is a finite dimensional vector
space over L, and IdL is the identity map on L).
(c) Suppose that α and β are elements of L which are integral over A. Let (M, ϕ), and (N, ψ) be the
free modules and A-module maps guaranteed by (b) above. Considering the A-module maps
ϕ ⊗ ψ and ϕ ⊗ IdN + IdM ⊗ψ from M ⊗A N to itself, prove that αβ and α + β are integral over A.
(d) The set of elements B in L which are integral over A is called the integral closure of A in L. Prove
B is a subring of L and that A ⊆ B.
A domain A is called integrally closed if the integral closure of A in its own quotient field K is A itself.
√
(e) Prove that the ring A = Z[ 5] is not integrally closed.
(f) Prove that A = Z is integrally closed.
III. — R IGHT
EXACTNESS
Let A be a ring.
(a) Say what it means for a functor F : A–Mod −→ A–Mod to be right exact.
(b) Is Symn : A–Mod −→ A–Mod right exact for n > 2?
(c) Is Λn : A–Mod −→ A–Mod right exact for n > 2?
IV. — I NTEGRALITY
OF
C HARACTERS
Let G be a finite group of order n. In this question all representations will be over C.
(a) Let ρ : G −→ GL(V ) be a representation of G. Explain why ρ(g)n = IdV for all g ∈ G.
(b) Show that all eigenvalues of ρ(g) are n-th roots of unity.
(c) Let B be the integral closure of Z in Q(ζn ), where ζn = e2πi/n . Show that χρ (g) ∈ B for all g ∈ G.
(Question II is relevant.)
(d) Suppose that χρ (g) ∈ Q. Show that χρ (g) ∈ Z. (See II again.)
V. — Q UESTIONS
ABOUT
Tor
Let A be any ring.
(a) Show that for any A-modules M, N1 , and N2 , that Tori (M, N1 ⊕N2 ) = Tori (M, N1 )⊕Tori (M, N2 )
for all i > 0.
Now let k be a field, A = k[x, y], and set M = A/(x, y) ∼
= k, where (x, y) is the ideal generated by x
and y.
(b) How do x and y act on M?
(c) Show that the complex below is a projective resolution of M :
A
h
δ2
A⊕A
δ1
A
(hy, −hx)
(f, g)
f x + gy
(d) Compute Tori (M, M) for all i > 0.
(e) Let N = A/(y). Compute Tori (M, N) for all i > 0.
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