Let us consider a variety $X$ over a field $k$ which is a finite field or an algebraic closure thereof. Let $\ell$ be a prime number different from the characteristic of $k$, and let $\Lambda = \mathbb Z/\ell^k\mathbb Z$ where $k\geq 1$. We have a "derived" category $D\_c^b(X,\Lambda)$ of bounded complexes of constructible $\Lambda$-adic sheaves on $X$. It is equipped with a 6 functors formalism.


Let $i:Z\hookrightarrow X$ be a closed immersion, and let $j:U \hookrightarrow X$ be the open complement. For any $K \in D\_c^b(X,\Lambda)$, I expect that we should have two distinguished triangles
$$i\_\*i^!(K) \to K \to \mathrm Rj\_\*j^\*(K) \xrightarrow{+1} \ldots$$
and
$$\mathrm Rj\_!j^\*(K) \to K \to i\_\*i^\*(K) \xrightarrow{+1} \ldots$$
Moreover, one triangle is the dual of the other via the dualizing functor $D\_X := \mathrm R\mathcal{Hom}(\cdot,f^!\Lambda)$ where $f:X\to \mathrm{Spec}(k)$ is the structure morphism. By this, I mean that one triangle is obtained by applying $D\_X$ to the other triangle for the complex $D\_X(K)$.


This is stated in the book *Weil conjectures, perverse sheaves and $\ell$-adic Fourier transform* by Kiehl and Weissauer (2001), section II.11, p.123-124 (see Lemma 11.1). However, in the way the paragraph is written, the authors seem to make the underlying assumption that $f$ is smooth. I feel like smoothness is not used in this specific section of the paragraph which I quoted above, and I'd like to ask confirmation of this fact. Besides, is there maybe another reference without this smoothness assumption?



Reference for localization distinguished triangles in the derived category of $\ell$-adic sheaves
Indeed $X$ does not have to be smooth.


These triangles are part of the yoga of "recollement". See Section 1.4 of Beilinson-Bernstein-Deligne. A quotable reference that works in great generality is (4.10) of Laszlo-Olsson.


