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Let $p:X\rightarrow Y$ be a finite cover of smooth curves, Consider a family $E$ of vector bundles over $X$ paramitrized by a scheme $T$,

Is it true that the determinant of the cohomology bundles of $E$ and $p_*E$ are the same?

Thanks

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    What do you mean by "the same"? They are bundles on different spaces. – 2017-01-20
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    The determinant bundle of $E$ is a line bundle over $T$; and the same thing for $\pi_*E$, they are both line bundles over $T$ – 2017-01-20

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Let $f:X \times T \to T$ and $g: Y \times T \to T$ be the projections. Then $f = g \circ p$. It follows that $$ R^if_*E \cong R^ig_*(p_*E), $$ hence a fortiori $$ \det(R^if_*E) \cong \det(R^ig_*(p_*E)). $$