Exercise 110.37.7. Pathologies in $\text{Proj}$. Give examples of $R$ as above such that

1. ${\mathcal O}_ X(1)$ is not an invertible ${\mathcal O}_ X$-module.

2. ${\mathcal O}_ X(1)$ is invertible, but the natural map ${\mathcal O}_ X(1) \otimes _{{\mathcal O}_ X} {\mathcal O}_ X(1) \to {\mathcal O}_ X(2)$ is NOT an isomorphism.

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