Lemma 21.53.1. Let $\mathcal{C}$ be a site with final object $X$. Let $\Lambda $ be a Noetherian ring. Let $K \in D^ b(\mathcal{C}, \Lambda )$ with $H^ i(K)$ locally constant sheaves of $\Lambda $-modules of finite type. Then there exists a covering $\{ U_ i \to X\} $ such that each $K|_{U_ i}$ is represented by a complex of locally constant sheaves of $\Lambda $-modules of finite type.
21.53 Complexes with locally constant cohomology sheaves
Locally constant sheaves are introduced in Modules on Sites, Section 18.43. Let $\mathcal{C}$ be a site. Let $\Lambda $ be a ring. We denote $D(\mathcal{C}, \Lambda )$ the derived category of the abelian category of $\underline{\Lambda }$-modules on $\mathcal{C}$.
Proof. Let $a \leq b$ be such that $H^ i(K) = 0$ for $i \not\in [a, b]$. By induction on $b - a$ we will prove there exists a covering $\{ U_ i \to X\} $ such that $K|_{U_ i}$ can be represented by a complex $\underline{M^\bullet }_{U_ i}$ with $M^ p$ a finite type $\Lambda $-module and $M^ p = 0$ for $p \not\in [a, b]$. If $b = a$, then this is clear. In general, we may replace $X$ by the members of a covering and assume that $H^ b(K)$ is constant, say $H^ b(K) = \underline{M}$. By Modules on Sites, Lemma 18.42.5 the module $M$ is a finite $\Lambda $-module. Choose a surjection $\Lambda ^{\oplus r} \to M$ given by generators $x_1, \ldots , x_ r$ of $M$.
By a slight generalization of Lemma 21.7.3 (details omitted) there exists a covering $\{ U_ i \to X\} $ such that $x_ i \in H^0(X, H^ b(K))$ lifts to an element of $H^ b(U_ i, K)$. Thus, after replacing $X$ by the $U_ i$ we reach the situation where there is a map $\underline{\Lambda ^{\oplus r}}[-b] \to K$ inducing a surjection on cohomology sheaves in degree $b$. Choose a distinguished triangle
\[ \underline{\Lambda ^{\oplus r}}[-b] \to K \to L \to \underline{\Lambda ^{\oplus r}}[-b + 1] \]
Now the cohomology sheaves of $L$ are nonzero only in the interval $[a, b - 1]$, agree with the cohomology sheaves of $K$ in the interval $[a, b - 2]$ and there is a short exact sequence
\[ 0 \to H^{b - 1}(K) \to H^{b - 1}(L) \to \underline{\mathop{\mathrm{Ker}}(\Lambda ^{\oplus r} \to M)} \to 0 \]
in degree $b - 1$. By Modules on Sites, Lemma 18.43.5 we see that $H^{b - 1}(L)$ is locally constant of finite type. By induction hypothesis we obtain an isomorphism $\underline{M^\bullet } \to L$ in $D(\mathcal{C}, \underline{\Lambda })$ with $M^ p$ a finite $\Lambda $-module and $M^ p = 0$ for $p \not\in [a, b - 1]$. The map $L \to \Lambda ^{\oplus r}[-b + 1]$ gives a map $\underline{M^{b - 1}} \to \underline{\Lambda ^{\oplus r}}$ which locally is constant (Modules on Sites, Lemma 18.43.3). Thus we may assume it is given by a map $M^{b - 1} \to \Lambda ^{\oplus r}$. The distinguished triangle shows that the composition $M^{b - 2} \to M^{b - 1} \to \Lambda ^{\oplus r}$ is zero and the axioms of triangulated categories produce an isomorphism
\[ \underline{M^ a \to \ldots \to M^{b - 1} \to \Lambda ^{\oplus r}} \longrightarrow K \]
in $D(\mathcal{C}, \Lambda )$. $\square$
Let $\mathcal{C}$ be a site. Let $\Lambda $ be a ring. Using the morphism $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (pt)$ we see that there is a functor $D(\Lambda ) \to D(\mathcal{C}, \Lambda )$, $K \mapsto \underline{K}$.
