21.46 Tor dimension
In this section we take a closer look at resolutions by flat modules.
Definition 21.46.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$ with $a \leq b$.
We say $E$ has tor-amplitude in $[a, b]$ if $H^ i(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{F}) = 0$ for all $\mathcal{O}$-modules $\mathcal{F}$ and all $i \not\in [a, b]$.
We say $E$ has finite tor dimension if it has tor-amplitude in $[a, b]$ for some $a, b$.
We say $E$ locally has finite tor dimension if for any object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ such that $E|_{U_ i}$ has finite tor dimension for all $i$.
An $\mathcal{O}$-module $\mathcal{F}$ has tor dimension $\leq d$ if $\mathcal{F}[0]$ viewed as an object of $D(\mathcal{O})$ has tor-amplitude in $[-d, 0]$.
Note that if $E$ as in the definition has finite tor dimension, then $E$ is an object of $D^ b(\mathcal{O})$ as can be seen by taking $\mathcal{F} = \mathcal{O}$ in the definition above.
Lemma 21.46.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet $ be a bounded above complex of flat $\mathcal{O}$-modules with tor-amplitude in $[a, b]$. Then $\mathop{\mathrm{Coker}}(d_{\mathcal{E}^\bullet }^{a - 1})$ is a flat $\mathcal{O}$-module.
Proof.
As $\mathcal{E}^\bullet $ is a bounded above complex of flat modules we see that $\mathcal{E}^\bullet \otimes _\mathcal {O} \mathcal{F} = \mathcal{E}^\bullet \otimes _\mathcal {O}^{\mathbf{L}} \mathcal{F}$ for any $\mathcal{O}$-module $\mathcal{F}$. Hence for every $\mathcal{O}$-module $\mathcal{F}$ the sequence
\[ \mathcal{E}^{a - 2} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^{a - 1} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^ a \otimes _\mathcal {O} \mathcal{F} \]
is exact in the middle. Since $\mathcal{E}^{a - 2} \to \mathcal{E}^{a - 1} \to \mathcal{E}^ a \to \mathop{\mathrm{Coker}}(d^{a - 1}) \to 0$ is a flat resolution this implies that $\text{Tor}_1^\mathcal {O}(\mathop{\mathrm{Coker}}(d^{a - 1}), \mathcal{F}) = 0$ for all $\mathcal{O}$-modules $\mathcal{F}$. This means that $\mathop{\mathrm{Coker}}(d^{a - 1})$ is flat, see Lemma 21.17.15.
$\square$
Lemma 21.46.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$ with $a \leq b$. The following are equivalent
$E$ has tor-amplitude in $[a, b]$.
$E$ is represented by a complex $\mathcal{E}^\bullet $ of flat $\mathcal{O}$-modules with $\mathcal{E}^ i = 0$ for $i \not\in [a, b]$.
Proof.
If (2) holds, then we may compute $E \otimes _\mathcal {O}^\mathbf {L} \mathcal{F} = \mathcal{E}^\bullet \otimes _\mathcal {O} \mathcal{F}$ and it is clear that (1) holds.
Assume that (1) holds. We may represent $E$ by a bounded above complex of flat $\mathcal{O}$-modules $\mathcal{K}^\bullet $, see Section 21.17. Let $n$ be the largest integer such that $\mathcal{K}^ n \not= 0$. If $n > b$, then $\mathcal{K}^{n - 1} \to \mathcal{K}^ n$ is surjective as $H^ n(\mathcal{K}^\bullet ) = 0$. As $\mathcal{K}^ n$ is flat we see that $\mathop{\mathrm{Ker}}(\mathcal{K}^{n - 1} \to \mathcal{K}^ n)$ is flat (Modules on Sites, Lemma 18.28.10). Hence we may replace $\mathcal{K}^\bullet $ by $\tau _{\leq n - 1}\mathcal{K}^\bullet $. Thus, by induction on $n$, we reduce to the case that $K^\bullet $ is a complex of flat $\mathcal{O}$-modules with $\mathcal{K}^ i = 0$ for $i > b$.
Set $\mathcal{E}^\bullet = \tau _{\geq a}\mathcal{K}^\bullet $. Everything is clear except that $\mathcal{E}^ a$ is flat which follows immediately from Lemma 21.46.2 and the definitions.
$\square$
Lemma 21.46.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a \in \mathbf{Z}$. The following are equivalent
$E$ has tor-amplitude in $[a, \infty ]$.
