The Stacks project

Exercise 111.1.6. Algebra. (Silly and should be easy.)

  1. Give an example of a ring $A$ and a nonsplit short exact sequence of $A$-modules

    \[ 0 \to M_1 \to M_2 \to M_3 \to 0. \]

  2. Give an example of a nonsplit sequence of $A$-modules as above and a faithfully flat $A \to B$ such that

    \[ 0 \to M_1\otimes _ A B \to M_2\otimes _ A B \to M_3\otimes _ A B \to 0. \]

    is split as a sequence of $B$-modules.

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View Exercise 111.1.6 as pdf