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Tag 0947

Chapter 21: Cohomology on Sites > Section 21.41: Weakly contractible objects

Proposition 21.41.2. Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \mathop{\rm Ob}\nolimits(\mathcal{C})$ such that every $U \in \mathcal{B}$ is weakly contractible and every object of $\mathcal{C}$ has a covering by elements of $\mathcal{B}$. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Then

  1. A complex $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ of $\mathcal{O}$-modules is exact, if and only if $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact for all $U \in \mathcal{B}$.
  2. Every object $K$ of $D(\mathcal{O})$ is a derived limit of its canonical truncations: $K = R\mathop{\rm lim}\nolimits \tau_{\geq -n} K$.
  3. Given an inverse system $\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$ with surjective transition maps, the projection $\mathop{\rm lim}\nolimits \mathcal{F}_n \to \mathcal{F}_1$ is surjective.
  4. Products are exact on $\textit{Mod}(\mathcal{O})$.
  5. Products on $D(\mathcal{O})$ can be computed by taking products of any representative complexes.
  6. If $(\mathcal{F}_n)$ is an inverse system of $\mathcal{O}$-modules, then $R^p\mathop{\rm lim}\nolimits \mathcal{F}_n = 0$ for all $p > 1$ and $$ R^1\mathop{\rm lim}\nolimits \mathcal{F}_n = \mathop{\rm Coker}(\prod \mathcal{F}_n \to \prod \mathcal{F}_n) $$ where the map is $(x_n) \mapsto (x_n - f(x_{n + 1}))$.
  7. If $(K_n)$ is an inverse system of objects of $D(\mathcal{O})$, then there are short exact sequences $$ 0 \to R^1\mathop{\rm lim}\nolimits H^{p - 1}(K_n) \to H^p(R\mathop{\rm lim}\nolimits K_n) \to \mathop{\rm lim}\nolimits H^p(K_n) \to 0 $$

Proof. Proof of (1). If the sequence is exact, then evaluating at any weakly contractible element of $\mathcal{C}$ gives an exact sequence by Lemma 21.41.1. Conversely, assume that $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact for all $U \in \mathcal{B}$. Let $V$ be an object of $\mathcal{C}$ and let $s \in \mathcal{F}_2(V)$ be an element of the kernel of $\mathcal{F}_2 \to \mathcal{F}_3$. By assumption there exists a covering $\{U_i \to V\}$ with $U_i \in \mathcal{B}$. Then $s|_{U_i}$ lifts to a section $s_i \in \mathcal{F}_1(U_i)$. Thus $s$ is a section of the image sheaf $\mathop{\rm Im}(\mathcal{F}_1 \to \mathcal{F}_2)$. In other words, the sequence $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ is exact.

Proof of (2). This holds by Lemma 21.22.10 with $d = 0$.

Proof of (3). Let $(\mathcal{F}_n)$ be a system as in (2) and set $\mathcal{F} = \mathop{\rm lim}\nolimits \mathcal{F}_n$. If $U \in \mathcal{B}$, then $\mathcal{F}(U) = \mathop{\rm lim}\nolimits \mathcal{F}_n(U)$ surjects onto $\mathcal{F}_1(U)$ as all the transition maps $\mathcal{F}_{n + 1}(U) \to \mathcal{F}_n(U)$ are surjective. Thus $\mathcal{F} \to \mathcal{F}_1$ is surjective by Sites, Definition 7.11.1 and the assumption that every object has a covering by elements of $\mathcal{B}$.

Proof of (4). Let $\mathcal{F}_{i, 1} \to \mathcal{F}_{i, 2} \to \mathcal{F}_{i, 3}$ be a family of exact sequences of $\mathcal{O}$-modules. We want to show that $\prod \mathcal{F}_{i, 1} \to \prod \mathcal{F}_{i, 2} \to \prod \mathcal{F}_{i, 3}$ is exact. We use the criterion of (1). Let $U \in \mathcal{B}$. Then $$ (\prod \mathcal{F}_{i, 1})(U) \to (\prod \mathcal{F}_{i, 2})(U) \to (\prod \mathcal{F}_{i, 3})(U) $$ is the same as $$ \prod \mathcal{F}_{i, 1}(U) \to \prod \mathcal{F}_{i, 2}(U) \to \prod \mathcal{F}_{i, 3}(U) $$ Each of the sequences $\mathcal{F}_{i, 1}(U) \to \mathcal{F}_{i, 2}(U) \to \mathcal{F}_{i, 3}(U)$ are exact by (1). Thus the displayed sequences are exact by Homology, Lemma 12.29.1. We conclude by (1) again.

