Q:

A clarification of an argument in "Perfectoid spaces"

The page 50 of (the arXiv version of) the above-mentioned paper of P. Scholze says "Now the Poincare duality pairing implies that $H^i(Y_{\mathbb{C}_p, et}, \bar{\mathbb{Q}}_l)$ is a direct summand of $H^i(Z'_{\mathbb{C}_p^\#, et}, \bar{\mathbb{Q}}_l)$." Surely I am missing something elementary but how exactly Poincare duality for etale cohomology implies that? 

A:

Let there be a map $f: X \rightarrow Y$ between two proper smooth varieties of the same dimension $n$, then we get a morphism $f^*:H^i(Y) \rightarrow H^i(X)$, where we assume the coefficient field is of char $0$ and algebraically closed. 
If $f^*$ induces an isomorphism between top degree (i.e degree $2n$) cohomology groups (which is satisfied by Lemma $9.8$ in that paper), then $f^*:H^i(Y) \rightarrow H^i(X)$ has a section constructed as follows: for any $s \in H^i(X)$, regard it as an element in $H^{2n-i}(X) ^*$  by Poincare duality, and pull back it to $H^{2n-i}(Y)^*$ along $f^*:H^{2n-i}(Y) \rightarrow H^{2n-i}(X)$, then by Poincare duality you get an element in $H^{i}(Y)$.

