Let $\mathcal{C}$ be a site. Consider the category $\mathcal{C} \times \mathbf{N}$ with $\mathop{\mathrm{Mor}}\nolimits ((U, n), (V, m)) = \emptyset $ if $n > m$ and $\mathop{\mathrm{Mor}}\nolimits ((U, n), (V, m)) = \mathop{\mathrm{Mor}}\nolimits (U, V)$ else. We endow this with the structure of a site by letting coverings be families $\{ (U_ i, n) \to (U, n)\} $ such that $\{ U_ i \to U\} $ is a covering of $\mathcal{C}$. Then the reader verifies immediately that sheaves on $\mathcal{C} \times \mathbf{N}$ are the same thing as inverse systems of sheaves on $\mathcal{C}$. In particular $\textit{Ab}(\mathcal{C} \times \mathbf{N})$ is inverse systems of abelian sheaves on $\mathcal{C}$. Consider now the functor
which takes an inverse system to its limit. This is nothing but $g_*$ where $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C} \times \mathbf{N}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is the morphism of topoi associated to the continuous and cocontinuous functor $\mathcal{C} \times \mathbf{N} \to \mathcal{C}$. (Observe that $g^{-1}$ assigns to a sheaf on $\mathcal{C}$ the corresponding constant inverse system.)
By the general machinery explained above we obtain a derived functor
As indicated this functor is often denoted $R\mathop{\mathrm{lim}}\nolimits $.
On the other hand, the continuous and cocontinuous functors $\mathcal{C} \to \mathcal{C} \times \mathbf{N}$, $U \mapsto (U, n)$ define morphisms of topoi $i_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C} \times \mathbf{N})$. Of course $i_ n^{-1}$ is the functor which picks the $n$th term of the inverse system. Thus there are transformations of functors $i_{n + 1}^{-1} \to i_ n^{-1}$. Hence given $K \in D(\mathcal{C} \times \mathbf{N})$ we get $K_ n = i_ n^{-1}K \in D(\mathcal{C})$ and maps $K_{n + 1} \to K_ n$. In Derived Categories, Definition 13.34.1 we have defined the notion of a homotopy limit
We claim the two notions agree (as far as it makes sense).
Lemma 21.23.1. Let $\mathcal{C}$ be a site. Let $K$ be an object of $D(\mathcal{C} \times \mathbf{N})$. Set $K_ n = i_ n^{-1}K$ as above. Then in $D(\mathcal{C})$.
\[ R\mathop{\mathrm{lim}}\nolimits K \cong R\mathop{\mathrm{lim}}\nolimits K_ n \]
Proof. To calculate $R\mathop{\mathrm{lim}}\nolimits $ on an object $K$ of $D(\mathcal{C} \times \mathbf{N})$ we choose a K-injective representative $\mathcal{I}^\bullet $ whose terms are injective objects of $\textit{Ab}(\mathcal{C} \times \mathbf{N})$, see Injectives, Theorem 19.12.6. We may and do think of $\mathcal{I}^\bullet $ as an inverse system of complexes $(\mathcal{I}_ n^\bullet )$ and then we see that
where the right hand side is the termwise inverse limit.
Let $\mathcal{J} = (\mathcal{J}_ n)$ be an injective object of $\textit{Ab}(\mathcal{C} \times \mathbf{N})$. The morphisms $(U, n) \to (U, n + 1)$ are monomorphisms of $\mathcal{C} \times \mathbf{N}$, hence $\mathcal{J}(U, n + 1) \to \mathcal{J}(U, n)$ is surjective (Lemma 21.12.6). It follows that $\mathcal{J}_{n + 1} \to \mathcal{J}_ n$ is surjective as a map of presheaves.
Note that the functor $i_ n^{-1}$ has an exact left adjoint $i_{n, !}$. Namely, $i_{n, !}\mathcal{F}$ is the inverse system $\ldots 0 \to 0 \to \mathcal{F} \to \ldots \to \mathcal{F}$. Thus the complexes $i_ n^{-1}\mathcal{I}^\bullet = \mathcal{I}_ n^\bullet $ are K-injective by Derived Categories, Lemma 13.31.9.