$E$ can be represented by a K-flat complex $\mathcal{E}^\bullet $ of flat $\mathcal{O}$-modules with $\mathcal{E}^ i = 0$ for $i \not\in [a, \infty ]$.
Moreover, we can choose $\mathcal{E}^\bullet $ such that any pullback by a morphism of ringed sites is a K-flat complex with flat terms.
Proof.
The implication (2) $\Rightarrow $ (1) is immediate. Assume (1) holds. First we choose a K-flat complex $\mathcal{K}^\bullet $ with flat terms representing $E$, see Lemma 21.17.11. For any $\mathcal{O}$-module $\mathcal{M}$ the cohomology of
\[ \mathcal{K}^{n - 1} \otimes _\mathcal {O} \mathcal{M} \to \mathcal{K}^ n \otimes _\mathcal {O} \mathcal{M} \to \mathcal{K}^{n + 1} \otimes _\mathcal {O} \mathcal{M} \]
computes $H^ n(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{M})$. This is always zero for $n < a$. Hence if we apply Lemma 21.46.2 to the complex $\ldots \to \mathcal{K}^{a - 1} \to \mathcal{K}^ a \to \mathcal{K}^{a + 1}$ we conclude that $\mathcal{N} = \mathop{\mathrm{Coker}}(\mathcal{K}^{a - 1} \to \mathcal{K}^ a)$ is a flat $\mathcal{O}$-module. We set
\[ \mathcal{E}^\bullet = \tau _{\geq a}\mathcal{K}^\bullet = (\ldots \to 0 \to \mathcal{N} \to \mathcal{K}^{a + 1} \to \ldots ) \]
The kernel $\mathcal{L}^\bullet $ of $\mathcal{K}^\bullet \to \mathcal{E}^\bullet $ is the complex
\[ \mathcal{L}^\bullet = (\ldots \to \mathcal{K}^{a - 1} \to \mathcal{I} \to 0 \to \ldots ) \]
where $\mathcal{I} \subset \mathcal{K}^ a$ is the image of $\mathcal{K}^{a - 1} \to \mathcal{K}^ a$. Since we have the short exact sequence $0 \to \mathcal{I} \to \mathcal{K}^ a \to \mathcal{N} \to 0$ we see that $\mathcal{I}$ is a flat $\mathcal{O}$-module. Thus $\mathcal{L}^\bullet $ is a bounded above complex of flat modules, hence K-flat by Lemma 21.17.8. It follows that $\mathcal{E}^\bullet $ is K-flat by Lemma 21.17.7.
Proof of the final assertion. Let $f : (\mathcal{C}', \mathcal{O}') \to (\mathcal{C}, \mathcal{O})$ be a morphism of ringed sites. By Lemma 21.18.1 the complex $f^*\mathcal{K}^\bullet $ is K-flat with flat terms. The complex $f^*\mathcal{L}^\bullet $ is K-flat as it is a bounded above complex of flat $\mathcal{O}'$-modules. We have a short exact sequence of complexes of $\mathcal{O}'$-modules
\[ 0 \to f^*\mathcal{L}^\bullet \to f^*\mathcal{K}^\bullet \to f^*\mathcal{E}^\bullet \to 0 \]
because the short exact sequence $0 \to \mathcal{I} \to \mathcal{K}^ a \to \mathcal{N} \to 0$ of flat modules pulls back to a short exact sequence. By Lemma 21.17.7. the complex $f^*\mathcal{E}^\bullet $ is K-flat and the proof is complete.
$\square$
Lemma 21.46.5. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\mathcal{O}_\mathcal {D})$. If $E$ has tor amplitude in $[a, b]$, then $Lf^*E$ has tor amplitude in $[a, b]$.
Proof.
Assume $E$ has tor amplitude in $[a, b]$. By Lemma 21.46.3 we can represent $E$ by a complex of $\mathcal{E}^\bullet $ of flat $\mathcal{O}$-modules with $\mathcal{E}^ i = 0$ for $i \not\in [a, b]$. Then $Lf^*E$ is represented by $f^*\mathcal{E}^\bullet $. By Modules on Sites, Lemma 18.39.1 the module $f^*\mathcal{E}^ i$ are flat. Thus by Lemma 21.46.3 we conclude that $Lf^*E$ has tor amplitude in $[a, b]$.
$\square$
Lemma 21.46.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$.
