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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (2)
Comment #5833 by Haodong Yao on
Comment #5849 by Johan on