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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 Künneth 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.)