Definition 59.77.1. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. We denote *$D_{ctf}(X_{\acute{e}tale}, \Lambda )$* the full subcategory of $D_ c(X_{\acute{e}tale}, \Lambda )$ consisting of objects having locally finite tor dimension.

## 59.77 Tor finite with constructible cohomology

Let $X$ be a scheme and let $\Lambda $ be a Noetherian ring. An often used subcategory of the derived category $D_ c(X_{\acute{e}tale}, \Lambda )$ defined in Section 59.76 is the full subcategory consisting of objects having (locally) finite tor dimension. Here is the formal definition.

This is a strictly full, saturated triangulated subcategory of $D_ c(X_{\acute{e}tale}, \Lambda )$ and $D(X_{\acute{e}tale}, \Lambda )$. By our conventions, see Cohomology on Sites, Definition 21.44.1, we see that

if $X$ is quasi-compact. A good way to think about objects of $D_{ctf}(X_{\acute{e}tale}, \Lambda )$ is given in Lemma 59.77.3.

Remark 59.77.2. Objects in the derived category $D_{ctf}(X_{\acute{e}tale}, \Lambda )$ in some sense have better global properties than the perfect objects in $D(\mathcal{O}_ X)$. Namely, it can happen that a complex of $\mathcal{O}_ X$-modules is locally quasi-isomorphic to a finite complex of finite locally free $\mathcal{O}_ X$-modules, without being globally quasi-isomorphic to a bounded complex of locally free $\mathcal{O}_ X$-modules. The following lemma shows this does not happen for $D_{ctf}$ on a Noetherian scheme.

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

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

$K$ 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

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

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

which finishes the proof. $\square$

Remark 59.77.4. Let $\Lambda $ be a Noetherian ring. Let $X$ be a scheme. For a bounded complex $K^\bullet $ of constructible flat $\Lambda $-modules on $X_{\acute{e}tale}$ each stalk $K^ p_{\overline{x}}$ is a finite projective $\Lambda $-module. Hence the stalks of the complex are perfect complexes of $\Lambda $-modules.

Lemma 59.77.5. Let $\Lambda $ be a Noetherian ring. If $j : U \to X$ is an étale morphism of schemes, then

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

$j_!M \in D_{ctf}(X_{\acute{e}tale}, \Lambda )$ if $M \in D_{ctf}(U_{\acute{e}tale}, \Lambda )$ and the morphism $j$ is quasi-compact and quasi-separated.

**Proof.**
Perhaps the easiest way to prove this lemma is to reduce to the case where $X$ is affine and then apply Lemma 59.77.3 to translate it into a statement about finite complexes of flat constructible sheaves of $\Lambda $-modules where the result follows from Lemma 59.73.1.
$\square$

Lemma 59.77.6. Let $\Lambda $ be a Noetherian ring. Let $f : X \to Y$ be a morphism of schemes. If $K \in D_{ctf}(Y_{\acute{e}tale}, \Lambda )$ then $Lf^*K \in D_{ctf}(X_{\acute{e}tale}, \Lambda )$.

**Proof.**
Apply Lemma 59.77.3 to reduce this to a question about finite complexes of flat constructible sheaves of $\Lambda $-modules. Then the statement follows as $f^{-1} = f^*$ is exact and Lemma 59.71.5.
$\square$

Lemma 59.77.7. Let $X$ be a connected scheme. Let $\Lambda $ be a Noetherian ring. Let $K \in D_{ctf}(X_{\acute{e}tale}, \Lambda )$ have locally constant cohomology sheaves. Then there exists a finite complex of finite projective $\Lambda $-modules $M^\bullet $ and an étale covering $\{ U_ i \to X\} $ such that $K|_{U_ i} \cong \underline{M^\bullet }|_{U_ i}$ in $D(U_{i, {\acute{e}tale}}, \Lambda )$.

**Proof.**
Choose an étale covering $\{ U_ i \to X\} $ such that $K|_{U_ i}$ is constant, say $K|_{U_ i} \cong \underline{M_ i^\bullet }_{U_ i}$ for some finite complex of finite $\Lambda $-modules $M_ i^\bullet $. See Cohomology on Sites, Lemma 21.51.1. Observe that $U_ i \times _ X U_ j$ is empty if $M_ i^\bullet $ is not isomorphic to $M_ j^\bullet $ in $D(\Lambda )$. For each complex of $\Lambda $-modules $M^\bullet $ let $I_{M^\bullet } = \{ i \in I \mid M_ i^\bullet \cong M^\bullet \text{ in }D(\Lambda )\} $. As étale morphisms are open we see that $U_{M^\bullet } = \bigcup _{i \in I_{M^\bullet }} \mathop{\mathrm{Im}}(U_ i \to X)$ is an open subset of $X$. Then $X = \coprod U_{M^\bullet }$ is a disjoint open covering of $X$. As $X$ is connected only one $U_{M^\bullet }$ is nonempty. As $K$ is in $D_{ctf}(X_{\acute{e}tale}, \Lambda )$ we see that $M^\bullet $ is a perfect complex of $\Lambda $-modules, see More on Algebra, Lemma 15.74.2. Hence we may assume $M^\bullet $ is a finite complex of finite projective $\Lambda $-modules.
$\square$

## 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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)