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?
Best Answer
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.