In the end of the *[Voevodsky’s lectures on cross functors](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/2015_transfer_from_ps_delnotes01.pdf)*, P. Deligne considers a couple of axioms which define (using the vocabulary of Ayoub's thesis) a stable homotopical 2-functor. Among them, we have that


1. **(Homotopy invariance)** If $p$ is the projection $\mathbb{A}^1\_X\to X$, the adjunction morphism $\operatorname{id}\to p\_\*p^\*$ is an isomorphism;
2. **(Stability)** If $s$ is the zero-section of $p$, then $p\_\#\circ s\_+$ is an equivalence of categories. (Where $p\_\#$ is the left adjoint of $p^\*$, which exists since $p$ is smooth.)


He then affirms that the two axioms above are well known in the $\ell$-adic setting. My first question then is: how are they proven? (I think a description of the proof would be nice for the MO community, but I would also be happy with a reference.)


The axiom of homotopy invariance surely axiomatises what its name describes: in Sheaves and Manifolds, M. Kashiwara and P. Shapira deduce the homotopy invariance of sheaf cohomology from the fact that the projection $X\times [0,1]\to X$ satisfies the axiom above. The axiom 1 then refers to this.


Now, I don't really understand whats the role of the axiom 2 (of "stability"). (Perhaps because I don't really have much of an intuition for $p\_\#$.) So my second question is: how should one think about this axiom? Perhaps it is more intuitive in the $\ell$-adic context?



Homotopy invariance of $\ell$-adic cohomology
Proof of homotopy invariance: This follows from a base change/Kunneth type statement and the calculation of the cohomology of $\mathbb A^1$.


Specifically, Lemma 7.6.7 of Lei Fu's etale cohomology theory, specialized to $S$ a point, $f$ the map from $X$ to a point, $g$ the map from $\mathbb A^1$ to a point, so that $g' = p$, $K$ an arbitrary complex on $X$, and $L$ the constant sheaf, implies that $p\_\* p^\* K = K \otimes f^\* g\_\* \mathbb Z\_\ell$ as soon as $K$ is strongly locally acyclic relative to $f$, which it is by Lemma 9.3.4.


Then $g\_\* \mathbb Z\_\ell$ is the cohomology of the affine line, which is simply $\mathbb Z\_\ell$, so $K \otimes f^\* g\_\* \mathbb Z\_\ell = K \otimes f^\* \mathbb Z\_\ell = K \otimes \mathbb Z\_\ell= K$.


Proof of stability: As Deligne notes in the very last paragraph, in the etale setting $p\_\#$ is $p\_!$ up to a shift and twist: For a smooth morphism, $p^\*$ and $p^!$ agree up to a shift and twist, and $p^!$ has a left adjoint $p\_!$, which thus agrees with $p\_\#$ up to the dual shift and twist.


Since shifting and twisting are equivalences of categories, it suffices to check that $p\_! s\_\*$ is an equivalence of categories. Now $s\_\*$ of any sheaf is compactly supported over the base $X$ (since a section is compact), which means $p\_! s\_\* = p\_\* s\_\*$, and $p\_\* s\_\*$ is the identity, and thus an equivalence, by the Leray spectral sequence.


Not sure on the motivation.


