The Stacks project

Remark 45.3.5. Let $X$ and $Y$ be smooth projective schemes over $k$. Assume $X$ is equidimensional of dimension $d$ and $Y$ is equidimensional of dimension $e$. Then the isomorphism $X \times Y \to Y \times X$ switching the factors determines an isomorphism

\[ \text{Corr}^ r(X, Y) \longrightarrow \text{Corr}^{d - e + r}(Y, X),\quad c \longmapsto c^ t \]

called the transpose. It acts on cycles as well as cycle classes. An example which is sometimes useful, is the transpose $[\Gamma _ f]^ t = [\Gamma _ f^ t]$ of the graph of a morphism $f : Y \to X$.

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