The Stacks project

57.74 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 $D(X_{\acute{e}tale}, \Lambda )$.

Definition 57.74.1. 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.

This definition makes sense by Lemma 57.70.6 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 $D_ c(X_{\acute{e}tale}, \Lambda )$.

Lemma 57.74.2. Let $\Lambda $ be a Noetherian ring. If $j : U \to X$ is an étale morphism of schemes, then

  1. $K|_ U \in D_ c(U_{\acute{e}tale}, \Lambda )$ if $K \in D_ c(X_{\acute{e}tale}, \Lambda )$, and

  2. $j_!M \in D_ c(X_{\acute{e}tale}, \Lambda )$ 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 57.72.1. $\square$

Lemma 57.74.3. 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 57.70.5. $\square$

Lemma 57.74.4. 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 18.30.5 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 57.70.9 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 57.74.5. 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

  1. $K$ is in $D_ c(X_{\acute{e}tale}, \Lambda )$,

  2. $K$ 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,

  3. $K$ 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 57.74.4. 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 57.74.4. 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 57.74.6. 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 $D_ c^-(X_{\acute{e}tale}, \Lambda )$.


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