Exercise 4, § 2

Commutative Algebra, I, 41

(a) Give an example of an exact sequence

0 \to E' \to E \to E'' \to 0

which does not split and all of whose terms are flat modules (cf. Algebra, Chapter VII, § 3, Exercise 8 (b)). (b) From (a) deduce an example of an exact sequence

0 \to E' \to E \to E'' \to 0

which does not split and whose terms are right A-modules which are not flat, such that for every left A-module F the sequence

0 \to E' \otimes F + E \otimes F \to E'' \otimes F \to 0

is exact (use Lemma 2 of no. 1).

Solution

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