The Stacks project

22.36 Characterizing compact objects

Compact objects of additive categories are defined in Derived Categories, Definition 13.37.1. In this section we characterize compact objects of the derived category of a differential graded algebra.

Remark 22.36.1. Let $(A, \text{d})$ be a differential graded algebra. Is there a characterization of those differential graded $A$-modules $P$ for which we have

\[ \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M) = \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(P, M) \]

for all differential graded $A$-modules $M$? Let $\mathcal{D} \subset K(A, \text{d})$ be the full subcategory whose objects are the objects $P$ satisfying the above. Then $\mathcal{D}$ is a strictly full saturated triangulated subcategory of $K(A, \text{d})$. If $P$ is projective as a graded $A$-module, then to see where $P$ is an object of $\mathcal{D}$ it is enough to check that $\mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M) = 0$ whenever $M$ is acyclic. However, in general it is not enough to assume that $P$ is projective as a graded $A$-module. Example: take $A = R = k[\epsilon ]$ where $k$ is a field and $k[\epsilon ] = k[x]/(x^2)$ is the ring of dual numbers. Let $P$ be the object with $P^ n = R$ for all $n \in \mathbf{Z}$ and differential given by multiplication by $\epsilon $. Then $\text{id}_ P \in \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, P)$ is a nonzero element but $P$ is acyclic.

Remark 22.36.2. Let $(A, \text{d})$ be a differential graded algebra. Let us say a differential graded $A$-module $M$ is finite if $M$ is generated, as a right $A$-module, by finitely many elements. If $P$ is a differential graded $A$-module which is finite graded projective, then we can ask: Does $P$ give a compact object of $D(A, \text{d})$? Presumably, this is not true in general, but we do not know a counter example. However, if $P$ is also an object of the category $\mathcal{D}$ of Remark 22.36.1, then this is the case (this follows from the fact that direct sums in $D(A, \text{d})$ are given by direct sums of modules; details omitted).

Lemma 22.36.3. Let $(A, \text{d})$ be a differential graded algebra. Let $E$ be a compact object of $D(A, \text{d})$. Let $P$ be a differential graded $A$-module which has a finite filtration

\[ 0 = F_{-1}P \subset F_0P \subset F_1P \subset \ldots \subset F_ nP = P \]

by differential graded submodules such that

\[ F_{i + 1}P/F_ iP \cong \bigoplus \nolimits _{j \in J_ i} A[k_{i, j}] \]

as differential graded $A$-modules for some sets $J_ i$ and integers $k_{i, j}$. Let $E \to P$ be a morphism of $D(A, \text{d})$. Then there exists a differential graded submodule $P' \subset P$ such that $F_{i + 1}P \cap P'/(F_ iP \cap P')$ is equal to $\bigoplus _{j \in J'_ i} A[k_{i, j}]$ for some finite subsets $J'_ i \subset J_ i$ and such that $E \to P$ factors through $P'$.

Proof. We will prove by induction on $-1 \leq m \leq n$ that there exists a differential graded submodule $P' \subset P$ such that

  1. $F_ mP \subset P'$,

  2. for $i \geq m$ the quotient $F_{i + 1}P \cap P'/(F_ iP \cap P')$ is isomorphic to $\bigoplus _{j \in J'_ i} A[k_{i, j}]$ for some finite subsets $J'_ i \subset J_ i$, and

  3. $E \to P$ factors through $P'$.

The base case is $m = n$ where we can take $P' = P$.

Induction step. Assume $P'$ works for $m$. For $i \geq m$ and $j \in J'_ i$ let $x_{i, j} \in F_{i + 1}P \cap P'$ be a homogeneous element of degree $k_{i, j}$ whose image in $F_{i + 1}P \cap P'/(F_ iP \cap P')$ is the generator in the summand corresponding to $j \in J_ i$. The $x_{i, j}$ generate $P'/F_ mP$ as an $A$-module. Write

\[ \text{d}(x_{i, j}) = \sum x_{i', j'} a_{i, j}^{i', j'} + y_{i, j} \]

with $y_{i, j} \in F_ mP$ and $a_{i, j}^{i', j'} \in A$. There exists a finite subset $J'_{m - 1} \subset J_{m - 1}$ such that each $y_{i, j}$ maps to an element of the submodule $\bigoplus _{j \in J'_{m - 1}} A[k_{m - 1, j}]$ of $F_ mP/F_{m - 1}P$. Let $P'' \subset F_ mP$ be the inverse image of $\bigoplus _{j \in J'_{m - 1}} A[k_{m - 1, j}]$ under the map $F_ mP \to F_ mP/F_{m - 1}P$. Then we see that the $A$-submodule

\[ P'' + \sum x_{i, j}A \]

is a differential graded submodule of the type we are looking for. Moreover

\[ P'/(P'' + \sum x_{i, j}A) = \bigoplus \nolimits _{j \in J_{m - 1} \setminus J'_{m - 1}} A[k_{m - 1, j}] \]

Since $E$ is compact, the composition of the given map $E \to P'$ with the quotient map, factors through a finite direct subsum of the module displayed above. Hence after enlarging $J'_{m - 1}$ we may assume $E \to P'$ factors through $P'' + \sum x_{i, j}A$ as desired. $\square$

It is not true that every compact object of $D(A, \text{d})$ comes from a finite graded projective differential graded $A$-module, see Examples, Section 110.69.

