2021-10-05

Compact objects

An important part of establishing Morita equivalence between $R$ and $M_n(R)$ was that fact that the natural map $\bigoplus_{i \in I} \operatorname{Hom}(U, N_i) \to \operatorname{Hom}(U, \bigoplus_{i \in I} N_i)$ was an isomorphism.

This is equivalent to saying we can factor any $f: U \to \bigoplus_{i \in I} N_i$ as

where $|I_f| < \infty$.

Objects with the property are very useful. They go be different names depending on the context. We will be focused on triangulated categories.

Definition. Let $\mathcal T$ be a triangulated category. An object $X$ is called compact if the natural map $\bigoplus_{i \in I} \operatorname{Hom}(X,Y_i) \to \operatorname{Hom}(X, \bigoplus_{i \in I} Y_i)$ is isomorphism for any $Y_i, i \in I$.

The full subcategory of compact objects in $\mathcal T$ is denoted by $\mathcal T^c$.

Lemma. $\mathcal T^c$ is triangulated and closed under retracts.

Proof. (Expand to view)

We have $$ \operatorname{Hom}(X[n], \bigoplus Y_i) \cong \operatorname{Hom}(X, \bigoplus Y_i[n]) $$ so $X[n] \in \mathcal T^c$ for any $n \in \mathbb{Z}$. Next, take a map $X \to X^\prime$ of compact objects and complete it to a triangle $$ X \to X^\prime \to C \to X[1] $$ We get a commutative diagram

with exact rows and where two out of the three vertical maps are isomorphisms. Thus, the third is also. Finally, if $X^\prime$ is a retract of $X$, the map $$ \bigoplus_{i \in I} \operatorname{Hom}(X^\prime,Y_i) \to \operatorname{Hom}(X^\prime, \bigoplus_{i \in I} Y_i) $$ is a retract of an isomorphism. Hence it is an isomorphism. ■

Definition. We say $\mathcal T$ is compactly generated if it has small coproducts and there exists a set of compact objects $\mathcal C$ such that $\operatorname{Hom}(C,Y) = 0 \ \forall C \in \mathcal C$ implies $Y = 0$. The set $\mathcal C$ is called a set of compact generators.

Lemma. A map $X \to Y$ in $\mathcal T$ with $\operatorname{Hom}(C,X) \to \operatorname{Hom}(C,Y)$ an isomorphism for all $C$ in a set of compact generators is an isomorphism.

Proof. (Expand to view)

The map $X \to Y$ is an isomorphism if and only if $\operatorname{cone}(X \to Y) = 0$. Since we have a set of compact generators, we only need to check that $$ \operatorname{Hom}(C,\operatorname{cone}(X \to Y)) = 0 $$ for all $C$. But this is equivalent to the map $$ \operatorname{Hom}(C,X) \to \operatorname{Hom}(C,Y) $$ being an isomorphism for all $C$. ■

Example. $D(R)$ is compactly generated. Indeed, $\operatorname{Hom}_{D(R)}(R[n],Y) \cong H^{-n}Y$ If all of these are $0$, then $Y$ is acyclic and hence $0$ in $D(R)$.

Definition. Given a sequence of composable maps $X_0 \overset{f_0}{\to} X_1 \overset{f_1}{\to} X_2 \overset{f_2}{\to} \cdots$ the homotopy colimit is the cone over the morphism $\bigoplus X_i \to \bigoplus X_i$ induced by the maps $X_i \overset{1,-f_i}{\to} X_i \oplus X_{i+1}$

Lemma. If $C$ is compact, then the natural map $\operatorname{colim} \operatorname{Hom}(C,X_\bullet) \to \operatorname{Hom}(C,\operatorname{hocolim} X_\bullet)$ is an isomorphism.

Proof. (Expand to view)

We have a commutative diagram

where the left two vertical maps are isomorphisms and the rows are exact. Since the map $$ \bigoplus \operatorname{Hom}(C,X_i) \to \bigoplus \operatorname{Hom}(C,X_i) $$ is injective we see that we have a short exact sequence $$ 0 \to \operatorname{Hom}(C, \bigoplus X_i) \to \operatorname{Hom}(C, \bigoplus X_i) \to \operatorname{Hom}(C,\operatorname{hocolim} X_\bullet) \to 0 $$ also. Thus, the map $$ \operatorname{colim}\operatorname{Hom}(C,X_\bullet) \to \operatorname{Hom}(C,\operatorname{hocolim} X_\bullet) $$ is also an isomorphism. ■

Proposition. If $\mathcal C$ is a compact set of generators, then the smallest subcategory of $\mathcal T$ that

is closed under $\bigoplus$’s
is triangulated and
contains $\mathcal C$

is $\mathcal T$ itself.

