I am taking my first Algebraic Geometry class and I do not understand how to solve the following exercises.
1) Let $Q$ be the sheaf of sections of the map $f:S^1 \rightarrow S^1$ defined via complex coordinates as $f(z)=z^2$, i.e. $$ Q(\mathcal{U}):=\{s:\mathcal{U}\rightarrow S^1 | f \circ s(z)=z \}. $$ Show that $Q$ has no global sections.
2) Now let $Q_k$ be the sheaf which assigns to each open set $\mathcal{U}$ the $k$ vector space freely generated by the set $Q(\mathcal{U})$. By taking a carefully chosen cover of $S^1$, show that $$ F: \mathcal{U} \rightarrow Q_k(\mathcal{U}) \otimes Q_k(\mathcal{U}) $$ is not a sheaf.
Can anyone help me?
Thank you
Edit. 1) As @conditionalMethod suggested, I solved the first problem as follows. Assume that a global section $\tilde{s}$ exists. Then $\tilde{s}$ is a continuous function defined from $S^1$ to $S^1$ and it is such that $f(\tilde{s}(z))=z$. So $\deg(f)\cdot \deg(\tilde{s})=1$, but since $\deg(f)=2$ then we have $\deg(\tilde{s})\notin \mathbb{Z}$, which is a contradiction.
2) For the second problem my guessing is that, when tensoring the elements via pointwise multiplication, there will be some cross-multiple terms that will prevent the agreement on the intersections, but I do not understand how to choose the right cover.
