Lemma 21.30.3. In Situation 21.30.1 assume $(V_ n)$ holds. For $f : X \to Y$ in $\mathcal{P}$ and $\mathcal{F}$ in $\mathcal{A}_ X$ we have $R^ if_{\tau ', *}\epsilon _{X, *}\mathcal{F} = \epsilon _{Y, *}R^ if_{\tau , *}\mathcal{F}$ for $i \leq n$.
Proof. We will use the commutative diagram (21.30.0.1) without further mention. In particular have
\[ Rf_{\tau ', *}R\epsilon _{X, *}\mathcal{F} = R\epsilon _{Y, *}Rf_{\tau , *}\mathcal{F} \]
Assumption $(V_ n)$ tells us that $\epsilon _{X, *}\mathcal{F} \to R\epsilon _{X, *}\mathcal{F}$ is an isomorphism in degrees $\leq n$. Hence $Rf_{\tau ', *}\epsilon _{X, *}\mathcal{F} \to Rf_{\tau ', *}R\epsilon _{X, *}\mathcal{F}$ is an isomorphism in degrees $\leq n$. We conclude that
\[ R^ if_{\tau ', *}\epsilon _{X, *}\mathcal{F} \to H^ i(R\epsilon _{Y, *}Rf_{\tau , *}\mathcal{F}) \]
is an isomorphism for $i \leq n$. We will prove the lemma by looking at the second page of the spectral sequence of Lemma 21.14.7 for $R\epsilon _{Y, *}Rf_{\tau , *}\mathcal{F}$. Here is a picture:
\[ \begin{matrix} \ldots
& \ldots
& \ldots
& \ldots
\\ \epsilon _{Y, *}R^2f_{\tau , *}\mathcal{F}
& R^1\epsilon _{Y, *}R^2f_{\tau , *}\mathcal{F}
& R^2\epsilon _{Y, *}R^2f_{\tau , *}\mathcal{F}
& \ldots
\\ \epsilon _{Y, *}R^1f_{\tau , *}\mathcal{F}
& R^1\epsilon _{Y, *}R^1f_{\tau , *}\mathcal{F}
& R^2\epsilon _{Y, *}R^1f_{\tau , *}\mathcal{F}
& \ldots
\\ \epsilon _{Y, *}f_{\tau , *}\mathcal{F}
& R^1\epsilon _{Y, *}f_{\tau , *}\mathcal{F}
& R^2\epsilon _{Y, *}f_{\tau , *}\mathcal{F}
& \ldots
\end{matrix} \]
Let $(C_ m)$ be the hypothesis: $R^ if_{\tau ', *}\epsilon _{X, *}\mathcal{F} = \epsilon _{Y, *}R^ if_{\tau , *}\mathcal{F}$ for $i \leq m$. Observe that $(C_0)$ holds. We will show that $(C_{m - 1}) \Rightarrow (C_ m)$ for $m < n$. Namely, if $(C_{m - 1})$ holds, then for $n \geq p > 0$ and $q \leq m - 1$ we have
\begin{align*} R^ p\epsilon _{Y, *}R^ qf_{\tau , *}\mathcal{F} & = R^ p\epsilon _{Y, *} \epsilon _ Y^{-1} \epsilon _{Y, *} R^ qf_{\tau , *}\mathcal{F} \\ & = R^ p\epsilon _{Y, *} \epsilon _ Y^{-1}R^ qf_{\tau ', *}\epsilon _{X, *}\mathcal{F} = 0 \end{align*}
First equality as $\epsilon _ Y^{-1}\epsilon _{Y, *} = \text{id}$, the second by $(C_{m - 1})$, and the final by by $(V_ n)$ because $\epsilon _ Y^{-1}R^ qf_{\tau ', *}\epsilon _{X, *}\mathcal{F}$ is in $\mathcal{A}_ Y$ by (4). Looking at the spectral sequence we see that $E_2^{0, m} = \epsilon _{Y, *}R^ mf_{\tau , *}\mathcal{F}$ is the only nonzero term $E_2^{p, q}$ with $p + q = m$. Recall that $\text{d}_ r^{p, q} : E_ r^{p, q} \to E_ r^{p + r, q - r + 1}$. Hence there are no nonzero differentials $\text{d}_ r^{p, q}$, $r \geq 2$ either emanating or entering this spot. We conclude that $H^ m(R\epsilon _{Y, *}Rf_{\tau , *}\mathcal{F}) = \epsilon _{Y, *}R^ mf_{\tau , *}\mathcal{F}$ which implies $(C_ m)$ by the discussion above.
Finally, assume $(C_{n - 1})$. The same analysis shows that $E_2^{0, n} = \epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F}$ is the only nonzero term $E_2^{p, q}$ with $p + q = n$. We do still have no nonzero differentials entering this spot, but there can be a nonzero differential emanating it. Namely, the map $d_{n + 1}^{0, n} : \epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F} \to R^{n + 1}\epsilon _{Y, *}f_{\tau , *}\mathcal{F}$. We conclude that there is an exact sequence
\[ 0 \to R^ nf_{\tau ', *}\epsilon _{X, *}\mathcal{F} \to \epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F} \to R^{n + 1}\epsilon _{Y, *}f_{\tau , *}\mathcal{F} \]
By (4) and (3) the sheaf $R^ nf_{\tau ', *}\epsilon _{X, *}\mathcal{F}$ satisfies the sheaf property for $\tau $-coverings as does $\epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F}$ (use the description of $\epsilon _*$ in Section 21.27). However, the $\tau $-sheafification of the $\tau '$-sheaf $R^{n + 1}\epsilon _{Y, *}f_{\tau , *}\mathcal{F}$ is zero (by locality of cohomology; use Lemmas 21.7.3 and 21.7.4). Thus $R^ nf_{\tau ', *}\epsilon _{X, *}\mathcal{F} \to \epsilon _{Y, *}R^ nf_{\tau , *}\mathcal{F}$ has to be an isomorphism and the proof is complete. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)