Proof. (Expand to view)

Let $$ U_0 := \bigcup_{C \in \mathcal C} \operatorname{Hom}(C,X) $$ Set $$ X_0 := \bigoplus_{(C,f) \in U_0} C $$ This comes with a natural evaluation map $$ X_0 \to X. $$ We work not by induction having constructed $\nu_i : X_i \to X$. We let $$ U_i := \bigcup_{C \in \mathcal C} \lbrace f : C \to X_i \mid \nu_i f = 0 \rbrace $$ Let $$ K_i := \bigoplus_{(C,f) \in U_i} C $$ and set $X_{i+1} = \operatorname{cone}(K_i \to X_i)$. The composition $K_i \to X_i \to X$ is $0$ so there exists a $X_{i+1} \to X$

We let $X = \operatorname{hocolim} X_\bullet$. Using the above lemma, it suffices to check that $$ \operatorname{Hom}(C,\operatorname{hocolim}X_\bullet) \to \operatorname{Hom}(C,X) $$ is any isomorphism for any $C \in \mathcal C$. From the other lemma, this is the the same as $$ \operatorname{colim}\operatorname{Hom}(C,X_\bullet) \to \operatorname{Hom}(C,X) $$ being an isomorphism. Any element of the colimit is represented by some $f: C \to X_i$. If $\nu f : C \to X$ is $0$, then $f \in U_i$ be definition and, in fact, $f = 0$ in the colimit. Given any $g: C \to X$, then $g \in U_0$. So there is some $C \to X_0$ whose composition with $X_0 \to X$ is $g$. Thus, we have an isomorphism. ■

If we have a compact generating set $\mathcal C$, we can also use it characterize all compact objects in $\mathcal T$. The following is due to Neeman.

Proposition. Let $\mathcal C$ be a set of compact generators for $\mathcal T$. The smallest triangulated subcategory of $\mathcal T$

containing $\mathcal C$ and
closed under retracts

is $\mathcal T^c$.

Proof. (Expand to view)

Pick some $X \in \mathcal T^c$. From the proof the proposition, we know that $$ X \cong \operatorname{hocolim} X_\bullet $$ where each $X_i$ is obtained as a cone over a map $K_{i-1} \to X_{i-1}$ with $K_{i-1}$ a sum of elements of $\mathcal C$. We also know that $$ 1_X \in \operatorname{Hom}(X,X) \cong \operatorname{colim}\operatorname{Hom}(X,X_i) $$ Thus, there exists $X \to X_i$ such that

So $X$ is a retract of $X_i$. The composition $X \to X_i \to K_{i-1}[1]$ factors through a finite summand of $K_{i-1}$. Denote that by $K_{i-1}^\prime$ and let $X^\prime_{i-1}[1]$ be the cone over $X \to K_{i-1}^\prime[1]$. Now repeat the procedure to produce

From repeated applications of the octahedral axiom, the cone of the map $X^\prime_l \to X$ is sequence of cones over the cones of $X^\prime_{j-1} \to X^\prime_j$. Each of these cones is a finite sum of objects from $\mathcal C$ by construction. Thus the cone over $X_l \to X$ for any $l$ lies in the smallest triangulated subcategory containing the objects of $\mathcal C$. From the commutative square

we see that $X^\prime_{-1} \to X$ is $0$ as $X \to X_i \to X$ is the identity. Thus, $X$ and $X^\prime_{-1}$ are summands over the cone of $X_{-1}^\prime \to X$. ■

Definition. A complex $P$ in $D(R)$ is called perfect if it is quasi-isomorphic to a bounded complex of finitely generated projectives. The full subcategory of perfect objects is denoted by $\operatorname{perf} R$.

Proposition. We have $D(R)^c = \operatorname{perf} R$

Proof. (Expand to view)

It is easy to see that $$ \operatorname{perf} R \subseteq D(R)^c $$ since compact objects are closed under finite direct sums, summands, and cones. But $\operatorname{perf} R$ - contains $R[n]$ for any $n \in \mathbb{Z}$ (a set of compact generators) - is triangulated and - is closed under retracts. Thus, the previous proposition says $$ D(R)^c \subseteq \operatorname{perf} R $$ ■

Finally it useful to introduce some notation for generation.

Definition. Given a subcategory $\mathcal S$ of a triangulated category $\mathcal T$, we let $\langle \mathcal S \rangle$ denote the smallest triangulated subcategory of $\mathcal T$ containing $\mathcal S$.

We also let $\overline{\langle \mathcal S \rangle}$ denote the smallest triangulated, closed under retracts, containing $\mathcal S$.