Lemma 21.53.2. Let $\mathcal{C}$ be a site with final object $X$. Let $\Lambda $ be a ring. Let
$K$ a perfect object of $D(\Lambda )$,
a finite complex $K^\bullet $ of finite projective $\Lambda $-modules representing $K$,
$\mathcal{L}^\bullet $ a complex of sheaves of $\Lambda $-modules, and
$\varphi : \underline{K} \to \mathcal{L}^\bullet $ a map in $D(\mathcal{C}, \Lambda )$.
Then there exists a covering $\{ U_ i \to X\} $ and maps of complexes $\alpha _ i : \underline{K}^\bullet |_{U_ i} \to \mathcal{L}^\bullet |_{U_ i}$ representing $\varphi |_{U_ i}$.
Proof. Follows immediately from Lemma 21.44.8. $\square$
Lemma 21.53.3. Let $\mathcal{C}$ be a site with final object $X$. Let $\Lambda $ be a ring. Let $K, L$ be objects of $D(\Lambda )$ with $K$ perfect. Let $\varphi : \underline{K} \to \underline{L}$ be map in $D(\mathcal{C}, \Lambda )$. There exists a covering $\{ U_ i \to X\} $ such that $\varphi |_{U_ i}$ is equal to $\underline{\alpha _ i}$ for some map $\alpha _ i : K \to L$ in $D(\Lambda )$.
Proof. Follows from Lemma 21.53.2 and Modules on Sites, Lemma 18.43.3. $\square$
Lemma 21.53.4. Let $\mathcal{C}$ be a site. Let $\Lambda $ be a Noetherian ring. Let $K, L \in D^-(\mathcal{C}, \Lambda )$. If the cohomology sheaves of $K$ and $L$ are locally constant sheaves of $\Lambda $-modules of finite type, then the cohomology sheaves of $K \otimes _\Lambda ^\mathbf {L} L$ are locally constant sheaves of $\Lambda $-modules of finite type.
Proof. We'll prove this as an application of Lemma 21.53.1. Note that $H^ i(K \otimes _\Lambda ^\mathbf {L} L)$ is the same as $H^ i(\tau _{\geq i - 1}K \otimes _\Lambda ^\mathbf {L} \tau _{\geq i - 1}L)$. Thus we may assume $K$ and $L$ are bounded. By Lemma 21.53.1 we may assume that $K$ and $L$ are represented by complexes of locally constant sheaves of $\Lambda $-modules of finite type. Then we can replace these complexes by bounded above complexes of finite free $\Lambda $-modules. In this case the result is clear. $\square$
Lemma 21.53.5. Let $\mathcal{C}$ be a site. Let $\Lambda $ be a Noetherian ring. Let $I \subset \Lambda $ be an ideal. Let $K \in D^-(\mathcal{C}, \Lambda )$. If the cohomology sheaves of $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ are locally constant sheaves of $\Lambda /I$-modules of finite type, then the cohomology sheaves of $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$ are locally constant sheaves of $\Lambda /I^ n$-modules of finite type for all $n \geq 1$.
Proof. Recall that the locally constant sheaves of $\Lambda $-modules of finite type form a weak Serre subcategory of all $\underline{\Lambda }$-modules, see Modules on Sites, Lemma 18.43.5. Thus the subcategory of $D(\mathcal{C}, \Lambda )$ consisting of complexes whose cohomology sheaves are locally constant sheaves of $\Lambda $-modules of finite type forms a strictly full, saturated triangulated subcategory of $D(\mathcal{C}, \Lambda )$, see Derived Categories, Lemma 13.17.1. Next, consider the distinguished triangles
\[ K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}} \to K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^{n + 1}} \to K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n} \to K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}}[1] \]
and the isomorphisms
\[ K \otimes _\Lambda ^\mathbf {L} \underline{I^ n/I^{n + 1}} = \left(K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}\right) \otimes _{\Lambda /I}^\mathbf {L} \underline{I^ n/I^{n + 1}} \]
Combined with Lemma 21.53.4 we obtain the result. $\square$
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