Exercise 111.37.7 (02A3)—The Stacks project

Table of contents
Part 9: Miscellany
Chapter 111: Exercises
Section 111.37: Proj and projective schemes
Exercise 111.37.7 (cite)

Exercise 111.37.7. Pathologies in $\text{Proj}$. Give examples of $R$ as above such that

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