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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