Advanced Algebra II
Homework 3 due on Mar. 23, 2007 (1) Determine Zm⊗ZZn.
(2) Let M = M1⊕ M2. Prove that M is flat if and only if both Mi are flat.
What can you say if M = ⊕i∈IMi with general index set I.
(3) Let R be a local ring and M is a finitely generated flat R- module. Then M is free.
In fact, if {x1, ..., xn} ⊂ M such that they form a basis in M/mM over R/m, then it forms a basis of M.
(4) Let 0 → M1 → M2 → M3 → 0 be an exact sequence of R- modules. If both M1, M3 are finitely generated then so is M2. (5) * Tensor product commutes with direct limit. That is, lim−→(Mi⊗N) ∼=
(lim−→Mi)⊗N.
(6) A ring S is said to be an R-algebra if there is a ring homomor- phism R → S. Show that S is an R-module.
(7) Let S be a flat R-algebra and M be a flat S-module. Then M is a flat R-module.
(8) * Complete the exercises and incomplete proofs in the note.
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