The Stacks project

Lemma 59.77.3. Let $\Lambda $ be a Noetherian ring. Let $X$ be a quasi-compact and quasi-separated scheme. Let $K \in D(X_{\acute{e}tale}, \Lambda )$. The following are equivalent

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

  2. $K$ can be represented by a finite complex of constructible flat sheaves of $\Lambda $-modules.

In fact, if $K$ has tor amplitude in $[a, b]$ then we can represent $K$ by a complex $\mathcal{F}^ a \to \ldots \to \mathcal{F}^ b$ with $\mathcal{F}^ p$ a constructible flat sheaf of $\Lambda $-modules.

Proof. It is clear that a finite complex of constructible flat sheaves of $\Lambda $-modules has finite tor dimension. It is also clear that it is an object of $D_ c(X_{\acute{e}tale}, \Lambda )$. Thus we see that (2) implies (1).

Assume (1). Choose $a, b \in \mathbf{Z}$ such that $H^ i(K \otimes _\Lambda ^\mathbf {L} \mathcal{G}) = 0$ if $i \not\in [a, b]$ for all sheaves of $\Lambda $-modules $\mathcal{G}$. We will prove the final assertion holds by induction on $b - a$. If $a = b$, then $K = H^ a(K)[-a]$ is a flat constructible sheaf and the result holds. Next, assume $b > a$. 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 Lemma 59.73.6 we can find finitely many affine schemes $U_ i$ étale over $X$ and a surjection $\bigoplus j_{U_ i!}\underline{\Lambda }_{U_ i} \to H^ b(K)$. After replacing $U_ i$ by standard étale coverings $\{ U_{ij} \to U_ i\} $ we may assume this surjection lifts to a map $\mathcal{F} = \bigoplus j_{U_ i!}\underline{\Lambda }_{U_ i} \to \mathop{\mathrm{Ker}}(\mathcal{K}^ b \to \mathcal{K}^{b + 1})$. This map determines a distinguished triangle

\[ \mathcal{F}[-b] \to K \to L \to \mathcal{F}[-b + 1] \]

in $D(X_{\acute{e}tale}, \Lambda )$. Since $D_{ctf}(X_{\acute{e}tale}, \Lambda )$ is a triangulated subcategory we see that $L$ is in it too. In fact $L$ has tor amplitude in $[a, b - 1]$ as $\mathcal{F}$ surjects onto $H^ b(K)$ (details omitted). By induction hypothesis we can find a finite complex $\mathcal{F}^ a \to \ldots \to \mathcal{F}^{b - 1}$ of flat constructible sheaves of $\Lambda $-modules representing $L$. The map $L \to \mathcal{F}[-b + 1]$ corresponds to a map $\mathcal{F}^ b \to \mathcal{F}$ annihilating the image of $\mathcal{F}^{b - 1} \to \mathcal{F}^ b$. Then it follows from axiom TR3 that $K$ is represented by the complex

\[ \mathcal{F}^ a \to \ldots \to \mathcal{F}^{b - 1} \to \mathcal{F}^ b \]

which finishes the proof. $\square$

Comments (2)

Comment #7811 by R on

The hypothesis is that has locally finite tor dimension, but you seem to use (and prove) that has finite tor dimension. Are they equivalent in this case? (If I understand correctly, Tag 21.46.5 shows that finite tor dimension implies locally finite tor dimension, though this is never spelled out.)

Comment #7812 by on

For quasi-compact spaces, being locally finite tor dimension is the same as having finite tor dimension. Maybe this should be explained in Section 21.46 as well.

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