Picard group of derived category of sheaves

Question

Let $X$ be a topological space and $R$ be a commutative ring with unit, $D(X,R)$ is the derived category of unbounded complexes of sheaves of $R$-modules. Moreover we suppose that $X$ is a stratified space and $D_c(X,R)\subset D(X,R)$ is the derived category of complexes of sheaves with constructible cohomology sheaves. The categories $D(X,R)$ and $D_c(X,R)$ are both closed symmetric monoidal categories, thus we can speak of their Picard groups as the group of isomorphism classes of invertible objects for the derived tensor product.

I was wondering if we know some computations of the Picard group of $D(X,R)$ or of $D_c(X,R)$?

Answer 1

Thanks to Drew Heard's comment I was able to find answers to my questions. In his paper "Picard groups of derived categories" H. Fausk proves the following theorem (see Theorem 4.2).

Theorem: Let $(\mathcal{E},\mathcal{O})$ be a commutative unital ringed Grothendieck topos with enough points such that for all points $p$ of $\mathcal{E}$ the ring $\mathcal{O}_p$ has a connected prime ideal spectrum. Then there is a natural split short exact sequence: $$0\rightarrow Pic(\mathcal{O}-Mod)\rightarrow Pic(D(\mathcal{O}-Mod))\rightarrow C(pt(\mathcal{E}))\rightarrow 0$$

Let us apply this theorem to the case of a connected and locally connected topological space $X$, together with the constant sheaf $\underline{R}$ (i.e. $\mathcal{O}=\underline{R}$) and let us suppose that $R$ is an integral domain (its prime ideal spectrum is connected), then we have the corollaries.

Corollary A: We have a split short exact sequence $$0\rightarrow Pic(\underline{R}-Mod)\rightarrow Pic(D(X,R))\rightarrow \mathbb{Z}\rightarrow 0$$where the section is the morphism that sends the integer $n\in\mathbb{Z}$ to the constant sheaf $\underline{R}[n]$.

Corollary B: Moreover if $X$ is stratified then we have a split short exact sequence $$0\rightarrow Pic(\underline{R}-Mod_c)\rightarrow Pic(D_c(X,R))\rightarrow \mathbb{Z}\rightarrow 0$$ where $\underline{R}-Mod_c$ is the category of constructible sheaves of $\underline{R}$-modules.

Let us look to the case of a point then we have a split short exact sequence (with the hypothesis that $Spec(R)$ connected):

$$0\rightarrow Pic(R)\rightarrow Pic(D(R)) \rightarrow \mathbb{Z}\rightarrow 0.$$