Proof of (5). Follows from (4) and (slightly generalized) Derived Categories, Lemma 13.32.2.

Proof of (6) and (7). We refer to Section 21.22 for a discussion of derived and homotopy limits and their relationship. By Derived Categories, Definition 13.32.1 we have a distinguished triangle $$ R\mathop{\rm lim}\nolimits K_n \to \prod K_n \to \prod K_n \to R\mathop{\rm lim}\nolimits K_n[1] $$ Taking the long exact sequence of cohomology sheaves we obtain $$ H^{p - 1}(\prod K_n) \to H^{p - 1}(\prod K_n) \to H^p(R\mathop{\rm lim}\nolimits K_n) \to H^p(\prod K_n) \to H^p(\prod K_n) $$ Since products are exact by (4) this becomes $$ \prod H^{p - 1}(K_n) \to \prod H^{p - 1}(K_n) \to H^p(R\mathop{\rm lim}\nolimits K_n) \to \prod H^p(K_n) \to \prod H^p(K_n) $$ Now we first apply this to the case $K_n = \mathcal{F}_n[0]$ where $(\mathcal{F}_n)$ is as in (6). We conclude that (6) holds. Next we apply it to $(K_n)$ as in (7) and we conclude (7) holds. $\square$

    The code snippet corresponding to this tag is a part of the file sites-cohomology.tex and is located in lines 9517–9551 (see updates for more information).

    \begin{proposition}
    \label{proposition-enough-weakly-contractibles}
    Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \Ob(\mathcal{C})$
    such that every $U \in \mathcal{B}$ is weakly contractible and
    every object of $\mathcal{C}$ has a covering by elements of $\mathcal{B}$.
    Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Then
    \begin{enumerate}
    \item A complex $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$
    of $\mathcal{O}$-modules is exact, if and only if
    $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$
    is exact for all $U \in \mathcal{B}$.
    \item Every object $K$ of $D(\mathcal{O})$ is a derived limit
    of its canonical truncations: $K = R\lim \tau_{\geq -n} K$.
    \item Given an inverse system
    $\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$
    with surjective transition maps, the projection
    $\lim \mathcal{F}_n \to \mathcal{F}_1$ is surjective.
    \item Products are exact on $\textit{Mod}(\mathcal{O})$.
    \item Products on $D(\mathcal{O})$ can be computed by taking
    products of any representative complexes.
    \item If $(\mathcal{F}_n)$ is an inverse system of $\mathcal{O}$-modules,
    then $R^p\lim \mathcal{F}_n = 0$ for all $p > 1$ and
    $$
    R^1\lim \mathcal{F}_n  =
    \Coker(\prod \mathcal{F}_n \to \prod \mathcal{F}_n)
    $$
    where the map is $(x_n) \mapsto (x_n - f(x_{n + 1}))$.
    \item If $(K_n)$ is an inverse system of objects of $D(\mathcal{O})$,
    then there are short exact sequences
    $$
    0 \to R^1\lim H^{p - 1}(K_n) \to H^p(R\lim K_n) \to
    \lim H^p(K_n) \to 0
    $$
    \end{enumerate}
    \end{proposition}
    
    \begin{proof}
    Proof of (1). If the sequence is exact, then evaluating at any
    weakly contractible element of $\mathcal{C}$ gives an exact
    sequence by Lemma \ref{lemma-w-contractible}. Conversely, assume that
    $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$
    is exact for all $U \in \mathcal{B}$.
    Let $V$ be an object of $\mathcal{C}$ and let
    $s \in \mathcal{F}_2(V)$ be an element of the kernel of
    $\mathcal{F}_2 \to \mathcal{F}_3$. By assumption there exists
    a covering $\{U_i \to V\}$ with $U_i \in \mathcal{B}$.
    Then $s|_{U_i}$ lifts to a section $s_i \in \mathcal{F}_1(U_i)$.
    Thus $s$ is a section of the image sheaf
    $\Im(\mathcal{F}_1 \to \mathcal{F}_2)$.
    In other words, the sequence
    $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$
    is exact.
    
