Math 689 Commutative and Homological Algebra Homework Assignment 5

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Math 689 Commutative and Homological Algebra
Homework Assignment 5
Due Wednesday December 16
Let R be an arbitrary ring with 1 6= 0.
1. (An algebraic Mayer-Vietoris sequence.) Let R be a ring, let A, B be submodules of
a left R-module M , and let C be another left R-module. Prove that there is a long
exact sequence
0 → HomR (A + B, C) → HomR (A, C) ⊕ HomR (B, C) → HomR (A ∩ B, C) →
Ext1R (A + B, C) → Ext1R (A, C) ⊕ Ext1R (B, C) → Ext1 (A ∩ B, C) → · · ·
2. Let R = Z/4Z.
(a) Let M = Z/2Z, considered as an R-module via the quotient map from R to
M∼
= R/(2). Show that pdR (M ) = ∞ by computing ExtnR (M, M ) for all n.
(b) Show that gldim(R) 6= dim(R) (where dim(R) is the Krull dimension of R).
3. Let R = k[x, y], I = (x, y), and consider the R-module k ∼
= R/I. Use the Koszul
resolution of k to find ExtnR (k, R) for all n. As a consequence, find depthI (R).
For #4,5 below, an augmented algebra R over a field k is an algebra over k together with an
algebra homomorphism ε : R → k. Consider k to be an R-module via this map ε, and let
M
H∗ (R, k) :=
ExtnR (k, k).
n≥0
4. Let A, B be augmented algebras over k. Assume they are both finite dimensional as
k-vector spaces. Define their tensor product algebra to be A ⊗k B as a vector space,
with multiplication determined by
(a ⊗ b)(a0 ⊗ b0 ) := aa0 ⊗ bb0
for all a, a0 ∈ A; b, b0 ∈ B.
Prove that
H∗ (A ⊗k B, k) ∼
= H∗ (A, k) ⊗k H∗ (B, k)
as graded k-vector spaces. (Hint: Let P· be a projective resolution of k as an A-module,
and Q· a projective resolution of k as a B-module, and use the Ku╠łnneth Theorem to
show that P· ⊗k Q· is a projective resolution of k as a A ⊗k B-module.)
5. Let R = k[x, y]/(x2 , y 2 ). Compute H∗ (R, k). (Hint: There is an isomorphism of algebras, R ∼
= k[x]/(x2 ) ⊗k k[y]/(y 2 ). Recall the free k[x]/(x2 )-resolution of k used in
Homework Assignment 4, #5, and apply #4 above.)
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