61.29 A suitable derived category

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

1. $K$ is derived complete with respect to $I$,

2. $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ 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

1. $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ and $H^ i(K) = 0$ for $i \not\in [a, b]$,

2. each $K_ n$ has tor amplitude in $[a, b]$,

3. each $K_ n$ has constructible cohomology sheaves,

4. 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

$(\underline{M^ a})^\wedge \to \ldots \to (\underline{M^ b})^\wedge$

in $D(U_{i, {pro\text{-}\acute{e}tale}}, \Lambda )$ for some maps of sheaves of $\Lambda$-modules $(\underline{M^ i})^\wedge \to (\underline{M^{i + 1}})^\wedge$.

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

$\mathcal{F} \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I} = \mathcal{F}/I\mathcal{F}$

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

$\rho |_{U_ i} \bmod I : \underline{\Lambda /I}^{\oplus t} \longrightarrow \underline{\overline{M}_ i}$

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

$\underline{M_ i}^\wedge \to (\underline{\Lambda }^\wedge )^{\oplus t} \to \mathcal{F}|_{U_ i}$

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

$0 \to \mathcal{K}/I\mathcal{K} \to \underline{\overline{M}_ i} \to \mathcal{F}/I\mathcal{F} \to 0$

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

$\tilde\rho : (\underline{\Lambda }^\wedge )^{\oplus t}[-b] \longrightarrow K$

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

$(\underline{\Lambda }^\wedge )^{\oplus t}[-b] \to K \to L \to (\underline{\Lambda }^\wedge )^{\oplus t}[-b + 1]$

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

$(\underline{M^ a})^\wedge \to \ldots \to (\underline{M^{b - 1}})^\wedge$

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 $K \in D(X_{pro\text{-}\acute{e}tale}, \Lambda )$.

1. 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

$\underline{M^ a}^\wedge \to \ldots \to \underline{M^ b}^\wedge$
2. 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.

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

$H^0(X, K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ n)$

has the Mittag-Leffler condition.

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

$K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^{n + 1}} \to K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$

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

$H^0(X, K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ n) \to H^0(X, K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ m) \to H^0(X, K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ k)$

is isomorphic to

$H^0(X, \underline{M^\bullet /I^ nM^\bullet }) \to H^0(X, \underline{M^\bullet /I^ mM^\bullet }) \to H^0(X, \underline{M^\bullet /I^ kM^\bullet })$

Namely, choose any isomorphism

$\underline{M^\bullet /I^ nM^\bullet } \to K \otimes _\Lambda ^\mathbf {L} \Lambda /I^ n$

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

$H^0(M^\bullet /I^ nM^\bullet )$

has Mittag-Leffler. Set $A = \Lambda ^\wedge$ and consider the spectral sequence

$\text{Tor}_{-p}^ A(H^ q(M^\bullet ), A/I^ nA) \Rightarrow H^{p + q}(M^\bullet /I^ nM^\bullet )$

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

$\underline{M^\bullet }^\wedge = \mathop{\mathrm{lim}}\nolimits \underline{M^\bullet /I^ nM^\bullet } = R\mathop{\mathrm{lim}}\nolimits \underline{M^\bullet /I^ nM^\bullet }$

Let $E^\bullet$ be the dual complex to $M^\bullet$, see More on Algebra, Lemma 15.74.15 and its proof. Consider the objects

$H_ n = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\Lambda /I^ n}(\underline{M^\bullet /I^ nM^\bullet }, K_ n) = \underline{E^\bullet /I^ nE^\bullet } \otimes _{\Lambda /I^ n}^\mathbf {L} K_ n$

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

\begin{align*} H_ n|_{W_ t} & \cong R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\Lambda /I^ n}(\underline{M^\bullet /I^ nM^\bullet }, \underline{M^\bullet /I^ nM^\bullet }) \\ & = \underline{ \text{Tot}(E^\bullet /I^ nE^\bullet \otimes _{\Lambda /I^ n} M^\bullet /I^ nM^\bullet ) } \end{align*}

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

$H^0(W_ t, H_1) = \mathop{\mathrm{Hom}}\nolimits (\underline{M^\bullet /IM^\bullet }, K_1)$

we find an element $(\varphi _{t, n})$ in the inverse limit which produces an isomorphism mod $I$. Then

$R\mathop{\mathrm{lim}}\nolimits \varphi _{t, n} : \underline{M^\bullet }^\wedge |_{W_ t} = R\mathop{\mathrm{lim}}\nolimits \underline{M^\bullet /I^ nM^\bullet }|_{W_ t} \longrightarrow R\mathop{\mathrm{lim}}\nolimits K_ n|_{W_ t} = K|_{W_ t}$

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. $\square$

[1] Proof: by Algebra, Lemma 10.32.7 we can lift $\overline{p}_ i$ to a compatible system of projectors $p_{i, n} : (\Lambda /I^ n)^{\oplus t} \to (\Lambda /I^ n)^{\oplus t}$ and then we set $p_ i = \mathop{\mathrm{lim}}\nolimits p_{i, n}$ which works because $\Lambda ^\wedge = \mathop{\mathrm{lim}}\nolimits \Lambda /I^ n$.
[2] This may be nonstandard notation
[3] This may be nonstandard notation.

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