Because we chose our K-injective complex to have injective terms we conclude that
is a short exact sequence of complexes of abelian sheaves as it is a short exact sequence of complexes of abelian presheaves. Moreover, the products in the middle and the right represent the products in $D(\mathcal{C})$, see Injectives, Lemma 19.13.4 and its proof (this is where we use that $\mathcal{I}_ n^\bullet $ is K-injective). Thus $R\mathop{\mathrm{lim}}\nolimits K$ is a homotopy limit of the inverse system $(K_ n)$ by definition of homotopy limits in triangulated categories. $\square$
Lemma 21.23.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The functors $R\Gamma (\mathcal{C}, -)$ and $R\Gamma (U, -)$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ commute with $R\mathop{\mathrm{lim}}\nolimits $. Moreover, there are short exact sequences for any inverse system $(K_ n)$ in $D(\mathcal{O})$ and $m \in \mathbf{Z}$. Similar for $H^ m(\mathcal{C}, R\mathop{\mathrm{lim}}\nolimits K_ n)$.
\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{m - 1}(U, K_ n) \to H^ m(U, R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits H^ m(U, K_ n) \to 0 \]
Proof. The first statement follows from Injectives, Lemma 19.13.6. Then we may apply More on Algebra, Remark 15.86.10 to $R\mathop{\mathrm{lim}}\nolimits R\Gamma (U, K_ n) = R\Gamma (U, R\mathop{\mathrm{lim}}\nolimits K_ n)$ to get the short exact sequences. $\square$
Lemma 21.23.3. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Then $Rf_*$ commutes with $R\mathop{\mathrm{lim}}\nolimits $, i.e., $Rf_*$ commutes with derived limits.
Proof. Let $(K_ n)$ be an inverse system of objects of $D(\mathcal{O})$. By induction on $n$ we may choose actual complexes $\mathcal{K}_ n^\bullet $ of $\mathcal{O}$-modules and maps of complexes $\mathcal{K}_{n + 1}^\bullet \to \mathcal{K}_ n^\bullet $ representing the maps $K_{n + 1} \to K_ n$ in $D(\mathcal{O})$. In other words, there exists an object $K$ in $D(\mathcal{C} \times \mathbf{N})$ whose associated inverse system is the given one. Next, consider the commutative diagram
of morphisms of topoi. It follows that $R\mathop{\mathrm{lim}}\nolimits R(f \times 1)_*K = Rf_* R\mathop{\mathrm{lim}}\nolimits K$. Working through the definitions and using Lemma 21.23.1 we obtain that $R\mathop{\mathrm{lim}}\nolimits (Rf_*K_ n) = Rf_*(R\mathop{\mathrm{lim}}\nolimits K_ n)$.
Alternate proof in case $\mathcal{C}$ has enough points. Consider the defining distinguished triangle
in $D(\mathcal{O})$. Applying the exact functor $Rf_*$ we obtain the distinguished triangle
in $D(\mathcal{O}')$. Thus we see that it suffices to prove that $Rf_*$ commutes with products in the derived category (which are not just given by products of complexes, see Injectives, Lemma 19.13.4). However, since $Rf_*$ is a right adjoint by Lemma 21.19.1 this follows formally (see Categories, Lemma 4.24.5). Caution: Note that we cannot apply Categories, Lemma 4.24.5 directly as $R\mathop{\mathrm{lim}}\nolimits K_ n$ is not a limit in $D(\mathcal{O})$. $\square$
Remark 21.23.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_ n)$ be an inverse system in $D(\mathcal{O})$. Set $K = R\mathop{\mathrm{lim}}\nolimits K_ n$. For each $n$ and $m$ let $\mathcal{H}^ m_ n = H^ m(K_ n)$ be the $m$th cohomology sheaf of $K_ n$ and similarly set $\mathcal{H}^ m = H^ m(K)$. Let us denote $\underline{\mathcal{H}}^ m_ n$ the presheaf Similarly we set $\underline{\mathcal{H}}^ m(U) = H^ m(U, K)$. By Lemma 21.20.3 we see that $\mathcal{H}^ m_ n$ is the sheafification of $\underline{\mathcal{H}}^ m_ n$ and $\mathcal{H}^ m$ is the sheafification of $\underline{\mathcal{H}}^ m$. Here is a diagram In general it may not be the case that $\mathop{\mathrm{lim}}\nolimits \mathcal{H}^ m_ n$ is the sheafification of $\mathop{\mathrm{lim}}\nolimits \underline{\mathcal{H}}^ m_ n$. If $U \in \mathcal{C}$, then we have short exact sequences by Lemma 21.23.2.
