Math 654
Homework #2
January 27, 2013
Due Tuesday, February 5, in class.
Justify all of your work.
Definitions:
• Recall throughout that “ring” means “ring with 1”.
• For a ring R, let R M denote the category of left R-modules, whose objects are left R-modules
and whose morphisms are R-module homomorphisms. Fix now two rings R and S, and fix a
covariant functor
F : R M → S M.
• We say that F is an exact functor if for any exact sequence of left R-modules
ψ
φ
M −→ N −→ P,
the sequence
F (ψ)
F (φ)
F (M ) −→ F (N ) −→ F (P )
is an exact sequence of S-modules.
• We say that F is a left exact functor if for any exact sequence of left R-modules
ψ
φ
0 −→ M −→ N −→ P,
the sequence
F (ψ)
F (φ)
0 −→ F (M ) −→ F (N ) −→ F (P )
is an exact sequence of S modules.
• We say that F is a right exact functor if for any exact sequence of left R-modules
ψ
φ
M −→ N −→ P −→ 0,
the sequence
F (ψ)
F (φ)
F (M ) −→ F (N ) −→ F (P ) −→ 0
is an exact sequence of S modules.
• For example, in problem 11(d) on p. 179, we showed that the torsion submodule functor is
left exact.
• Similar definitions of exact, left exact, and right exact can be made for contravariant functors
with the appropriate changes.
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B1. For two rings R and S, suppose we have a covariant functor F : R M → S M. Prove that F
is an exact functor if and only if F is both left exact and right exact.
B2. Let G be a group, and let Z[G] be the group ring of G over Z. A left Z[G]-module is sometimes
called a (left) G-module.
(a) Let M be a G-module. Show that the induced function
G × M → M,
(σ, m) 7→ σm
defines a left action of G on M .
(b) Let M and N be G-modules. Show that a function f : N → M is a G-module homomorphism (i.e., a Z[G]-module homomorphism) if and only if f is a homomorphism of
abelian groups and f (σn) = σf (n) for all n ∈ N and σ ∈ G.
(c) Let M be a G-module, and set
M G := {m ∈ M | ∀σ ∈ G, σm = m},
the set of G-invariants of M . Show that M G is an abelian subgroup of M under +.
B3. Continue with the situation of B2. Let A denote the category of abelian groups, and let G M
denote the category of G-modules.
(a) Define a covariant functor F : G M → A such that F (M ) = M G .
(b) Show that F is left exact.
(c) Show that F is not right exact by working out the following counterexample. Let t be
a variable, and let G = {tn | n ∈ Z} be the infinite cyclic group generated by t. As
we showed last semester, Z[G] = Z[t, t−1 ]. Now let M = Z[G], considered a G-module
under left-multiplication. Let N be the set
N = {n ∈ M | n = m(t − 1) for some m ∈ M } = Z[t, t−1 ](t − 1).
i. Show that N is a G-submodule of M .
ii. Show that as abelian groups, N/M ∼
= Z, and that the action of G on Z, induced by
this isomorphism, is trivial (i.e., σa = a for all σ ∈ G, a ∈ Z).
iii. Use the resulting exact sequence of G-modules,
0 → N → M → Z → 0,
to show that F is not right exact.
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