The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

20.42 Tor dimension

In this section we take a closer look at resolutions by flat modules.

Definition 20.42.1. 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 $a \leq b$.

  1. 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 $i \not\in [a, b]$.

  2. We say $E$ has finite tor dimension if it has tor-amplitude in $[a, b]$ for some $a, b$.

  3. 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 20.42.2. 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 20.27.15. $\square$

Lemma 20.42.3. 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

  1. $E$ has tor-amplitude in $[a, b]$.

  2. $E$ 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 20.27. 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 17.16.8). 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 20.42.2 and the definitions. $\square$

Lemma 20.42.4. 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 20.42.3 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 17.18.2 the modules $f^*\mathcal{E}^ i$ are flat. Thus by Lemma 20.42.3 we conclude that $Lf^*E$ has tor amplitude in $[a, b]$. $\square$

Lemma 20.42.5. 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

  1. $E$ has tor-amplitude in $[a, b]$.

  2. 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 20.42.4. For the converse, note that taking stalks commutes with tensor products (Modules, Lemma 17.15.1). 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 20.42.6. 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 $a, b \in \mathbf{Z}$.

  1. If $K$ has tor-amplitude in $[a + 1, b + 1]$ and $L$ has tor-amplitude in $[a, b]$ then $M$ has tor-amplitude in $[a, b]$.

  2. If $K$ and $M$ have tor-amplitude in $[a, b]$, then $L$ has tor-amplitude in $[a, b]$.

  3. 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 20.42.7. 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 20.42.8. 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. $\square$


Comments (0)


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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08CF. Beware of the difference between the letter 'O' and the digit '0'.