# Homotopy invariance of $\ell$-adic cohomology

In the end of the Voevodsky’s lectures on cross functors, 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?

• About axiom 2 : cohomological purity implies that $s^!p^*\Lambda \simeq \Lambda(-1)[-2]$ (Milne's étale cohomology , theorem 6.1). We even have (by remark 5.2 in loc. cit.) that for any locally constant étale $\Lambda$-sheaf $F$ on $S$, $s^!p^*F \simeq F\otimes\Lambda(-1)$, therefore the functor $s^!p^*$ coincides with twisting by (-1). It is natural to ask that twisting is invertible, and as $p_\sharp s_*$ is left adjoint to $s^!p^*$ it also equal its inverse, thus condition 2 is ''twisting by 1 is an invertible functor''. May 9 at 15:48

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.

• Dear @WillSawin, the Lemma 7.6.7 of Lei Fu's book needs $g$ to be an open immersion, but the Theorem 7.6.9 works in the needed generality. Thank you for your answer :) May 9 at 15:48
• It would probably be better to consider $\mathbb{Z}_l$-linear (co?)homology in your answer. I believe that your arguments work in this setting without difficulty, whereas $\mathbb{Z}_l$-cohomology contains more information than $\mathbb{Q}_l$-one and it factors through motives with integer coefficients as well. May 11 at 19:37
• @MikhailBondarko Sure! Interestingly I looked to see which is used in the document linked in the question and as far as I can see it does not specify. May 11 at 20:08