    \medskip\noindent
    Proof of (2). This holds by Lemma \ref{lemma-is-limit-dimension} with $d = 0$.
    
    \medskip\noindent
    Proof of (3). Let $(\mathcal{F}_n)$ be a system as in (2) and set
    $\mathcal{F} = \lim \mathcal{F}_n$. If $U \in \mathcal{B}$, then
    $\mathcal{F}(U) = \lim \mathcal{F}_n(U)$
    surjects onto $\mathcal{F}_1(U)$ as all the transition maps
    $\mathcal{F}_{n + 1}(U) \to \mathcal{F}_n(U)$ are surjective.
    Thus $\mathcal{F} \to \mathcal{F}_1$ is surjective by
    Sites, Definition \ref{sites-definition-sheaves-injective-surjective}
    and the assumption that every object
    has a covering by elements of $\mathcal{B}$.
    
    \medskip\noindent
    Proof of (4). Let
    $\mathcal{F}_{i, 1} \to \mathcal{F}_{i, 2} \to \mathcal{F}_{i, 3}$
    be a family of exact sequences of $\mathcal{O}$-modules.
    We want to show that
    $\prod \mathcal{F}_{i, 1} \to \prod \mathcal{F}_{i, 2} \to
    \prod \mathcal{F}_{i, 3}$ is exact. We use the criterion of (1).
    Let $U \in \mathcal{B}$. Then
    $$
    (\prod \mathcal{F}_{i, 1})(U) \to
    (\prod \mathcal{F}_{i, 2})(U) \to
    (\prod \mathcal{F}_{i, 3})(U)
    $$
    is the same as
    $$
    \prod \mathcal{F}_{i, 1}(U) \to
    \prod \mathcal{F}_{i, 2}(U) \to
    \prod \mathcal{F}_{i, 3}(U)
    $$
    Each of the sequences
    $\mathcal{F}_{i, 1}(U) \to \mathcal{F}_{i, 2}(U) \to \mathcal{F}_{i, 3}(U)$
    are exact by (1). Thus the displayed sequences are exact by
    Homology, Lemma \ref{homology-lemma-product-abelian-groups-exact}.
    We conclude by (1) again.
    
    \medskip\noindent
    Proof of (5). Follows from (4) and (slightly generalized)
    Derived Categories, Lemma \ref{derived-lemma-products}.
    
    \medskip\noindent
    Proof of (6) and (7). We refer to Section \ref{section-derived-limits}
    for a discussion of derived and homotopy limits and their relationship.
    By Derived Categories, Definition \ref{derived-definition-derived-limit}
    we have a distinguished
    triangle
    $$
    R\lim K_n \to \prod K_n \to \prod K_n \to R\lim K_n[1]
    $$
    Taking the long exact sequence of cohomology sheaves we obtain
    $$
    H^{p - 1}(\prod K_n) \to H^{p - 1}(\prod K_n) \to
    H^p(R\lim K_n) \to H^p(\prod K_n) \to H^p(\prod K_n)
    $$
    Since products are exact by (4) this becomes
    $$
    \prod H^{p - 1}(K_n) \to \prod H^{p - 1}(K_n) \to
    H^p(R\lim K_n) \to \prod H^p(K_n) \to \prod H^p(K_n)
    $$
    Now we first apply this to the case $K_n = \mathcal{F}_n[0]$
    where $(\mathcal{F}_n)$ is as in (6). We conclude that (6) holds.
    Next we apply it to $(K_n)$ as in (7) and we conclude (7) holds.
    \end{proof}

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