Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one.

For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a coherent locally free sheaf $E$ on $X$ we define a line bundle $Det(E)$ on $Y$ by $Det(E):=det(p_{*}E)\otimes det(R^1p_{*}E)^{-1}$.

Now we want to define a pairing $<.,.> : Pic(X)\times Pic(X)\rightarrow Pic(Y)$ by $<L,M>:=Det(L\otimes M)\otimes Det(L)^{-1}\otimes Det(M)^{-1}\otimes Det(\mathcal{O}_X)$

How can I see that this pairing satisfies $<L_1\otimes L_2,M>\cong <L_1,M>\otimes <L_2,M>$?

I think this pairing is also called Deligne bracket and this definition here is equivalent to the usual definition using local sections $s,t$ of $L$ and $M$ to get a local section $<s,t>$ of $<L,M>$. With this one can easily see that the pairing is compatible with tensor products. But i cannot seem to prove this using the determinant definition.

Is there some trick that helps? Or is it just not possible?

Edit: We may assume that every coherent sheaf on $X$ and $S$ has a finite locally free resolution. Furthermore i would like this isomorphism to be invariant under base change, but if this is not possible, what is the correct definition of $Det$ for this to be possible?

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