Suppose $X$ is a smooth projective complex variety, connected of dimension $n$.

Let $a$ be an algebraic correspondence in $A^n(X\times X)$, the group of cycles modulo homological equivalence in $H^{2n}((X\times X)(\mathbf{C}),\mathbf{Q})$. This is a ring with the usual composition of correspondences.

$a$ acts on the cohomology of $X$ by the Künneth formula, as the graded linear map of degree zero $a^*$.

- Is $a^*$ compatible with cup products?
- Do we have, for another algebraic correspondence $b$, $(a\circ b)^*=b^*\circ a^*$?

I expect the answer to the first question to be no, and the answer to the second question to be yes. It would be great to collect a few examples.

For instance, I’m thinking if $X$ is an abelian variety and $f$ is the graph of multiplication by $N>1$, then $H^2\times H^i\to H^{i+2}$ should give counterexamples to the first question for $i\le n-1$ by taking the characteristic polynomial $p(T)$ of $f^*$ on $H^{i+2}$ and choosing $a=p(f)$.