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*}

13.33 Generators of triangulated categories

In this section we briefly introduce a few of the different notions of a generator for a triangulated category. Our terminology is taken from [BvdB] (except that we use “saturated” for what they call “épaisse”, see Definition 13.6.1).

Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. Denote $\langle E \rangle _1$ the strictly full subcategory of $\mathcal{D}$ consisting of objects in $\mathcal{D}$ isomorphic to direct summands of finite direct sums

\[ \bigoplus \nolimits _{i = 1, \ldots , r} E[n_ i] \]

of shifts of $E$. For $n > 1$ let $\langle E \rangle _ n$ denote the full subcategory of $\mathcal{D}$ consisting of objects of $\mathcal{D}$ isomorphic to direct summands of objects $X$ which fit into a distinguished triangle

\[ A \to X \to B \to A[1] \]

where $A$ is an object of $\langle E \rangle _1$ and $B$ an object of $\langle E \rangle _{n - 1}$. Each of the categories $\langle E \rangle _ n$ is a strictly full additive subcategory of $\mathcal{D}$ preserved under shifts and under taking summands. But, $\langle E \rangle _ n$ is not necessarily closed under “taking cones”, hence not necessarily a triangulated subcategory.

Lemma 13.33.1. Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. The subcategory

\[ \langle E \rangle = \bigcup \nolimits _ n \langle E \rangle _ n \]

is a strictly full, saturated, triangulated subcategory of $\mathcal{D}$ and it is the smallest such subcategory of $\mathcal{D}$ containing the object $E$.

Proof. To prove this it suffices to show: if $A \in \langle E \rangle _ a$ and $B \in \langle E \rangle _ b$ and if $A \to X \to B \to A[1]$ is a distinguished triangle, then $X \in \langle E \rangle _{a + b}$. We omit the details. $\square$

Definition 13.33.2. Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$.

  1. We say $E$ is a classical generator of $\mathcal{D}$ if the smallest strictly full, saturated, triangulated subcategory of $\mathcal{D}$ containing $E$ is equal to $\mathcal{D}$, in other words, if $\langle E \rangle = \mathcal{D}$.

  2. We say $E$ is a strong generator of $\mathcal{D}$ if $\langle E \rangle _ n = \mathcal{D}$ for some $n \geq 1$.

  3. We say $E$ is a weak generator or a generator of $\mathcal{D}$ if for any nonzero object $K$ of $\mathcal{D}$ there exists an integer $n$ and a nonzero map $E \to K[n]$.

This definition can be generalized to the case of a family of objects.

Lemma 13.33.3. Let $\mathcal{D}$ be a triangulated category. Let $E, K$ be objects of $\mathcal{D}$. The following are equivalent

  1. $\mathop{\mathrm{Hom}}\nolimits (E, K[i]) = 0$ for all $i \in \mathbf{Z}$,

  2. $\mathop{\mathrm{Hom}}\nolimits (E', K) = 0$ for all $E' \in \langle E \rangle $.

Proof. The implication (2) $\Rightarrow $ (1) is immediate. Conversely, assume (1). Then $\mathop{\mathrm{Hom}}\nolimits (X, K) = 0$ for all $X$ in $\langle E \rangle _1$. Arguing by induction on $n$ and using Lemma 13.4.2 we see that $\mathop{\mathrm{Hom}}\nolimits (X, K) = 0$ for all $X$ in $\langle E \rangle _ n$. $\square$

Lemma 13.33.4. Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. If $E$ is a classical generator of $\mathcal{D}$, then $E$ is a generator.

Proof. Assume $E$ is a classical generator. Let $K$ be an object of $\mathcal{D}$ such that $\mathop{\mathrm{Hom}}\nolimits (E, K[i]) = 0$ for all $i \in \mathbf{Z}$. By Lemma 13.33.3 $\mathop{\mathrm{Hom}}\nolimits (E', K) = 0$ for all $E'$ in $\langle E \rangle $. However, since $\mathcal{D} = \langle E \rangle $ we conclude that $\text{id}_ K = 0$, i.e., $K = 0$. $\square$

Remark 13.33.5. Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. Let $T$ be a property of objects of $\mathcal{D}$. Suppose that

  1. if $K_ i \in D(A)$, $i = 1, \ldots , r$ with $T(K_ i)$ for $i = 1, \ldots , r$, then $T(\bigoplus K_ i)$,

  2. if $K \to L \to M \to K[1]$ is a distinguished triangle and $T$ holds for two, then $T$ holds for the third object,

  3. if $T(K \oplus L)$ then $T(K)$ and $T(L)$, and

  4. $T(E[n])$ holds for all $n$.

Then $T$ holds for all objects of $\langle E \rangle $.


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 09SI. Beware of the difference between the letter 'O' and the digit '0'.