If $K$ has tor-amplitude in $[a + 1, b + 1]$ and $L$ has tor-amplitude in $[a, b]$ then $M$ has tor-amplitude in $[a, b]$.
If $K$ and $M$ have tor-amplitude in $[a, b]$, then $L$ has tor-amplitude in $[a, b]$.
If $L$ has tor-amplitude in $[a + 1, b + 1]$ and $M$ has tor-amplitude in $[a, b]$, then $K$ has tor-amplitude in $[a + 1, b + 1]$.
Proof.
Omitted. Hint: This just follows from the long exact cohomology sequence associated to a distinguished triangle and the fact that $- \otimes _\mathcal {O}^{\mathbf{L}} \mathcal{F}$ preserves distinguished triangles. The easiest one to prove is (2) and the others follow from it by translation.
$\square$
Lemma 21.46.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. If $K$ has tor-amplitude in $[a, b]$ and $L$ has tor-amplitude in $[c, d]$ then $K \otimes _\mathcal {O}^\mathbf {L} L$ has tor amplitude in $[a + c, b + d]$.
Proof.
Omitted. Hint: use the spectral sequence for tors.
$\square$
Lemma 21.46.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $a, b \in \mathbf{Z}$. For $K$, $L$ objects of $D(\mathcal{O})$ if $K \oplus L$ has tor amplitude in $[a, b]$ so do $K$ and $L$.
Proof.
Clear from the fact that the Tor functors are additive.
$\square$
Lemma 21.46.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. Let $K$ be an object of $D(\mathcal{O})$.
If $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}$ is bounded above, then $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$ is uniformly bounded above for all $n$.
If $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}$ as an object of $D(\mathcal{O}/\mathcal{I})$ has tor amplitude in $[a, b]$, then $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$ as an object of $D(\mathcal{O}/\mathcal{I}^ n)$ has tor amplitude in $[a, b]$ for all $n$.
Proof.
Proof of (1). Assume that $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}$ is bounded above, say $H^ i(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}) = 0$ for $i > b$. Note that we have distinguished triangles
\[ K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1} \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^{n + 1} \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1}[1] \]
and that
\[ K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1} = \left( K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}\right) \otimes _{\mathcal{O}/\mathcal{I}}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1} \]
By induction we conclude that $H^ i(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n) = 0$ for $i > b$ for all $n$.
Proof of (2). Assume $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}$ as an object of $D(\mathcal{O}/\mathcal{I})$ has tor amplitude in $[a, b]$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}/\mathcal{I}^ n$-modules. Then we have a finite filtration
\[ 0 \subset \mathcal{I}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{I}\mathcal{F} \subset \mathcal{F} \]
whose successive quotients are sheaves of $\mathcal{O}/\mathcal{I}$-modules. Thus to prove that $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$ has tor amplitude in $[a, b]$ it suffices to show $H^ i(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n \otimes _{\mathcal{O}/\mathcal{I}^ n}^\mathbf {L} \mathcal{G})$ is zero for $i \not\in [a, b]$ for all $\mathcal{O}/\mathcal{I}$-modules $\mathcal{G}$. Since
\[ \left(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n\right) \otimes _{\mathcal{O}/\mathcal{I}^ n}^\mathbf {L} \mathcal{G} = \left(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}\right) \otimes _{\mathcal{O}/\mathcal{I}}^\mathbf {L} \mathcal{G} \]
for every sheaf of $\mathcal{O}/\mathcal{I}$-modules $\mathcal{G}$ the result follows.
$\square$
Lemma 21.46.10. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$.
If $E$ has tor amplitude in $[a, b]$, then for every point $p$ of the site $\mathcal{C}$ the object $E_ p$ of $D(\mathcal{O}_ p)$ has tor amplitude in $[a, b]$.
If $\mathcal{C}$ has enough points, then the converse is true.
Proof.
Proof of (1). This follows because taking stalks at $p$ is the same as pulling back by the morphism of ringed sites $(p, \mathcal{O}_ p) \to (\mathcal{C}, \mathcal{O})$ and hence we can apply Lemma 21.46.5.
Proof of (2). If $\mathcal{C}$ has enough points, then we can check vanishing of $H^ i(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{F})$ at stalks, see Modules on Sites, Lemma 18.14.4. Since $H^ i(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{F})_ p = H^ i(E_ p \otimes _{\mathcal{O}_ p}^\mathbf {L} \mathcal{F}_ p)$ we conclude.
$\square$
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