Let $X, Y, Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. Consider the etale homology as defined in Bloch-Ogus (http://www.numdam.org/article/ASENS_1974_4_7_2_181_0.pdf: note that they work with torsion coefficients, but everything works more or less the same with $\ell$-adic coefficients, see https://arxiv.org/pdf/0712.1712). There is a Leray spectral sequence for $H_i(X, \mathbb{Q}_{\ell}(j))$ whose $E_2$ terms are cohomology groups of sheaves on $Y$: simply note that $H_i(X, \mathbb{Q}_{\ell}(j)) \cong H^{2d-i}(X, \mathbb{Q}_{\ell}(d-j))$ (where $d = \dim X$) and apply the Leray spectral sequence for etale cohomology. However, it is not transparent how the pushforward $g_*$ from the homology of $Z$ to the homology of $X$ interacts with the respective Leray spectral sequences for $h$ and $f$. Is there a Leray spectral sequence for etale homology whose $E_2$ page is compatible with proper pushforward? The question is similar to Leray spectral sequence and pullbacks. It’s worth noting that I am in a highly degenerate situation where the $E_2$ page of the Leray spectral sequence for $h$ has only one nonzero term.

Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. Consider the etale homology as defined in Bloch–Ogus (Gersten's conjecture and the homology of schemes: note that they work with torsion coefficients, but everything works more or less the same with $\ell$-adic coefficients, see Li - The Étale Homology and The Cycle Maps in Adic Coefficients). There is a Leray spectral sequence for $H_i(X, \mathbb{Q}_{\ell}(j))$ whose $E_2$ terms are cohomology groups of sheaves on $Y$: simply note that $H_i(X, \mathbb{Q}_{\ell}(j)) \cong H^{2d-i}(X, \mathbb{Q}_{\ell}(d-j))$ (where $d = \dim X$) and apply the Leray spectral sequence for étale cohomology. However, it is not transparent how the pushforward $g_*$ from the homology of $Z$ to the homology of $X$ interacts with the respective Leray spectral sequences for $h$ and $f$. Is there a Leray spectral sequence for étale homology whose $E_2$ page is compatible with proper pushforward? The question is similar to Leray spectral sequence and pullbacks. It’s worth noting that I am in a highly degenerate situation where the $E_2$ page of the Leray spectral sequence for $h$ has only one nonzero term.

Let $X, Y, Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. Consider the etale homology as defined in Bloch-Ogus (http://www.numdam.org/article/ASENS_1974_4_7_2_181_0.pdf: note that they work with torsion coefficients, but everything works more or less the same with $\ell$-adic coefficients, see https://arxiv.org/pdf/0712.1712). There is a Leray spectral sequence for $H_i(X, \mathbb{Q}_{\ell}(j))$ whose $E_2$ terms are cohomology groups of sheaves on $Y$: simply note that $H_i(X, \mathbb{Q}_{\ell}(j)) \cong H^{2d-i}(X, \mathbb{Q}_{\ell}(d-j))$ (where $d = \dim X$) and apply the Leray spectral sequence for etale cohomology. However, it is not transparent how the pushforward $g_*$ from the homology of $Z$ to the homology of $X$ interacts with the respective Leray spectral sequences for $h$ and $f$. Is there a Leray spectral sequence for etale homology whose $E_2$ pages are compatible with proper pushforward? The question is similar to Leray spectral sequence and pullbacks. It’s worth noting that I am in a highly degenerate situation where the $E_2$ page of the Leray spectral sequence for $h$ has only one nonzero term.

Let $X, Y, Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. Consider the etale homology as defined in Bloch-Ogus (http://www.numdam.org/article/ASENS_1974_4_7_2_181_0.pdf: note that they work with torsion coefficients, but everything works more or less the same with $\ell$-adic coefficients, see https://arxiv.org/pdf/0712.1712). There is a Leray spectral sequence for $H_i(X, \mathbb{Q}_{\ell}(j))$ whose $E_2$ terms are cohomology groups of sheaves on $Y$: simply note that $H_i(X, \mathbb{Q}_{\ell}(j)) \cong H^{2d-i}(X, \mathbb{Q}_{\ell}(d-j))$ (where $d = \dim X$) and apply the Leray spectral sequence for etale cohomology. However, it is not transparent how the pushforward $g_*$ from the homology of $Z$ to the homology of $X$ interacts with the respective Leray spectral sequences for $h$ and $f$. Is there a Leray spectral sequence for etale homology whose $E_2$ page is compatible with proper pushforward? The question is similar to Leray spectral sequence and pullbacks. It’s worth noting that I am in a highly degenerate situation where the $E_2$ page of the Leray spectral sequence for $h$ has only one nonzero term.

Let $X, Y, Z$ be smooth, quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. Consider the etale homology as defined in Bloch-Ogus (http://www.numdam.org/article/ASENS_1974_4_7_2_181_0.pdf: note that they work with torsion coefficients, but everything works more or less the same with $\ell$-adic coefficients, see https://arxiv.org/pdf/0712.1712). There is a Leray spectral sequence for $H_i(X, \mathbb{Q}_{\ell}(j))$ whose $E_2$ terms are cohomology groups of sheaves on $Y$: simply note that $H_i(X, \mathbb{Q}_{\ell}(j)) \cong H^{2d-i}(X, \mathbb{Q}_{\ell}(d-j))$ (where $d = \dim X$) and apply the Leray spectral sequence for etale cohomology. However, it is not transparent how the pushforward $g_*$ from the homology of $Z$ to the homology of $X$ interacts with the respective Leray spectral sequences for $h$ and $f$. Is there a Leray spectral sequence for etale homology whose $E_2$ pages are compatible with proper pushforward? The question is similar to Leray spectral sequence and pullbacks. It’s worth noting that I am in a highly degenerate situation where the $E_2$ page of the Leray spectral sequence for $h$ has only one nonzero term.

Let $X, Y, Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. Consider the etale homology as defined in Bloch-Ogus (http://www.numdam.org/article/ASENS_1974_4_7_2_181_0.pdf: note that they work with torsion coefficients, but everything works more or less the same with $\ell$-adic coefficients, see https://arxiv.org/pdf/0712.1712). There is a Leray spectral sequence for $H_i(X, \mathbb{Q}_{\ell}(j))$ whose $E_2$ terms are cohomology groups of sheaves on $Y$: simply note that $H_i(X, \mathbb{Q}_{\ell}(j)) \cong H^{2d-i}(X, \mathbb{Q}_{\ell}(d-j))$ (where $d = \dim X$) and apply the Leray spectral sequence for etale cohomology. However, it is not transparent how the pushforward $g_*$ from the homology of $Z$ to the homology of $X$ interacts with the respective Leray spectral sequences for $h$ and $f$. Is there a Leray spectral sequence for etale homology whose $E_2$ pages are compatible with proper pushforward? The question is similar to Leray spectral sequence and pullbacks. It’s worth noting that I am in a highly degenerate situation where the $E_2$ page of the Leray spectral sequence for $h$ has only one nonzero term.