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, and our definition of $add(\mathcal{A})$ is different).
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
of shifts of $E$. It is clear that in the notation of Section 13.35 we have
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
where $A$ is an object of $\langle E \rangle _1$ and $B$ an object of $\langle E \rangle _{n - 1}$. In the notation of Section 13.35 we have
Each of the categories $\langle E \rangle _ n$ is a strictly full additive (by Lemma 13.35.3) subcategory of $\mathcal{D}$ preserved under shifts and under taking summands. But, $\langle E \rangle _ n$ is not necessarily closed under “taking cones” or “extensions”, hence not necessarily a triangulated subcategory. This will be true for the subcategory
as will be shown in the lemmas below.
Lemma 13.36.1. Let $\mathcal{T}$ be a triangulated category. Let $E$ be an object of $\mathcal{T}$. For $n \geq 1$ we have For $n, n' \geq 1$ we have $\langle E \rangle _{n + n'} = smd(\langle E \rangle _ n \star \langle E \rangle _{n'})$.
\[ \langle E \rangle _ n = smd(\langle E \rangle _1 \star \ldots \star \langle E \rangle _1) = smd({\langle E \rangle _1}^{\star n}) = \bigcup \nolimits _{m \geq 1} smd(add(E[-m, m])^{\star n}) \]
Proof. The left equality in the displayed formula follows from Lemmas 13.35.1 and 13.35.2 and induction. The middle equality is a matter of notation. Since $\langle E \rangle _1 = smd(add(E[-\infty , \infty ])])$ and since $E[-\infty , \infty ] = \bigcup _{m \geq 1} E[-m, m]$ we see from Remark 13.35.6 and Lemma 13.35.2 that we get the equality on the right. Then the final statement follows from the remark and the corresponding statement of Lemma 13.35.4. $\square$
Lemma 13.36.2. Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. The subcategory is a strictly full, saturated, triangulated subcategory of $\mathcal{D}$ and it is the smallest such subcategory of $\mathcal{D}$ containing the object $E$.
\[ \langle E \rangle = \bigcup \nolimits _ n \langle E \rangle _ n = \bigcup \nolimits _{n, m \geq 1} smd(add(E[-m, m])^{\star n}) \]
Proof. The equality on the right follows from Lemma 13.36.1. It is clear that $\langle E \rangle = \bigcup \langle E \rangle _ n$ contains $E$, is preserved under shifts, direct sums, direct summands. 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 $X \in \langle E \rangle _{a + b}$ by Lemma 13.36.1. Hence $\bigcup \langle E \rangle _ n$ is also preserved under extensions and it follows that it is a triangulated subcategory.
Finally, let $\mathcal{D}' \subset \mathcal{D}$ be a strictly full, saturated, triangulated subcategory of $\mathcal{D}$ containing $E$. Then $\mathcal{D}'[-\infty , \infty ] \subset \mathcal{D}'$, $add(\mathcal{D}) \subset \mathcal{D}'$, $smd(\mathcal{D}') \subset \mathcal{D}'$, and $\mathcal{D}' \star \mathcal{D}' \subset \mathcal{D}'$. In other words, all the operations we used to construct $\langle E \rangle $ out of $E$ preserve $\mathcal{D}'$. Hence $\langle E \rangle \subset \mathcal{D}'$ and this finishes the proof. $\square$
Definition 13.36.3. Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$.
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 $\langle E \rangle = \mathcal{D}$.
We say $E$ is a strong generator of $\mathcal{D}$ if $\langle E \rangle _ n = \mathcal{D}$ for some $n \geq 1$.
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 13.36.4. Let $\mathcal{D}$ be a triangulated category. Let $E, K$ be objects of $\mathcal{D}$. The following are equivalent
$\mathop{\mathrm{Hom}}\nolimits (E, K[i]) = 0$ for all $i \in \mathbf{Z}$,
$\mathop{\mathrm{Hom}}\nolimits (E', K) = 0$ 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 13.36.5. 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 13.36.4 $\mathop{\mathrm{Hom}}\nolimits (E', K) = 0$ for all $E'$ in $\langle E \rangle $. However, since $\mathcal{D} = \langle E \rangle $ we conclude that $\text{id}_ K = 0$, i.e., $K = 0$. $\square$
Lemma 13.36.6. Let $\mathcal{D}$ be a triangulated category which has a strong generator. Let $E$ be an object of $\mathcal{D}$. If $E$ is a classical generator of $\mathcal{D}$, then $E$ is a strong generator.
Proof. Let $E'$ be an object of $\mathcal{D}$ such that $\mathcal{D} = \langle E' \rangle _ n$. Since $\mathcal{D} = \langle E \rangle $ we see that $E' \in \langle E \rangle _ m$ for some $m \geq 1$ by Lemma 13.36.2. Then $\langle E' \rangle _1 \subset \langle E \rangle _ m$ hence
as desired. Here we used Lemma 13.36.1. $\square$
Remark 13.36.7. 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
if $K_ i \in D(A)$, $i = 1, \ldots , r$ with $T(K_ i)$ for $i = 1, \ldots , r$, then $T(\bigoplus K_ i)$,
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,
if $T(K \oplus L)$ then $T(K)$ and $T(L)$, and
$T(E[n])$ holds for all $n$.
Then $T$ holds for all objects of $\langle E \rangle $.
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)