MATH 610, TOPICS IN ALGEBRA, HOMEWORK 2.
Solution of the mandatory exercises is due to Friday, June 23
1. Recommended exercises
1. Exercise. Recall that for a R-module M its annihilator consists of
Ann(M ) = {r ∈ R | rm = 0 for all m ∈ M }
Check that if M1 , M2 are two submodules of an R-module M , then Ann(M1 +M2 ) =
Ann(M1 ) ∩ Ann(M2 )
2. Exercise. Suppose M is a finitely generated module over a local ring R and m
denotes the unieque maximal ideal of R.
• Prove that elements x1 , . . . , xn ∈ M are generators of M over R if and only
if their classes x1 , . . . , xn ∈ M ⊗R R/m generate M ⊗R R/m as a vector
space over R/m. Hint: Use Nakayama lemma.
• Is it possible to drop the assumption that M is finitely generated?
Recall that a sequence of R-modules and homomorphisms
fi+1
fi
. . . → Mi+1 → Mi → Mi−i → . . .
is called exact at Mi if ker(fi ) = im(fi+1 ). A sequence is exact if it is exact at
every term, where there is incoming and outgoing morphisms.
3. Exercise. Prove that
• the homomorphism f : M → N is injective if and only if the sequence
f
0 → M → N is exact
• the homomorphism f : M → N is surjective if and only if the sequence
f
M → N → 0 is exact
• f : M1 → M2 is a homomorphism and M3 = M2 /f (M1 ) if and only if the
sequence
M1 → M 2 → M3 → 0
is exact (sequences if this form are called right exact)
• M1 is a submodule of M2 and f : M2 /M2 → M3 is an injective homomorphism if and only if the sequence
0 → M1 → M2 → M3
is exact (sequences if this form are called left exact)
• M1 is a submodule of M2 and M3 = M2 /M1 if and only if the sequence
0 → M1 → M2 → M3 → 0
is exact (sequences of this form are called short exact)
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MATH 610, TOPICS IN ALGEBRA, HOMEWORK 2.
4. Exercise. (Split exact sequences)
α
β
Suppose 0 → M1 → M2 → M3 → 0 is a short exact sequence of R-modules.
Prove that the following are equivalent:
• There is a retraction of β, i.e. a map f : M3 → M2 such that β ◦ f = idM3
• There is a retraction of α, i.e. a map g : M2 → M1 such that g ◦ α = idM1
• M2 splits as direct sum of M1 and M3 , i.e. there is an isomorphism of
∼
=
R-modules M2 → M1 ⊕ M3 that fits into the commutative diagram
/ M3
/ M2
M1
:
$ M1 ⊕ M3
We will say that a short exact sequence splits if it satisfies one of the equivalent
consitions above.
5. Exercise. Let M be an R-module. Prove that the following are equivalent:
• M is projective R-module
• every short exact sequence 0 → A → B → M → 0 splits.
6. Exercise. Prove that an R-module M is finitely generated if and only if there
is an exact sequence Rn → M → 0 for some n ∈ N.
7. Exercise. Recall that an R-module M is called finitely presented if there is a
right exact sequence of the form Rm → Rn → M → 0 for some n, m ∈ N. Prove
that a projective module M over R is finitely presented if and only if it is finitely
generated.
8. Exercise. Suppose that 0 → A → B → C → 0 is a split short exact sequence.
Prove that for any R-module M the sequence
0 → A ⊗R M → B ⊗R M → C ⊗R M → 0
is also split short exact sequence.
9. Exercise. Suppose φ : R → S is a ring homomorphism. Recall that any for any
R-module M the tensor product S ⊗R M has a natural strucuture of S-module, and
every S module N also has a structure of R-module by scalar restriction. Prove
that ther is an isomorphism
(S ⊗R M ) ⊗S N ∼
= M ⊗R N
10. Exercise. Suppose that R → S is a ring homomorphism. Let M, N be Rmodules. Prove that
∼ (S ⊗R M ) ⊗S (S ⊗R N )
S ⊗R (M ⊗R N ) =
11. Exercise. Recall that for any R-module M and ideal I in R there is an isomorphism M ⊗R R/I ∼
= M/IM .
