I am very new to the world of almost mathematics and I am curious about the following:
Fix an almost mathematics situation $(R,I)$ throughout. Very generally, the almost module category comes with a tensor product, can one determine the picard group (of tensor invertible objects) in there?
More precisely, here is an example I wish to understand. Let $R=\mathbb{F}_p[t^{\frac{1}{p^\infty}}]$ and $I\subset R$ be the ideal spanned as a vector space by all positive powers of $t$ then $R/I=\mathbb{F}_p$. What is the picard group of $\text{Mod}^a(R,I)$? I would also love to know picard group of the (unbounded) derived category $D(\text{Mod}^a(R,I))$.
As a subquestion, note that both the ring and the ideal are graded by the abelian group $M=\mathbb{Z}[\frac{1}{p}]$ - the element $t^a$ has grading $a$. I would also love to understand picard group of $M$-graded almost module category of $(R,I)$ and its derived category. I guess grading simplifies the question a little bit.
(For unbounded derived category of almost modules and tensor products, I follow the definition as in Bogdan Zavyalov's paper.)
Thanks a lot!