Proposition 22.36.4. Let $(A, \text{d})$ be a differential graded algebra. Let $E$ be an object of $D(A, \text{d})$. Then the following are equivalent

  1. $E$ is a compact object,

  2. $E$ is a direct summand of an object of $D(A, \text{d})$ which is represented by a differential graded module $P$ which has a finite filtration $F_\bullet $ by differential graded submodules such that $F_ iP/F_{i - 1}P$ are finite direct sums of shifts of $A$.

Proof. Assume $E$ is compact. By Lemma 22.20.4 we may assume that $E$ is represented by a differential graded $A$-module $P$ with property (P). Consider the distinguished triangle

\[ \bigoplus F_ iP \to \bigoplus F_ iP \to P \xrightarrow {\delta } \bigoplus F_ iP[1] \]

coming from the admissible short exact sequence of Lemma 22.20.1. Since $E$ is compact we have $\delta = \sum _{i = 1, \ldots , n} \delta _ i$ for some $\delta _ i : P \to F_ iP[1]$. Since the composition of $\delta $ with the map $\bigoplus F_ iP[1] \to \bigoplus F_ iP[1]$ is zero (Derived Categories, Lemma 13.4.1) it follows that $\delta = 0$ (follows as $\bigoplus F_ iP \to \bigoplus F_ iP$ maps the summand $F_ iP$ via the difference of $\text{id}$ and the inclusion map into $F_{i - 1}P$). Thus we see that the identity on $E$ factors through $\bigoplus F_ iP$ in $D(A, \text{d})$ (by Derived Categories, Lemma 13.4.11). Next, we use that $P$ is compact again to see that the map $E \to \bigoplus F_ iP$ factors through $\bigoplus _{i = 1, \ldots , n} F_ iP$ for some $n$. In other words, the identity on $E$ factors through $\bigoplus _{i = 1, \ldots , n} F_ iP$. By Lemma 22.36.3 we see that the identity of $E$ factors as $E \to P \to E$ where $P$ is as in part (2) of the statement of the lemma. In other words, we have proven that (1) implies (2).

Assume (2). By Derived Categories, Lemma 13.37.2 it suffices to show that $P$ gives a compact object. Observe that $P$ has property (P), hence we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(P, M) = \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M) \]

for any differential graded module $M$ by Lemma 22.22.3. As direct sums in $D(A, \text{d})$ are given by direct sums of graded modules (Lemma 22.22.4) we reduce to showing that $\mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M)$ commutes with direct sums. Using that $K(A, \text{d})$ is a triangulated category, that $\mathop{\mathrm{Hom}}\nolimits $ is a cohomological functor in the first variable, and the filtration on $P$, we reduce to the case that $P$ is a finite direct sum of shifts of $A$. Thus we reduce to the case $P = A[k]$ which is clear. $\square$

Lemma 22.36.5. Let $(A, \text{d})$ be a differential graded algebra. For every compact object $E$ of $D(A, \text{d})$ there exist integers $a \leq b$ such that $\mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(E, M) = 0$ if $H^ i(M) = 0$ for $i \in [a, b]$.

Proof. Observe that the collection of objects of $D(A, \text{d})$ for which such a pair of integers exists is a saturated, strictly full triangulated subcategory of $D(A, \text{d})$. Thus by Proposition 22.36.4 it suffices to prove this when $E$ is represented by a differential graded module $P$ which has a finite filtration $F_\bullet $ by differential graded submodules such that $F_ iP/F_{i - 1}P$ are finite direct sums of shifts of $A$. Using the compatibility with triangles, we see that it suffices to prove it for $P = A$. In this case $\mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(A, M) = H^0(M)$ and the result holds with $a = b = 0$. $\square$

If $(A, \text{d})$ is just an algebra placed in degree $0$ with zero differential or more generally lives in only a finite number of degrees, then we do obtain the more precise description of compact objects.

Lemma 22.36.6. Let $(A, \text{d})$ be a differential graded algebra. Assume that $A^ n = 0$ for $|n| \gg 0$. Let $E$ be an object of $D(A, \text{d})$. The following are equivalent

  1. $E$ is a compact object, and

  2. $E$ can be represented by a differential graded $A$-module $P$ which is finite projective as a graded $A$-module and satisfies $\mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M) = \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(P, M)$ for every differential graded $A$-module $M$.

Proof. Let $\mathcal{D} \subset K(A, \text{d})$ be the triangulated subcategory discussed in Remark 22.36.1. Let $P$ be an object of $\mathcal{D}$ which is finite projective as a graded $A$-module. Then $P$ represents a compact object of $D(A, \text{d})$ by Remark 22.36.2.

