In the paper " On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne" by Richard Pink, he wrote in the first page that
To obtain finer information one would like to split up the total cohomology using algebraic cycles and to describe the Galois representations on the individual factors. The process of splitting can be described in terms of correspondences; hence one wants to use a Lefschetz trace formula for the twist of a correspondence by Frobenius.
I already see some sort of examples on modular curves (and Shimura varieties): One obtain the Galois representation corresponding to a cusp form by a "cut out" process from the cohomology of modular curves via the action of the Hecke algebra, which can be interpreted as Hecke correspondences. I am curious on the purely geometric picture: namely on a "good enough" (smooth, projective, etc) algebraic variety $X$ over a field $k$, we can define the $\mathbb{Q}$-algebra of correspondence $\mathrm{Corr}(X/k)$, generated by isomorphism classes of diagrams $$X\xleftarrow{c_1}Z\xrightarrow{c_2}X$$ where $c_1$ is proper and $c_2$ is finite etale. This algebra acts on the etale cohomology groups by $c_1^*c_{2!}$. So my question is
What can we say about the structure of the $\mathbb{Q}$-algebra of correspondences? Is there a general description of the decomposition of the etale cohomology of $X$ under the action of the algebra of correspondences? Is there a reference on this?
Intuitively, the action of the correspondence algebra commutes with the Galois action, so should give smaller Galois representations as in the modular curve case.
