In SGA 5 Exposé VII, at the beginning of §2, Jouanolou lets $X$ and $Y$ denote two schemes, $f:X\rightarrow Y$ a morphism, and $A$ the ring $\mathbf{Z}/\nu\mathbf{Z}$ where $\nu$ is an integer prime to the residual characteristics of $X$ and $Y$. He defines an arrow $$Rf_*(A_X)\otimes^L Rf_*(A_X)\to Rf_*(A_X)\tag{*}$$ by first defining an arrow $$f_*C^\cdot(X,A_X)\otimes f_*C^\cdot(X,A_X)\to f_*C^\cdot(X,A_X)$$ where $C^\cdot(X,A_X)$ is the Godement resolution of the sheaf $A_X$ so that $f_*C^\cdot(X,A_X)$ computes $Rf_*(A_X)$. He then writes that one can obtain (*) using flat resolutions of the components on the left. Under some finiteness hypotheses that assure that $Rf_*(A_X)$ is in $D^-(X)$, I see this no problem. Actually, using the result of Spaltenstein published in ’88, one can use a K-flat resolution to define $\otimes^L$ on all of $D(X)\times D(X)$, but this definition hadn’t been made by the time Jouanolou’s exposé was published. So, what did Jouanolou have in mind here? Thanks for your help!
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