20.48 Tor dimension
In this section we take a closer look at resolutions by flat modules.
Definition 20.48.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $a, b \in \mathbf{Z}$ with $a \leq b$.
We say $E$ has tor-amplitude in $[a, b]$ if $H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F}) = 0$ for all $\mathcal{O}_ X$-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 there exists an open covering $X = \bigcup U_ i$ such that $E|_{U_ i}$ has finite tor dimension for all $i$.
An $\mathcal{O}_ X$-module $\mathcal{F}$ has tor dimension $\leq d$ if $\mathcal{F}[0]$ viewed as an object of $D(\mathcal{O}_ X)$ has tor-amplitude in $[-d, 0]$.
Note that if $E$ as in the definition has finite tor dimension, then $E$ is an object of $D^ b(\mathcal{O}_ X)$ as can be seen by taking $\mathcal{F} = \mathcal{O}_ X$ in the definition above.
Lemma 20.48.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $ be a bounded above complex of flat $\mathcal{O}_ X$-modules with tor-amplitude in $[a, b]$. Then $\mathop{\mathrm{Coker}}(d_{\mathcal{E}^\bullet }^{a - 1})$ is a flat $\mathcal{O}_ X$-module.
Proof.
As $\mathcal{E}^\bullet $ is a bounded above complex of flat modules we see that $\mathcal{E}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{F} = \mathcal{E}^\bullet \otimes _{\mathcal{O}_ X}^{\mathbf{L}} \mathcal{F}$ for any $\mathcal{O}_ X$-module $\mathcal{F}$. Hence for every $\mathcal{O}_ X$-module $\mathcal{F}$ the sequence
\[ \mathcal{E}^{a - 2} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{E}^{a - 1} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{E}^ a \otimes _{\mathcal{O}_ X} \mathcal{F} \]
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 $\text{Tor}_1^{\mathcal{O}_ X}(\mathop{\mathrm{Coker}}(d^{a - 1}), \mathcal{F}) = 0$ for all $\mathcal{O}_ X$-modules $\mathcal{F}$. This means that $\mathop{\mathrm{Coker}}(d^{a - 1})$ is flat, see Lemma 20.26.16.
$\square$
Lemma 20.48.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_ X)$. 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 $\mathcal{O}_ X$-modules with $\mathcal{E}^ i = 0$ for $i \not\in [a, b]$.
Proof.
If (2) holds, then we may compute $E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F} = \mathcal{E}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{F}$ and it is clear that (1) holds.
Assume that (1) holds. We may represent $E$ by a bounded above complex of flat $\mathcal{O}_ X$-modules $\mathcal{K}^\bullet $, see Section 20.26. 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, Lemma 17.17.8). 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 $\mathcal{O}_ X$-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 20.48.2 and the definitions.
$\square$
Lemma 20.48.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $E$ be an object of $D(\mathcal{O}_ Y)$. 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 20.48.3 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, Lemma 17.20.2 the modules $f^*\mathcal{E}^ i$ are flat. Thus by Lemma 20.48.3 we conclude that $Lf^*E$ has tor amplitude in $[a, b]$.
$\square$
Lemma 20.48.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $a, b \in \mathbf{Z}$ with $a \leq b$. The following are equivalent
$E$ has tor-amplitude in $[a, b]$.
for every $x \in X$ the object $E_ x$ of $D(\mathcal{O}_{X, x})$ has tor-amplitude in $[a, b]$.
Proof.
Taking stalks at $x$ is the same thing as pulling back by the morphism of ringed spaces $(x, \mathcal{O}_{X, x}) \to (X, \mathcal{O}_ X)$. Hence the implication (1) $\Rightarrow $ (2) follows from Lemma 20.48.4. For the converse, note that taking stalks commutes with tensor products (Modules, Lemma 17.16.1). Hence
\[ (E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F})_ x = E_ x \otimes _{\mathcal{O}_{X, x}}^\mathbf {L} \mathcal{F}_ x \]
On the other hand, taking stalks is exact, so
\[ H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F})_ x = H^ i((E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F})_ x) = H^ i(E_ x \otimes _{\mathcal{O}_{X, x}}^\mathbf {L} \mathcal{F}_ x) \]
and we can check whether $H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{F})$ is zero by checking whether all of its stalks are zero (Modules, Lemma 17.3.1). Thus (2) implies (1).
$\square$
Lemma 20.48.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O}_ X)$. 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 $- \otimes _{\mathcal{O}_ X}^{\mathbf{L}} \mathcal{F}$ preserves distinguished triangles. The easiest one to prove is (2) and the others follow from it by translation.
$\square$
Lemma 20.48.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_ X)$. If $K$ has tor-amplitude in $[a, b]$ and $L$ has tor-amplitude in $[c, d]$ then $K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ has tor amplitude in $[a + c, b + d]$.
Proof.
Omitted. Hint: use the spectral sequence for tors.
$\square$
Lemma 20.48.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $a, b \in \mathbf{Z}$. For $K$, $L$ objects of $D(\mathcal{O}_ X)$ 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.
$\square$
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