Homework 2 corrections

Question 3.

$(a)\nRightarrow (b)$ ===> So one needs to add more assumptions.

example. $(\mathbb{\mathbb{Z}}\oplus \mathbb{\mathbb{Z}}/2\mathbb{\mathbb{Z}}){\otimes }_{\mathbb{\mathbb{Z}}}2\mathbb{\mathbb{Z}}\simeq \mathbb{\mathbb{Z}}$ but $\mathbb{\mathbb{Z}}\oplus \mathbb{\mathbb{Z}}/2\mathbb{\mathbb{Z}}$ is not a projective $\mathbb{\mathbb{Z}}$-module.

This question is the exercise 3.1 of the K-book chapter 1 on page 23, also look at pages 15-23 for the review of line bundles and the picard group.