Math 689 Commutative and Homological Algebra
Homework Assignment 4
Due Friday November 20
Let R be an arbitrary ring with 1 6= 0.
1. Prove that chain homotopy is an equivalence relation.
2. Let B be a left R-module and let x be a nonzerodivisor of R. Show that TorR
1 (R/xR, B) =
{b ∈ B | xb = 0}.
3. Prove that an R-module Q is injective if and only if Ext1R (R/I, Q) = 0 for all left
ideals I. (Hint: Use Baer’s Criterion and a long exact sequence for Ext.)
n
4. Let k be a field, R = k[x], and I an ideal of R. Find TorR
n (R/I, R/I) and ExtR (R/I, R/I)
for all n ≥ 0.
5. Let k be a field and R = k[x]/(x2 ). Consider k to be an R-module on which x acts as
∼
multiplication by 0. Show that TorR
n (k, k) = k for all n ≥ 0.
6. Let k be a field, r a positive integer, r ≥ 2, and R = k[x]/(xr ). Consider k to be an
R-module on which x acts as multiplication by 0. Find ExtnR (k, k) for all n ≥ 0.
7. Let k be a field and q ∈ k × . Let R = kq [x, y], that is, R is the k-algebra generated
by x, y subject to the relation yx = qxy. (Note that as a k-vector space, R has basis
{xi y j | i, j ≥ 0}.) Consider k to be an R-module on which x and y each act as 0.
(a) Show that the following is a free resolution of k as an R-module:
β
α
0 → R −→ R ⊕ R −→ R → k → 0
qy
where α =
and β = (x y).
−x
(b) Find ExtnR (k, k) for all n ≥ 0.
8. Consider the extension of Z/2Z by Z/2Z:
α
β
0 −→ Z/2Z −→ Z/4Z −→ Z/2Z −→ 0,
where α is inclusion of Z/2Z as a subgroup of Z/4Z, and β is projection of Z/4Z onto
its quotient by this subgroup. Find the element of Ext1Z (Z/2Z, Z/2Z) corresponding
to this extension.