If $X$ and $Y$ are varieties over an algebraically closed field, then in the corresponding derived category of complexes we have $RH_{et}(X\times Y,\mathbb{Z}/l^n\mathbb{Z})\cong RH_{et}(X,\mathbb{Z}/l^n \mathbb{Z})\otimes RH_{et}(Y,\mathbb{Z}/l^n\mathbb{Z})$. Where can I found a proof of this fact whose 'idea' would be clear (I don't want to assume that $X$ and $Y$ are smooth, and I don't want to consider cohomology with compact support)? Does the Leray spectral sequence help here? In order to compute the stalks of the corresponding (higher) direct images, one has to consider $Y$ being a strictly henselian scheme; how can this be done?

And what happens when the base field is no longer algebraically closed? In particular, I would like to understand completely the case when $X$ is singular, but $Y$ is just $G_m$ (an affine line minus one point). This particular case is related with the question The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base

If $X$ and $Y$ are varieties over an algebraically closed field, then in the corresponding derived category of complexes we have $RH_{et}(X\times Y,\mathbb{Z}/l^n\mathbb{Z})\cong RH_{et}(X,\mathbb{Z}/l^n \mathbb{Z})\otimes RH_{et}(Y,\mathbb{Z}/l^n\mathbb{Z})$. Where can I found a proof of this fact whose 'idea' would be clear (I don't want to assume that $X$ and $Y$ are smooth, and I don't want to consider cohomology with compact support)? Does the Leray spectral sequence help here? In order to compute the stalks of the corresponding (higher) direct images, one has to consider $Y$ being a strictly henselian scheme; how can this be done?

And what happens when the base field is no longer algebraically closed? In particular, I would like to understand completely the case when $X$ is singular, but $Y$ is just $G_m$ (an affine line minus one point). This particular case is related with the question The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base