Group law on invertible fractional ideals on a scheme I have some questions about the group of invertible fractional ideals on a scheme $X$.


*

*What is the group law ? In Görtz-Wedhorn book (paragraph 11.12), it is written that product of two invertible fractional ideals $\mathcal{I}, \mathcal{J}$ on $X$ is
the submodule $\mathcal{IJ}$ of $\mathcal{K}_X$ (the sheaf of total fractions of $X$).
I didn't find the definition of $\mathcal{IJ}$.

*What is the inverse of a invertible fractional ideal $\mathcal{I}$ ?
 A: $\mathcal{K}_X$ is a sheaf of $\mathcal{O}_X$ algebras. Thus for sections of submodulus of $\mathcal{K}_X$ the notion of product is naturally defined over any open set. So for any $U$ there is a defined product $\mathcal{I}(U)\mathcal{J}(U)$.  This gives you the desired sheaf. 
A: The answer to your question is, at least implicitly, given on the same page by stating that
$$\phi: \operatorname{Div}(X) \to \{\text{invertible fractional ideals on }X\},\quad D \mapsto \mathcal{I}_X(D)$$
is an isomorphism of abelian groups. Thus we can see what the group law in the right hand side is by examine what the group law of the left hand side is:
By definition the sum $D+E$ of two divisors $D$ and $E$ respectively given by configurations $(U_i,f_i)_{i \in I}$ and $(V_j,g_j)_{j \in J}$ is the divisor given by the configuration $$(U_i \cap V_j,(f_i)_{\mid U_i \cap V_j}\cdot (g_j)_{\mid U_i \cap V_j})_{(i,j) \in I \times J}.$$
By definition of $\phi$ this yields that the product of $\mathcal{I}$ and $\mathcal{J}$ is given by the sheaf defined by
$$
(\mathcal{I}\mathcal{J})_{\mid U_i \cap V_j} = (f_i)_{\mid U_i \cap V_j}\cdot (g_j)_{\mid U_i \cap V_j} \mathcal{O}_{\mid U_i \cap V_j}.
$$
All this should come down to the fact that this is the same as the sheafification of the presheaf $U \mapsto \mathcal{I}(U)\mathcal{J}(U)$ as hinted by @user68061.
This also provides that $\mathcal{I}^{-1}$ is given by $(\mathcal{I}^{-1})_{\mid U_i} = f_i^{-1}\mathcal{O}_{U_i}$ whenever $\mathcal{I}$ is given by $\mathcal{I}_{\mid U_i} = f_i\mathcal{O}_{U_i}$.
