Definition 59.76.1. Let X be a scheme. Let \Lambda be a Noetherian ring. We denote D_ c(X_{\acute{e}tale}, \Lambda ) the full subcategory of D(X_{\acute{e}tale}, \Lambda ) of complexes whose cohomology sheaves are constructible sheaves of \Lambda -modules.
59.76 Complexes with constructible cohomology
Let \Lambda be a ring. Denote D(X_{\acute{e}tale}, \Lambda ) the derived category of sheaves of \Lambda -modules on X_{\acute{e}tale}. We denote by D^ b(X_{\acute{e}tale}, \Lambda ) (respectively D^+, D^-) the full subcategory of bounded (resp. above, below) complexes in D(X_{\acute{e}tale}, \Lambda ).
This definition makes sense by Lemma 59.71.6 and Derived Categories, Section 13.17. Thus we see that D_ c(X_{\acute{e}tale}, \Lambda ) is a strictly full, saturated triangulated subcategory of D(X_{\acute{e}tale}, \Lambda ).
Lemma 59.76.2. Let \Lambda be a Noetherian ring. If j : U \to X is an étale morphism of schemes, then
K|_ U \in D_ c(U_{\acute{e}tale}, \Lambda ) if K \in D_ c(X_{\acute{e}tale}, \Lambda ), and
j_!M \in D_ c(X_{\acute{e}tale}, \Lambda ) if M \in D_ c(U_{\acute{e}tale}, \Lambda ) and the morphism j is quasi-compact and quasi-separated.
Proof. The first assertion is clear. The second follows from the fact that j_! is exact and Lemma 59.73.1. \square
Lemma 59.76.3. Let \Lambda be a Noetherian ring. Let f : X \to Y be a morphism of schemes. If K \in D_ c(Y_{\acute{e}tale}, \Lambda ) then Lf^*K \in D_ c(X_{\acute{e}tale}, \Lambda ).
Proof. This follows as f^{-1} = f^* is exact and Lemma 59.71.5. \square
Lemma 59.76.4. Let X be a quasi-compact and quasi-separated scheme. Let \Lambda be a Noetherian ring. Let K \in D(X_{\acute{e}tale}, \Lambda ) and b \in \mathbf{Z} such that H^ b(K) is constructible. Then there exist a sheaf \mathcal{F} which is a finite direct sum of j_{U!}\underline{\Lambda } with U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale}) affine and a map \mathcal{F}[-b] \to K in D(X_{\acute{e}tale}, \Lambda ) inducing a surjection \mathcal{F} \to H^ b(K).
Proof. Represent K by a complex \mathcal{K}^\bullet of sheaves of \Lambda -modules. Consider the surjection
\mathop{\mathrm{Ker}}(\mathcal{K}^ b \to \mathcal{K}^{b + 1}) \longrightarrow H^ b(K)
By Modules on Sites, Lemma 18.30.6 we may choose a surjection \bigoplus _{i \in I} j_{U_ i!} \underline{\Lambda } \to \mathop{\mathrm{Ker}}(\mathcal{K}^ b \to \mathcal{K}^{b + 1}) with U_ i affine. For I' \subset I finite, denote \mathcal{H}_{I'} \subset H^ b(K) the image of \bigoplus _{i \in I'} j_{U_ i!} \underline{\Lambda }. By Lemma 59.71.8 we see that \mathcal{H}_{I'} = H^ b(K) for some I' \subset I finite. The lemma follows taking \mathcal{F} = \bigoplus _{i \in I'} j_{U_ i!} \underline{\Lambda }. \square
Lemma 59.76.5. Let X be a quasi-compact and quasi-separated scheme. Let \Lambda be a Noetherian ring. Let K \in D^-(X_{\acute{e}tale}, \Lambda ). Then the following are equivalent
K is in D_ c(X_{\acute{e}tale}, \Lambda ),
K can be represented by a bounded above complex whose terms are finite direct sums of j_{U!}\underline{\Lambda } with U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale}) affine,
K can be represented by a bounded above complex of flat constructible sheaves of \Lambda -modules.
Proof. It is clear that (2) implies (3) and that (3) implies (1). Assume K is in D_ c^-(X_{\acute{e}tale}, \Lambda ). Say H^ i(K) = 0 for i > b. By induction on a we will construct a complex \mathcal{F}^ a \to \ldots \to \mathcal{F}^ b such that each \mathcal{F}^ i is a finite direct sum of j_{U!}\underline{\Lambda } with U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale}) affine and a map \mathcal{F}^\bullet \to K which induces an isomorphism H^ i(\mathcal{F}^\bullet ) \to H^ i(K) for i > a and a surjection H^ a(\mathcal{F}^\bullet ) \to H^ a(K). For a = b this can be done by Lemma 59.76.4. Given such a datum choose a distinguished triangle
\mathcal{F}^\bullet \to K \to L \to \mathcal{F}^\bullet [1]
Then we see that H^ i(L) = 0 for i \geq a. Choose \mathcal{F}^{a - 1}[-a +1] \to L as in Lemma 59.76.4. The composition \mathcal{F}^{a - 1}[-a +1] \to L \to \mathcal{F}^\bullet corresponds to a map \mathcal{F}^{a - 1} \to \mathcal{F}^ a such that the composition with \mathcal{F}^ a \to \mathcal{F}^{a + 1} is zero. By TR4 we obtain a map
(\mathcal{F}^{a - 1} \to \ldots \to \mathcal{F}^ b) \to K
in D(X_{\acute{e}tale}, \Lambda ). This finishes the induction step and the proof of the lemma. \square
Lemma 59.76.6. Let X be a scheme. Let \Lambda be a Noetherian ring. Let K, L \in D_ c^-(X_{\acute{e}tale}, \Lambda ). Then K \otimes _\Lambda ^\mathbf {L} L is in D_ c^-(X_{\acute{e}tale}, \Lambda ).
Proof. This follows from Lemmas 59.76.5 and 59.71.9. \square
Comments (2)
Comment #5833 by Haodong Yao on
Comment #5849 by Johan on