\[ U \longmapsto \underline{\mathcal{H}}^ m_ n(U) = H^ m(U, K_ n) \]
\[ \xymatrix{ K \ar@{=}[d] & \underline{\mathcal{H}}^ m \ar[d] \ar[r] & \mathcal{H}^ m \ar[d] \\ R\mathop{\mathrm{lim}}\nolimits K_ n & \mathop{\mathrm{lim}}\nolimits \underline{\mathcal{H}}^ m_ n \ar[r] & \mathop{\mathrm{lim}}\nolimits \mathcal{H}^ m_ n } \]
21.23.4.1
\begin{equation} \label{sites-cohomology-equation-ses-Rlim-over-U} 0 \to R^1\mathop{\mathrm{lim}}\nolimits \underline{\mathcal{H}}^{m - 1}_ n(U) \to \underline{\mathcal{H}}^ m(U) \to \mathop{\mathrm{lim}}\nolimits \underline{\mathcal{H}}^ m_ n(U) \to 0 \end{equation}
The following lemma applies to an inverse system of quasi-coherent modules with surjective transition maps on an algebraic space or an algebraic stack.
Lemma 21.23.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{F}_ n)$ be an inverse system of $\mathcal{O}$-modules. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$,
$H^ p(U, \mathcal{F}_ n) = 0$ for $p > 0$ and $U \in \mathcal{B}$,
the inverse system $\mathcal{F}_ n(U)$ has vanishing $R^1\mathop{\mathrm{lim}}\nolimits $ for $U \in \mathcal{B}$.
Then $R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$ and we have $H^ p(U, \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n) = 0$ for $p > 0$ and $U \in \mathcal{B}$.
Proof. Set $K_ n = \mathcal{F}_ n$ and $K = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$. Using the notation of Remark 21.23.4 and assumption (2) we see that for $U \in \mathcal{B}$ we have $\underline{\mathcal{H}}_ n^ m(U) = 0$ when $m \not= 0$ and $\underline{\mathcal{H}}_ n^0(U) = \mathcal{F}_ n(U)$. From Equation (21.23.4.1) and assumption (3) we see that $\underline{\mathcal{H}}^ m(U) = 0$ when $m \not= 0$ and equal to $\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n(U)$ when $m = 0$. Sheafifying using (1) we find that $\mathcal{H}^ m = 0$ when $m \not= 0$ and equal to $\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$ when $m = 0$. Hence $K = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$. Since $H^ m(U, K) = \underline{\mathcal{H}}^ m(U) = 0$ for $m > 0$ (see above) we see that the second assertion holds. $\square$
Lemma 21.23.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_ n)$ be an inverse system in $D(\mathcal{O})$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $m \in \mathbf{Z}$. Assume there exist an integer $n(V)$ and a cofinal system $\text{Cov}_ V$ of coverings of $V$ such that for $\{ V_ i \to V\} \in \text{Cov}_ V$
$R^1\mathop{\mathrm{lim}}\nolimits H^{m - 1}(V_ i, K_ n) = 0$, and
$H^ m(V_ i, K_ n) \to H^ m(V_ i, K_{n(V)})$ is injective for $n \geq n(V)$.
Then the map on sections $H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n)(V) \to H^ m(K_{n(V)})(V)$ is injective.
