I already posted this question on mathstackexchange [there](https://math.stackexchange.com/questions/4739910/composition-of-gysin-and-restriction-maps-on-ell-adic-cohomology), but I figured that it may have more replies here.




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I follow the notations of Milne's lectures notes on etale cohomology, most specifically the section titled "The Gysin map" in chapter 24, p. 145.


Let $k$ be an algebraically closed field, let $Y$ and $X$ be two non singular (separated) complete varieties of respective dimensions $d$ and $a$, and let $\pi: Y\hookrightarrow X$ be a closed immersion. In particular $a \geq d$ and let $c := a - d$ denote the codimension. Let $\Lambda = \mathbb Z / n\mathbb Z$ for some $n$ which is prime with the characteristic of $k$. Since the varieties are complete, I do not bother to distinguish between cohomology with or without compact support.


Functoriality of etale cohomology provides us with restriction maps $\pi^\*: \mathrm H^r(X,\Lambda) \to \mathrm H^r(Y,\Lambda)$. Taking duals and applying Poincaré duality as well as some Tate twist, we obtain maps $\pi\_\*:\mathrm H^r(Y,\Lambda) \to \mathrm H^{r+2c}(X,\Lambda(c))$. These are the Gysin maps.


Let $1\_Y$ denote the identity element of $\mathrm H^0(Y,\Lambda) \simeq \Lambda$. The element $\pi\_\*(1\_Y) \in \mathrm H^{2c}(X,\Lambda(c))$ is, by definition, the image of $Y$, seen as a prime cycle on $X$, via the cycle map $\mathrm{cl}\_X: \mathrm{CH}^c(X)\to \mathrm H^{2c}(X,\Lambda(c))$.


In Remark 24.2, Milne states the projection formula. As a particular case, for any $x\in \mathrm H^r(X,\Lambda)$, we have the identity
$$\pi\_\*\pi^\*(x) = \mathrm{cl}\_X(Y) \cup x \in \mathrm H^{r+2c}(X,\Lambda(c)).$$
(Simply take $y = 1\_Y \in \mathrm H^0(Y,\Lambda)$ in Milne's notations).


Thus, we have an explicit formula to describe $\pi\_\*\pi^\*: \mathrm H^r(X,\Lambda) \to \mathrm H^{r+2c}(X,\Lambda(c))$.


Is there similarly a formula to compute the reverse composition, ie. $\pi^\*\pi\_\*: \mathrm H^r(Y,\Lambda) \to \mathrm H^{r+2c}(Y,\Lambda(c))$? Would this map be similarly described as a cup product by some canonically defined element of the cohomology of $Y$?



Composition of Gysin and restriction maps on $\ell$-adic cohomology
Yes: $\pi ^\*\pi \_\*$ is the cup-product with $c\_c(N)$, the $c$-th Chern class of the normal bundle of $Y$ in $X$. This is Theorem VII.4.1 in SGA 5.


