Math 689 Commutative and Homological Algebra
Homework Assignment 3
Due Friday October 30
All rings are commutative with 1 6= 0.
1. Let A and B be rings, C = A × B, and P ∈ Spec C.
(a) Show that dim C = max{dim A, dim B}.
(b) Find an example for which ht P + coht P < dim C.
(Hint: See #7 on Homework Assignment 2.)
2. Let A be a ring and M1 , . . . , Mn be A-modules. Show that
Ass(M1 ⊕ · · · ⊕ Mn ) = Ass(M1 ) ∪ · · · ∪ Ass(Mn ).
3. Let M = Z ⊕ Z/2Z as a Z-module.
(a) Find Ass(M ).
(b) Find two submodules U and V of M for which U + V = M but Ass(M ) 6=
Ass(U ) ∪ Ass(V ).
4. Let f : A → B be a homomorphism of noetherian rings. Let a f : Spec B → Spec A
be the induced map. Let M be a finitely generated B-module, considered to be an
A-module via f . Show that a f (AssB (M )) = AssA (M ).
5. Let k be a field and A = k[x, y]. Show that (x2 , xy) = (x) ∩ (x2 , y) is a primary
decomposition of (x2 , xy). (Hint: See #4 on Homework Assignment 2.)
6. Let k be a field and A = k[x, y, z]. Let P1 = (x, y), P2 = (x, z), P3 = (x, y, z), and
I = P1 P2 . Show that I = P1 ∩ P2 ∩ P32 is a primary decomposition of I.