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An algebraic correspondence between two varieties, $V,W$ is a kind of multi-valued map from $V$ to $W$, or, in other words, a map from a covering $U$ of $V$ to $W$. Apparently, such a map gives a morphism from the cohomology of $W$ to that of $V$. How does this work? I know that we would get a map to the cohomology of $U$, by the functorial nature of cohomology. However, how do we then map this to an element of the cohomology of $V$?

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