To prove the converse, let $E$ be a compact object of $D(A, \text{d})$. Fix $a \leq b$ as in Lemma 22.36.5. After decreasing $a$ and increasing $b$ if necessary, we may also assume that $H^ i(E) = 0$ for $i \not\in [a, b]$ (this follows from Proposition 22.36.4 and our assumption on $A$). Moreover, fix an integer $c > 0$ such that $A^ n = 0$ if $|n| \geq c$.

By Proposition 22.36.4 we see that $E$ is a direct summand, in $D(A, \text{d})$, of a differential graded $A$-module $P$ which has a finite filtration $F_\bullet $ by differential graded submodules such that $F_ iP/F_{i - 1}P$ are finite direct sums of shifts of $A$. In particular, $P$ has property (P) and we have $\mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(P, M) = \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M)$ for any differential graded module $M$ by Lemma 22.22.3. In other words, $P$ is an object of the triangulated subcategory $\mathcal{D} \subset K(A, \text{d})$ discussed in Remark 22.36.1. Note that $P$ is finite free as a graded $A$-module.

Choose $n > 0$ such that $b + 4c - n < a$. Represent the projector onto $E$ by an endomorphism $\varphi : P \to P$ of differential graded $A$-modules. Consider the distinguished triangle

\[ P \xrightarrow {1 - \varphi } P \to C \to P[1] \]

in $K(A, \text{d})$ where $C$ is the cone of the first arrow. Then $C$ is an object of $\mathcal{D}$, we have $C \cong E \oplus E[1]$ in $D(A, \text{d})$, and $C$ is a finite graded free $A$-module. Next, consider a distinguished triangle

\[ C[1] \to C \to C' \to C[2] \]

in $K(A, \text{d})$ where $C'$ is the cone on a morphism $C[1] \to C$ representing the composition

\[ C[1] \cong E[1] \oplus E[2] \to E[1] \to E \oplus E[1] \cong C \]

in $D(A, \text{d})$. Then we see that $C'$ represents $E \oplus E[2]$. Continuing in this manner we see that we can find a differential graded $A$-module $P$ which is an object of $\mathcal{D}$, is a finite free as a graded $A$-module, and represents $E \oplus E[n]$.

Choose a basis $x_ i$, $i \in I$ of homogeneous elements for $P$ as an $A$-module. Let $d_ i = \deg (x_ i)$. Let $P_1$ be the $A$-submodule of $P$ generated by $x_ i$ and $\text{d}(x_ i)$ for $d_ i \leq a - c - 1$. Let $P_2$ be the $A$-submodule of $P$ generated by $x_ i$ and $\text{d}(x_ i)$ for $d_ i \geq b - n + c$. We observe

  1. $P_1$ and $P_2$ are differential graded submodules of $P$,

  2. $P_1^ t = 0$ for $t \geq a$,

  3. $P_1^ t = P^ t$ for $t \leq a - 2c$,

  4. $P_2^ t = 0$ for $t \leq b - n$,

  5. $P_2^ t = P^ t$ for $t \geq b - n + 2c$.

As $b - n + 2c \geq a - 2c$ by our choice of $n$ we obtain a short exact sequence of differential graded $A$-modules

\[ 0 \to P_1 \cap P_2 \to P_1 \oplus P_2 \xrightarrow {\pi } P \to 0 \]

Since $P$ is projective as a graded $A$-module this is an admissible short exact sequence (Lemma 22.16.1). Hence we obtain a boundary map $\delta : P \to (P_1 \cap P_2)[1]$ in $K(A, \text{d})$, see Lemma 22.7.2. Since $P = E \oplus E[n]$ and since $P_1 \cap P_2$ lives in degrees $(b - n, a)$ we find that $\mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(E \oplus E[n], (P_1 \cap P_2)[1])$ is zero. Therefore $\delta = 0$ as a morphism in $K(A, \text{d})$ as $P$ is an object of $\mathcal{D}$. By Derived Categories, Lemma 13.4.11 we can find a map $s : P \to P_1 \oplus P_2$ such that $\pi \circ s = \text{id}_ P + \text{d}h + h\text{d}$ for some $h : P \to P$ of degree $-1$. Since $P_1 \oplus P_2 \to P$ is surjective and since $P$ is projective as a graded $A$-module we can choose a homogeneous lift $\tilde h : P \to P_1 \oplus P_2$ of $h$. Then we change $s$ into $s + \text{d} \tilde h + \tilde h \text{d}$ to get $\pi \circ s = \text{id}_ P$. This means we obtain a direct sum decomposition $P = s^{-1}(P_1) \oplus s^{-1}(P_2)$. Since $s^{-1}(P_2)$ is equal to $P$ in degrees $\geq b - n + 2c$ we see that $s^{-1}(P_2) \to P \to E$ is a quasi-isomorphism, i.e., an isomorphism in $D(A, \text{d})$. This finishes the proof. $\square$


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