Proof. Let $\gamma \in H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n)(V)$ map to zero in $H^ m(K_{n(V)})(V)$. Since $H^ m(R\mathop{\mathrm{lim}}\nolimits K_ n)$ is the sheafification of $U \mapsto H^ m(U, R\mathop{\mathrm{lim}}\nolimits K_ n)$ (by Lemma 21.20.3) we can choose $\{ V_ i \to V\} \in \text{Cov}_ V$ and elements $\tilde\gamma _ i \in H^ m(V_ i, R\mathop{\mathrm{lim}}\nolimits K_ n)$ mapping to $\gamma |_{V_ i}$. Then $\tilde\gamma _ i$ maps to $\tilde\gamma _{i, n(V)} \in H^ m(V_ i, K_{n(V)})$. Using that $H^ m(K_{n(V)})$ is the sheafification of $U \mapsto H^ m(U, K_{n(V)})$ (by Lemma 21.20.3 again) we see that after replacing $\{ V_ i \to V\} $ by a refinement we may assume that $\tilde\gamma _{i, n(V)} = 0$ for all $i$. For this covering we consider the short exact sequences
of Lemma 21.23.2. By assumption (1) the group on the left is zero and by assumption (2) the group on the right maps injectively into $H^ m(V_ i, K_{n(V)})$. We conclude $\tilde\gamma _ i = 0$ and hence $\gamma = 0$ as desired. $\square$
Lemma 21.23.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, and
for every $V \in \mathcal{B}$ there exist a function $p(V, -) : \mathbf{Z} \to \mathbf{Z}$ and a cofinal system $\text{Cov}_ V$ of coverings of $V$ such that
for all $\{ V_ i \to V\} \in \text{Cov}_ V$ and all integers $p, m$ satisfying $p > p(V, m)$.
\[ H^ p(V_ i, H^{m - p}(E)) = 0 \]
Then the canonical map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.
Proof. Set $K_ n = \tau _{\geq -n}E$ and $K = R\mathop{\mathrm{lim}}\nolimits K_ n$. The canonical map $E \to K$ comes from the canonical maps $E \to K_ n = \tau _{\geq -n}E$. We have to show that $E \to K$ induces an isomorphism $H^ m(E) \to H^ m(K)$ of cohomology sheaves. In the rest of the proof we fix $m$. If $n \geq -m$, then the map $E \to \tau _{\geq -n}E = K_ n$ induces an isomorphism $H^ m(E) \to H^ m(K_ n)$. To finish the proof it suffices to show that for every $V \in \mathcal{B}$ there exists an integer $n(V) \geq -m$ such that the map $H^ m(K)(V) \to H^ m(K_{n(V)})(V)$ is injective. Namely, then the composition
is a bijection and the second arrow is injective, hence the first arrow is bijective. By property (1) this will imply $H^ m(E) \to H^ m(K)$ is an isomorphism. Set
so that in any case $n(V) \geq -m$. Claim: the maps
are isomorphisms for $n \geq n(V)$ and $\{ V_ i \to V\} \in \text{Cov}_ V$. The claim implies conditions (1) and (2) of Lemma 21.23.6 are satisfied and hence implies the desired injectivity. Recall (Derived Categories, Remark 13.12.4) that we have distinguished triangles
Looking at the asssociated long exact cohomology sequence the claim follows if
are zero for $n \geq n(V)$ and $\{ V_ i \to V\} \in \text{Cov}_ V$. This follows from our choice of $n(V)$ and the assumption in the lemma. $\square$
Lemma 21.23.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, and
for every $V \in \mathcal{B}$ there exist an integer $d_ V \geq 0$ and a cofinal system $\text{Cov}_ V$ of coverings of $V$ such that
\[ H^ p(V_ i, H^ q(E)) = 0 \text{ for } \{ V_ i \to V\} \in \text{Cov}_ V,\ p > d_ V, \text{ and }q < 0 \]
Then the canonical map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.
