Let $X$ be a scheme. It will turn out that for many schemes $X$ a suitable derived category of $\ell $-adic sheaves can be gotten by considering the derived complete objects $K$ of $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ with the property that $K \otimes _\Lambda ^\mathbf {L} \mathbf{F}_\ell $ is bounded with constructible cohomology sheaves. Here is the general definition.
Definition 61.29.1. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $X$ be a scheme. An object $K$ of $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ is called constructible if
$K$ is derived complete with respect to $I$,
$K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ has constructible cohomology sheaves and locally has finite tor dimension.
We denote $D_{cons}(X, \Lambda )$ the full subcategory of constructible $K$ in $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$.
Recall that with our conventions a complex of finite tor dimension is bounded (Cohomology on Sites, Definition 21.46.1). In fact, let's collect everything proved so far in a lemma.
Lemma 61.29.2. In the situation above suppose $K$ is in $D_{cons}(X, \Lambda )$ and $X$ is quasi-compact. Set $K_ n = K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$. There exist $a, b$ such that
$K = R\mathop{\mathrm{lim}}\nolimits K_ n$ and $H^ i(K) = 0$ for $i \not\in [a, b]$,
each $K_ n$ has tor amplitude in $[a, b]$,
each $K_ n$ has constructible cohomology sheaves,
each $K_ n = \epsilon ^{-1}L_ n$ for some $L_ n \in D_{ctf}(X_{\acute{e}tale}, \Lambda /I^ n)$ (Étale Cohomology, Definition 59.77.1).
Proof. By definition of local having finite tor dimension, we can find $a, b$ such that $K_1$ has tor amplitude in $[a, b]$. Part (2) follows from Cohomology on Sites, Lemma 21.46.9. Then (1) follows as $K$ is derived complete by the description of limits in Cohomology on Sites, Proposition 21.51.2 and the fact that $H^ b(K_{n + 1}) \to H^ b(K_ n)$ is surjective as $K_ n = K_{n + 1} \otimes ^\mathbf {L}_\Lambda \underline{\Lambda /I^ n}$. Part (3) follows from Lemma 61.27.6, Part (4) follows from Lemma 61.27.4 and the fact that $L_ n$ has finite tor dimension because $K_ n$ does (small argument omitted). $\square$
Lemma 61.29.3. Let $X$ be a weakly contractible affine scheme. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $K$ be an object of $D_{cons}(X, \Lambda )$ such that the cohomology sheaves of $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ are locally constant. Then there exists a finite disjoint open covering $X = \coprod U_ i$ and for each $i$ a finite collection of finite projective $\Lambda ^\wedge $-modules $M_ a, \ldots , M_ b$ such that $K|_{U_ i}$ is represented by a complex in $D(U_{i, {pro\text{-}\acute{e}tale}}, \Lambda )$ for some maps of sheaves of $\Lambda $-modules $(\underline{M^ i})^\wedge \to (\underline{M^{i + 1}})^\wedge $.
\[ (\underline{M^ a})^\wedge \to \ldots \to (\underline{M^ b})^\wedge \]
Proof. We freely use the results of Lemma 61.29.2. Choose $a, b$ as in that lemma. We will prove the lemma by induction on $b - a$. Let $\mathcal{F} = H^ b(K)$. Note that $\mathcal{F}$ is a derived complete sheaf of $\Lambda $-modules by Proposition 61.21.1. Moreover $\mathcal{F}/I\mathcal{F}$ is a locally constant sheaf of $\Lambda /I$-modules of finite type. Apply Lemma 61.28.7 to get a surjection $\rho : (\underline{\Lambda }^\wedge )^{\oplus t} \to \mathcal{F}$.
