About the determinant bundle

Question

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

Answer 1

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 $$ \det(R^if_*E) \cong \det(R^ig_*(p_*E)). $$