Cohomology unbounded on simplicial topoi

Spectral sequence also works in unbounded case...

more-algebra.tex

@@ -28853,6 +28853,41 @@ \section{Tricks with double complexes}
 This section continues the discussion in
 Homology, Section \ref{homology-section-double-complexes-abelian-groups}.
 
+\begin{lemma}
+\label{lemma-vanishing-coh-prod-totalization}
+Let $A_0^\bullet \to A_1^\bullet \to A_2^\bullet \to \ldots$
+be a complex of complexes of abelian groups. Assume
+$H^{-p}(A_p^\bullet) = 0$ for all $p \geq 0$.
+Set $A^{p, q} = A_p^q$ and view $A^{\bullet, \bullet}$ as a double complex.
+Then $H^0(\text{Tot}_\pi(A^{\bullet, \bullet})) = 0$.
+\end{lemma}
+
+\begin{proof}
+Denote $f_p : A_p^\bullet \to A_{p + 1}^\bullet$ the given maps
+of complexes. Recall that the differential on
+$\text{Tot}_\pi(A^{\bullet, \bullet})$ is given by
+$$
+\prod\nolimits_{p + q = n} (f^q_p + (-1)^p\text{d}^q_{A_p^\bullet})
+$$
+on elements in degree $n$.
+Let $\xi \in H^0(\text{Tot}_\pi(A^{\bullet, \bullet}))$ be a cohomology
+class. We will show $\xi$ is zero. Represent $\xi$ as the class of an cocycle
+$x = (x_p) \in \prod A^{p, -p}$.
+Since $\text{d}(x) = 0$ we find that
+$\text{d}_{A_0^\bullet}(x_0) = 0$.
+Since $H^0(A_0^\bullet) = 0$ there exists a $y_{-1} \in A^{0, -1}$ with
+$\text{d}_{A_0^\bullet}(y_{-1}) = x_0$.
+Then we see that $\text{d}_{A_1^\bullet}(x_1 + f_0(y_{-1})) = 0$.
+Since $H^{-1}(A_1^\bullet) = 0$ we can find a $y_{-2} \in A^{1, -2}$
+such that $-\text{d}_{A_1^\bullet}(y_{-2}) = x_1 + f_0(y_{-1})$.
+By induction we can find
+$y_{-p - 1} \in A^{p, -p - 1}$ such that
+$$
+(-1)^p\text{d}_{A_p^\bullet}(y_{-p - 1}) = x_p + f_{p - 1}(y_{-p})
+$$
+This implies that $\text{d}(y) = x$ where $y = (y_{-p - 1})$.
+\end{proof}
+
 \begin{lemma}
 \label{lemma-prod-qis-gives-qis}
 Let
@@ -28894,41 +28929,15 @@ \section{Tricks with double complexes}
 $\text{Tot}_\pi(A^{\bullet, \bullet}) \to \text{Tot}_\pi(B^{\bullet, \bullet})$
 (We omit the verification that the differentials agree.)
 Thus it suffices to show $\text{Tot}_\pi(A^{\bullet, \bullet})$ is
-acyclic if each $A_p^\bullet$ is acyclic.
-
-\medskip\noindent
-Denote $f_p : A_p^\bullet \to A_{p + 1}^\bullet$ the given maps
-of complexes. Recall that the differential on
-$\text{Tot}_\pi(A^{\bullet, \bullet})$ is given by
-$$
-\prod\nolimits_{p + q = n} (f^q_p + (-1)^p\text{d}^q_{A_p^\bullet})
-$$
-on elements in degree $n$.
-Let $\xi \in H^0(\text{Tot}_\pi(A^{\bullet, \bullet}))$ be a cohomology
-class. We will show $\xi$ is zero; the same argument works in other
-degrees. Represent $\xi$ as the class of an cocycle
-$x = (x_p) \in \prod A^{p, -p}$.
-Since $\text{d}(x) = 0$ we find that
-$\text{d}_{A_0^\bullet}(x_0) = 0$.
-Thus there exists a $y_{-1} \in A^{0, -1}$ with
-$\text{d}_{A_0^\bullet}(y_{-1}) = x_0$.
-Then we see that $\text{d}_{A_1^\bullet}(x_1 + f_0(y_{-1})) = 0$.
-Thus we can find a $y_{-2} \in A^{1, -2}$ such that
-$-\text{d}_{A_1^\bullet}(y_{-2}) = x_1 + f_0(y_{-1})$.
-By induction we can find
-$y_{-p - 1} \in A^{p, -p - 1}$ such that
-$$
-(-1)^p\text{d}_{A_p^\bullet}(y_{-p - 1}) = x_p + f_{p - 1}(y_{-p})
-$$
-This implies that $\text{d}(y) = x$ where $y = (y_{-p - 1})$.
+acyclic if each $A_p^\bullet$ is acyclic. This follows from
+Lemma \ref{lemma-vanishing-coh-prod-totalization}.
 \end{proof}
 