• Show that Z/2 ⊗Z Q = 0.
• More generally, A is called torsion group if for any a ∈ A there exists some
n 6= 0 in Z such that na = 0. Prove that an abelian group A is torsion if
and only if A ⊗Z Q = 0
12. Exercise. Suppose R → S is a ring homomorphism.
MATH 610, TOPICS IN ALGEBRA, HOMEWORK 2.
3
• Prove that for R-modules M, M (M ⊕ N ) ⊗R S ∼
= M ⊗R S ⊕ N ⊗R S
• Prove that for any projective R-module P the S-module P ⊗R S is projective
13. Exercise. Suppose N is a flat R-module. Prove that for any long exact sequence
f1
f2
f3
f4
. . . → M1 → M2 → M3 → M4 → . . . the sequence of tensor products is exact:
f1 ⊗id
f2 ⊗id
f3 ⊗id
f4 ⊗id
. . . → M 1 ⊗R N → M 2 ⊗R N → M 3 ⊗R N → M 4 ⊗ N → . . .
(Hint: The long exact sequence can be presented in the form
f1
M1
!
K2
/ M2
=
f2
!
/ M3
=
!
f3
K3
/ M4
=
f4
/ ...
K4
where Ki = ker(fi ) and each 0 → Ki → Mi → Ki+1 → 0 is a short exact sequence.)
14. Exercise. Suppose R is an integral domain. Recall that the fraction field F of R
is defined as the localization S −1 R where S = {r ∈ R | r 6= 0} is the multiplicative
system of all nonzero elements of R. Prove that for any multiplicative system S
such that 0 ∈
/ S the localization S −1 R is isomorphic to a subring of the field F .
15. Exercise. Suppose k is an algebraically closed field. R = k[x1 , . . . xn ], Let
k(x1 , . . . , xn ) denote the fraction field of R. For any a = (a1 , . . . , an ) ∈ k n let
ma = {f ∈ R | f (a) = 0} denote the corresponding maximal ideal. Prove that
the local ring Rm is isomorphic to a subring of k(x1 , . . . , xn ) consisting of fg with
g(a) 6= 0.
2. Mandatory exercises
16. Exercise. Suppose R is a ring, f ∈ R. Consider a multiplicative system S =
{f n | n ∈ N0 }. Prove that the ring S −1 R is isomorphic to the ring R[ f1 ] = R[x]/(1−
f x)
17. Exercise. (exactness of localization) Suppose R is a ring S ⊆ R is a multiplicative system and 0 → M1 → M2 → M3 → 0 is a short exact sequence of R-modules.
Show that 0 → S −1 M1 → S −1 M2 → S −1 M3 → 0 is a short exact sequence of
S −1 R-modules
18. Exercise. Suppose that M is a free R-module of rank n. Recall that if m1 , . . . , mn ∈
M is a sequence of linearly independent elements, then these elements do not necessarily form a basis of M (for example, R = M = Z and m1 = 2).
Prove that if m1 , . . . , mn ∈ M generate M as R-module, then m1 , . . . , mn will
be linearly independent (so, they will form a basis of M ).
Hint: Consider a short exact sequence
f
0 → K → Rn → M → 0
(∗)
where f sends standard basis element ei to mi . To prove that m1 , . . . , mn are linearly independent, it is sufficient to prove that K=0. Show that the exact sequence
(∗) splits. Show that the localization Km = 0 for every maximal ideal m (Use the
Nakayama lemma)
19. Exercise. (Rank invariance of free modules over a commutative ring)
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MATH 610, TOPICS IN ALGEBRA, HOMEWORK 2.
• Suppose R is a ring and there is an isomorphism f : Rn → Rm between
free modules of rank n and m. Prove that n = m. (Hint: Take a maximal
ideal m in R. Check that f induces an isomorphism between tensor products
Rn ⊗ R/m → Rm ⊗ R/m)
• Suppose that there is a surjective homomorphism f : Rn → Rm . Prove that
n > m.
• Assume that the ring R is an integral domain. Prove that if there is an
injection f : Rn → Rm , then m 6 n (Hint: consider the field of fractions
of R)