Proof. This follows from Lemma 21.23.7 with $p(V, m) = d_ V + \max (0, m)$. $\square$
Lemma 21.23.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Assume there exists a function $p(-) : \mathbf{Z} \to \mathbf{Z}$ and a subset $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$,
$H^ p(V, H^{m - p}(E)) = 0$ for $p > p(m)$ and $V \in \mathcal{B}$.
Then the canonical map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.
Proof. Apply Lemma 21.23.7 with $p(V, m) = p(m)$ and $\text{Cov}_ V$ equal to the set of coverings $\{ V_ i \to V\} $ with $V_ i \in \mathcal{B}$ for all $i$. $\square$
Lemma 21.23.10. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Assume there exists an integer $d \geq 0$ and a subset $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$,
$H^ p(V, H^ q(E)) = 0$ for $p > d$, $q < 0$, and $V \in \mathcal{B}$.
Then the canonical map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.
Proof. Apply Lemma 21.23.8 with $d_ V = d$ and $\text{Cov}_ V$ equal to the set of coverings $\{ V_ i \to V\} $ with $V_ i \in \mathcal{B}$ for all $i$. $\square$
The lemmas above can be used to compute cohomology in certain situations.
Lemma 21.23.11. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\mathcal{O})$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$,
$H^ p(U, H^ q(K)) = 0$ for all $p > 0$, $q \in \mathbf{Z}$, and $U \in \mathcal{B}$.
Then $H^ q(U, K) = H^0(U, H^ q(K))$ for $q \in \mathbf{Z}$ and $U \in \mathcal{B}$.
Proof. Observe that $K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} K$ by Lemma 21.23.10 with $d = 0$. Let $U \in \mathcal{B}$. By Equation (21.23.4.1) we get a short exact sequence
Condition (2) implies $H^ q(U, \tau _{\geq -n} K) = H^0(U, H^ q(\tau _{\geq -n} K))$ for all $q$ by using the spectral sequence of Derived Categories, Lemma 13.21.3. The spectral sequence converges because $\tau _{\geq -n}K$ is bounded below. If $n > -q$ then we have $H^ q(\tau _{\geq -n}K) = H^ q(K)$. Thus the systems on the left and the right of the displayed short exact sequence are eventually constant with values $H^0(U, H^{q - 1}(K))$ and $H^0(U, H^ q(K))$ and the lemma follows. $\square$
Here is another case where we can describe the derived limit.
Lemma 21.23.12. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_ n)$ be an inverse system of objects of $D(\mathcal{O})$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$,
for all $U \in \mathcal{B}$ and all $q \in \mathbf{Z}$ we have
$H^ p(U, H^ q(K_ n)) = 0$ for $p > 0$,
the inverse system $H^0(U, H^ q(K_ n))$ has vanishing $R^1\mathop{\mathrm{lim}}\nolimits $.
Then $H^ q(R\mathop{\mathrm{lim}}\nolimits K_ n) = \mathop{\mathrm{lim}}\nolimits H^ q(K_ n)$ for $q \in \mathbf{Z}$.
Proof. Set $K = R\mathop{\mathrm{lim}}\nolimits K_ n$. We will use notation as in Remark 21.23.4. Let $U \in \mathcal{B}$. By Lemma 21.23.11 and (2)(a) we have $H^ q(U, K_ n) = H^0(U, H^ q(K_ n))$. Using that the functor $R\Gamma (U, -)$ commutes with derived limits we have
where the final equality follows from More on Algebra, Remark 15.86.10 and assumption (2)(b). Thus $H^ q(U, K)$ is the inverse limit the sections of the sheaves $H^ q(K_ n)$ over $U$. Since $\mathop{\mathrm{lim}}\nolimits H^ q(K_ n)$ is a sheaf we find using assumption (1) that $H^ q(K)$, which is the sheafification of the presheaf $U \mapsto H^ q(U, K)$, is equal to $\mathop{\mathrm{lim}}\nolimits H^ q(K_ n)$. This proves the lemma. $\square$
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