 
 
 
 
 
-
 \section{Weakly \'etale ring maps}
 \label{section-weakly-etale}
 

spaces-simplicial.tex

@@ -1592,6 +1592,76 @@ \section{Cohomology on simplicial sites}
 \ref{homology-lemma-double-complex-gives-resolution}.
 \end{proof}
 
+\begin{remark}
+\label{remark-simplicial-cohomology-site}
+Assumptions and notation as in
+Lemma \ref{lemma-simplicial-sheaf-cohomology-site}
+except we do not require $K$ in $D(\mathcal{C}_{total})$
+to be bounded below. We claim there is a natural
+spectral sequence in this case also. Namely, suppose
+that $\mathcal{I}^\bullet$ is a K-injective complex of sheaves on
+$\mathcal{C}_{total}$ with injective terms representing $K$.
+We have
+\begin{align*}
+R\Gamma(\mathcal{C}_{total}, K)
+& =
+R\Hom(\mathbf{Z}, K) \\
+& =
+R\Hom(
+\ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}, K) \\
+& =
+\Gamma(\mathcal{C}_{total},
+\SheafHom^\bullet(
+\ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z},
+\mathcal{I}^\bullet)) \\
+& =
+\text{Tot}_\pi(A^{\bullet, \bullet})
+\end{align*}
+where $A^{\bullet, \bullet}$ is the double complex with terms
+$A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p)$ and $\text{Tot}_\pi$
+denotes the product totalization of this double complex.
+Namely, the first equality holds in any site. The second equality
+holds by Lemma \ref{lemma-simplicial-resolution-Z-site}.
+The third equality holds because $\mathcal{I}^\bullet$ is K-injective, see
+Cohomology on Sites, Sections \ref{sites-cohomology-section-hom-complexes}
+and \ref{sites-cohomology-section-internal-hom}.
+The final equality holds by the construction of $\SheafHom^\bullet$
+and the fact that $\Hom(g_{p!}\mathbf{Z}, \mathcal{I}^q) =
+\Gamma(\mathcal{C}_p, \mathcal{I}^q_p)$.
+Then we get our spectral sequence by viewing
+$\text{Tot}_\pi(A^{\bullet, \bullet})$ as a filtered
+complex with $F^i\text{Tot}^n_\pi(A^{\bullet, \bullet}) =
+\prod_{p + q = n,\ p \geq i} A^{p, q}$. The spectral sequence
+we obtain behaves like the spectral sequence $({}'E_r, {}'d_r)_{r \geq 0}$ in
+Homology, Section \ref{homology-section-double-complex}
+(where the case of the direct sum totalization
+is discussed) except for regularity, boundedness, convergence, and
+abutment issues. In particular we obtain $E_1^{p, q} = H^q(\mathcal{C}_p, K_p)$
+as in Lemma \ref{lemma-simplicial-sheaf-cohomology-site}.
+\end{remark}
+
+\begin{lemma}
+\label{lemma-simplicial-sheaf-cohomology-site-zero}
+In Situation \ref{situation-simplicial-site}. Let $K$ be an object of
+$D(\mathcal{C}_{total})$.
+\begin{enumerate}
+\item If $H^{-p}(\mathcal{C}_p, K_p) = 0$ for all $p \geq 0$, then
+$H^0(\mathcal{C}_{total}, K) = 0$.
+\item If $R\Gamma(\mathcal{C}_p, K_p) = 0$
+for all $p \geq 0$, then $R\Gamma(\mathcal{C}_{total}, K) = 0$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+With notation as in Remark \ref{remark-simplicial-cohomology-site}
+we see that $R\Gamma(\mathcal{C}_{total}, K)$ is
+represented by $\text{Tot}_\pi(A^{\bullet, \bullet})$.
+The assumption in (1) tells us that $H^{-p}(A^{p, \bullet}) = 0$.
+Thus the vanishing in (1) follows from
+More on Algebra, Lemma \ref{more-algebra-lemma-vanishing-coh-prod-totalization}.
+Part (2) follows from part (1) and taking shifts.
+\end{proof}
+
 \begin{lemma}
 \label{lemma-sanity-check}
 Let $\mathcal{C}$ be as in Situation \ref{situation-simplicial-site}.