Let $X$ be a scheme and $n$ be an integer invertible on $X$, consider the category $D^b(X,\mathbb{Z}/n)$ (resp. $D^b_c(X,\mathbb{Z}/n)$) of bounded chain complexes (up to quasi-isomorphisms) of sheaves of $\mathbb{Z}/n$-modules (w.r.t the étale topology) (resp. whose cohomology sheaves are constructible).
Question: can you have an explicit set of compact generators of $D^b(X)$ (in the sense of triangulated categories)? I guess that one of the following sets is what I am seeking for but I cannot find any reference about such a result.
- $f_!(\mathbb{Z}/n)(-d)[-2d]$ for $f: Y \longrightarrow X$ smooth of relative dimension $d$.
- $j_!(\mathbb{Z}/n)$ for $j: U \longrightarrow X$ étale.
and more generally, given a set $S$ inside a triangulated category $\mathcal{T}$, we denote by $\left <S \right>^{ct}$ the smallest triangulated full subcategory containing $S$ and stable under direct factors. What is the difference between $$\left <f_!(\mathbb{Z}/n)(-d)[-2d] \right>^{ct} \ \ \ \ \text{and} \ \ \ \ \left <j_!(\mathbb{Z}/n) \right>^{ct}$$ as subcategories of $D^b(X)$?
If we denote by $\left < \left< S \right > \right>$ by the smallest triangulated full subcategory containing $S$ and stable under arbitrary direct sums, then by the abstract theory (maybe due to Neeman) of triangulated categories, if $S$ consists of compact objects, and $$\mathcal{T} = \left < \left< S \right> \right>$$ then $\left <S \right>^{ct}$ is precisely the subcategory of compact objects of $\mathcal{T}$.
Let $\mathcal{T} = D^b(X)$, if one can show that either $$D^b(X) = \left < \left< f_!(\mathbb{Z}/n)(-d)[-2d] \right> \right> \ \ \ \ \text{or} \ \ \ \ D^b(X) = \left < \left< j_!(\mathbb{Z}/n) \right> \right>$$ then we know that one of $$\left <f_!(\mathbb{Z}/n)(-d)[-2d] \right>^{ct} \ \ \ \ \text{and} \ \ \ \ \left <j_!(\mathbb{Z}/n) \right>^{ct}$$ is the subcategory of compact objects of $D^b(X)$. In that case, can we show the following $$D^b_c(X) = \left <f_!(\mathbb{Z}/n)(-d)[-2d] \right>^{ct} \ \ \ \ \text{or} \ \ \ \ D^b_c(X) = \left <j_!(\mathbb{Z}/n) \right>^{ct}$$ Equivalently, $D^b_c(X)$ can be defined algebraically as the subcategory of compact objects of $D^b(X)$.
If $X$ is qcqs then any constructible sheaf $\mathcal{F} \in \operatorname{Sh}_{c}(X)$ can be expressed as a cokernel $$ \bigoplus_{k=1}^m\,j_{k,!}\underline{\mathbf{Z}/d_k} \to \bigoplus_{k=1}^{m'}\,j_{k,!}'\underline{\mathbf{Z}/d_k'} \to \mathcal{F} \to 0 $$ where $j_k:U_k \to X$ and $j_k' : U_k'\to X$ are étale morphisms - this can be seen by just using that any sheaf is a quotient of a direct sum of representables and that constructible sheaves are compact objects in $\operatorname{Sh}_{c}(X)$.
Thus, if $C^\bullet$ is a bounded chain complex of sheaves of $n$-torsion groups with constructible cohomology groups and $\mathcal{H}^d(C^\bullet)$ is the highest non-zero cohomology group, we can fit $C^\bullet$ into a distinguished triangle $$ \tau_{\leq d-1}C^\bullet \to C^\bullet \to \mathcal{H}^d(C^\bullet)[-d] \to \tau_{\leq d-1}C^\bullet [1] $$ where the left and right terms lie in $\langle \langle j_!\underline{\mathbf{Z}/n} \rangle \rangle$ by the above and by induction on the width of $C^\bullet$.
I hope this helps somewhat and I didn't mess something up:P I'm not sure about your first choice of potential compact generators but aren't these just a strict subset of the latter? So I guess both of these generate $D^b_c(X,\underline{\mathbf{Z}/n})$.