# Some questions on the basics of invertible sheaves

Let $X$ be a scheme. A $\mathcal O_X$-module $\mathcal L$ is called invertible if, for every point $x\in X$, there is an open neighborhood $U$ of $x$ and an isomorphism of $U$-modules $\mathcal O_X|_U \cong \mathcal L|_U$.

I am struggling with the basics of sheaf theory, and I have the following questions.

$1.$ Why is the tensor product of two invertible sheaves invertible?

$2$. Why is the pullback of an invertible sheaf by a morphism an invertible sheaf?

$3.$ Why is the twist $\mathcal O_X(k)$ on $X=\operatorname{Proj} B$ for a graded ring $B$ invertible, where $k$ is any integer, possibly negative?

I have a few jumbled thoughts.