If $a = b$, then $K = \mathcal{F}[-b]$. In this case we see that
As $X$ is weakly contractible and $\mathcal{F}/I\mathcal{F}$ locally constant, we can find a finite disjoint union decomposition $X = \coprod U_ i$ by affine opens $U_ i$ and $\Lambda /I$-modules $\overline{M}_ i$ such that $\mathcal{F}/I\mathcal{F}$ restricts to $\underline{\overline{M}_ i}$ on $U_ i$. After refining the covering we may assume the map
is equal to $\underline{\alpha _ i}$ for some surjective module map $\alpha _ i : \Lambda /I^{\oplus t} \to \overline{M}_ i$, see Modules on Sites, Lemma 18.43.3. Note that each $\overline{M}_ i$ is a finite $\Lambda /I$-module. Since $\mathcal{F}/I\mathcal{F}$ has tor amplitude in $[0, 0]$ we conclude that $\overline{M}_ i$ is a flat $\Lambda /I$-module. Hence $\overline{M}_ i$ is finite projective (Algebra, Lemma 10.78.2). Hence we can find a projector $\overline{p}_ i : (\Lambda /I)^{\oplus t} \to (\Lambda /I)^{\oplus t}$ whose image maps isomorphically to $\overline{M}_ i$ under the map $\alpha _ i$. We can lift $\overline{p}_ i$ to a projector $p_ i : (\Lambda ^\wedge )^{\oplus t} \to (\Lambda ^\wedge )^{\oplus t}$1. Then $M_ i = \mathop{\mathrm{Im}}(p_ i)$ is a finite $I$-adically complete $\Lambda ^\wedge $-module with $M_ i/IM_ i = \overline{M}_ i$. Over $U_ i$ consider the maps
By construction the composition induces an isomorphism modulo $I$. The source and target are derived complete, hence so are the cokernel $\mathcal{Q}$ and the kernel $\mathcal{K}$. We have $\mathcal{Q}/I\mathcal{Q} = 0$ by construction hence $\mathcal{Q}$ is zero by Lemma 61.28.6. Then
is exact by the vanishing of $\text{Tor}_1$ see at the start of this paragraph; also use that $\underline{\Lambda }^\wedge /I\overline{\Lambda }^\wedge $ by Modules on Sites, Lemma 18.42.4 to see that $\underline{M_ i}^\wedge /I\underline{M_ i}^\wedge = \underline{\overline{M}_ i}$. Hence $\mathcal{K}/I\mathcal{K} = 0$ by construction and we conclude that $\mathcal{K} = 0$ as before. This proves the result in case $a = b$.
If $b > a$, then we lift the map $\rho $ to a map
in $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$. This is possible as we can think of $K$ as a complex of $\underline{\Lambda }^\wedge $-modules by discussion in the introduction to Section 61.20 and because $X_{pro\text{-}\acute{e}tale}$ is weakly contractible hence there is no obstruction to lifting the elements $\rho (e_ s) \in H^0(X, \mathcal{F})$ to elements of $H^ b(X, K)$. Fitting $\tilde\rho $ into a distinguished triangle
we see that $L$ is an object of $D_{cons}(X, \Lambda )$ such that $L \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ has tor amplitude contained in $[a, b - 1]$ (details omitted). By induction we can describe $L$ locally as stated in the lemma, say $L$ is isomorphic to
The map $L \to (\underline{\Lambda }^\wedge )^{\oplus t}[-b + 1]$ corresponds to a map $(\underline{M^{b - 1}})^\wedge \to (\underline{\Lambda }^\wedge )^{\oplus t}$ which allows us to extend the complex by one. The corresponding complex is isomorphic to $K$ in the derived category by the properties of triangulated categories. This finishes the proof. $\square$
Motivated by what happens for constructible $\Lambda $-sheaves we introduce the following notion.
Definition 61.29.4. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $K \in D(X_{pro\text{-}\acute{e}tale}, \Lambda )$.
We say $K$ is adic lisse2 if there exists a finite complex of finite projective $\Lambda ^\wedge $-modules $M^\bullet $ such that $K$ is locally isomorphic to
We say $K$ is adic constructible3 if for every affine open $U \subset X$ there exists a decomposition $U = \coprod U_ i$ into constructible locally closed subschemes such that $K|_{U_ i}$ is adic lisse.
\[ \underline{M^ a}^\wedge \to \ldots \to \underline{M^ b}^\wedge \]
The difference between the local structure obtained in Lemma 61.29.3 and the structure of an adic lisse complex is that the maps $\underline{M^ i}^\wedge \to \underline{M^{i + 1}}^\wedge $ in Lemma 61.29.3 need not be constant, whereas in the definition above they are required to be constant.
Lemma 61.29.5. Let $X$ be a weakly contractible affine scheme. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $K$ be an object of $D_{cons}(X, \Lambda )$ such that $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$ is isomorphic in $D(X_{pro\text{-}\acute{e}tale}, \Lambda /I^ n)$ to a complex of constant sheaves of $\Lambda /I^ n$-modules. Then has the Mittag-Leffler condition.
\[ H^0(X, K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ n) \]
Proof. Say $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$ is isomorphic to $\underline{E_ n}$ for some object $E_ n$ of $D(\Lambda /I^ n)$. Since $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ has finite tor dimension and has finite type cohomology sheaves we see that $E_1$ is perfect (see More on Algebra, Lemma 15.74.2). The transition maps
locally come from (possibly many distinct) maps of complexes $E_{n + 1} \to E_ n$ in $D(\Lambda /I^{n + 1})$ see Cohomology on Sites, Lemma 21.53.3. For each $n$ choose one such map and observe that it induces an isomorphism $E_{n + 1} \otimes _{\Lambda /I^{n + 1}}^\mathbf {L} \Lambda /I^ n \to E_ n$ in $D(\Lambda /I^ n)$. By More on Algebra, Lemma 15.97.4 we can find a finite complex $M^\bullet $ of finite projective $\Lambda ^\wedge $-modules and isomorphisms $M^\bullet /I^ nM^\bullet \to E_ n$ in $D(\Lambda /I^ n)$ compatible with the transition maps.
