The Stacks project

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.

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.

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 21.46.1, we see that

\[ D_{ctf}(X_{\acute{e}tale}, \Lambda ) \subset D^ b_ c(X_{\acute{e}tale}, \Lambda ) \subset D^ b(X_{\acute{e}tale}, \Lambda ) \]

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

  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. $\square$

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

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

  2. $j_!M \in D_{ctf}(X_{\acute{e}tale}, \Lambda )$ 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 21.53.1. 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. $\square$

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