Now observe that for each finite collection of indices $n > m > k$ the triple of maps
is isomorphic to
Namely, choose any isomorphism
induces similar isomorphisms module $I^ m$ and $I^ k$ and we see that the assertion is true. Thus to prove the lemma it suffices to show that the system $H^0(X, \underline{M^\bullet /I^ nM^\bullet })$ has Mittag-Leffler. Since taking sections over $X$ is exact, it suffices to prove that the system of $\Lambda $-modules
has Mittag-Leffler. Set $A = \Lambda ^\wedge $ and consider the spectral sequence
By More on Algebra, Lemma 15.27.3 the pro-systems $\{ \text{Tor}_{-p}^ A(H^ q(M^\bullet ), A/I^ nA)\} $ are zero for $p > 0$. Thus the pro-system $\{ H^0(M^\bullet /I^ nM^\bullet )\} $ is equal to the pro-system $\{ H^0(M^\bullet )/I^ nH^0(M^\bullet )\} $ and the lemma is proved. $\square$
Lemma 61.29.6. Let $X$ be a connected scheme. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. If $K$ is in $D_{cons}(X, \Lambda )$ such that $K \otimes _\Lambda \underline{\Lambda /I}$ has locally constant cohomology sheaves, then $K$ is adic lisse (Definition 61.29.4).
Proof. Write $K_ n = K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$. We will use the results of Lemma 61.29.2 without further mention. By Cohomology on Sites, Lemma 21.53.5 we see that $K_ n$ has locally constant cohomology sheaves for all $n$. We have $K_ n = \epsilon ^{-1}L_ n$ some $L_ n$ in $D_{ctf}(X_{\acute{e}tale}, \Lambda /I^ n)$ with locally constant cohomology sheaves. By Étale Cohomology, Lemma 59.77.7 there exist perfect $M_ n \in D(\Lambda /I^ n)$ such that $L_ n$ is étale locally isomorphic to $\underline{M_ n}$. The maps $L_{n + 1} \to L_ n$ corresponding to $K_{n + 1} \to K_ n$ induces isomorphisms $L_{n + 1} \otimes _{\Lambda /I^{n + 1}}^\mathbf {L} \underline{\Lambda /I^ n} \to L_ n$. Looking locally on $X$ we conclude that there exist maps $M_{n + 1} \to M_ n$ in $D(\Lambda /I^{n + 1})$ inducing isomorphisms $M_{n + 1} \otimes _{\Lambda /I^{n + 1}} \Lambda /I^ n \to M_ n$, see Cohomology on Sites, Lemma 21.53.3. Fix a choice of such maps. By More on Algebra, Lemma 15.97.4 we can find a finite complex $M^\bullet $ of finite projective $\Lambda ^\wedge $-modules and isomorphisms $M^\bullet /I^ nM^\bullet \to M_ n$ in $D(\Lambda /I^ n)$ compatible with the transition maps. To finish the proof we will show that $K$ is locally isomorphic to
Let $E^\bullet $ be the dual complex to $M^\bullet $, see More on Algebra, Lemma 15.74.15 and its proof. Consider the objects
of $D(X_{pro\text{-}\acute{e}tale}, \Lambda /I^ n)$. Modding out by $I^ n$ defines a transition map $H_{n + 1} \to H_ n$. Set $H = R\mathop{\mathrm{lim}}\nolimits H_ n$. Then $H$ is an object of $D_{cons}(X, \Lambda )$ (details omitted) with $H \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n} = H_ n$. Choose a covering $\{ W_ t \to X\} _{t \in T}$ with each $W_ t$ affine and weakly contractible. By our choice of $M^\bullet $ we see that
Thus we may apply Lemma 61.29.5 to $H = R\mathop{\mathrm{lim}}\nolimits H_ n$. We conclude the system $H^0(W_ t, H_ n)$ satisfies Mittag-Leffler. Since for all $n \gg 1$ there is an element of $H^0(W_ t, H_ n)$ which maps to an isomorphism in
we find an element $(\varphi _{t, n})$ in the inverse limit which produces an isomorphism mod $I$. Then
is an isomorphism. This finishes the proof. $\square$
Proposition 61.29.7. Let $X$ be a Noetherian scheme. Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal. Let $K$ be an object of $D_{cons}(X, \Lambda )$. Then $K$ is adic constructible (Definition 61.29.4).
Proof. This is a consequence of Lemma 61.29.6 and the fact that a Noetherian scheme is locally connected (Topology, Lemma 5.9.6), and the definitions. $\square$
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 (0)