22.7 Admissible short exact sequences
An admissible short exact sequence is the analogue of termwise split exact sequences in the setting of differential graded modules.
Definition 22.7.1. Let $(A, \text{d})$ be a differential graded algebra.
A homomorphism $K \to L$ of differential graded $A$-modules is an admissible monomorphism if there exists a graded $A$-module map $L \to K$ which is left inverse to $K \to L$.
A homomorphism $L \to M$ of differential graded $A$-modules is an admissible epimorphism if there exists a graded $A$-module map $M \to L$ which is right inverse to $L \to M$.
A short exact sequence $0 \to K \to L \to M \to 0$ of differential graded $A$-modules is an admissible short exact sequence if it is split as a sequence of graded $A$-modules.
Thus the splittings are compatible with all the data except for the differentials. Given an admissible short exact sequence we obtain a triangle; this is the reason that we require our splittings to be compatible with the $A$-module structure.
Lemma 22.7.2. Let $(A, \text{d})$ be a differential graded algebra. Let $0 \to K \to L \to M \to 0$ be an admissible short exact sequence of differential graded $A$-modules. Let $s : M \to L$ and $\pi : L \to K$ be splittings such that $\mathop{\mathrm{Ker}}(\pi ) = \mathop{\mathrm{Im}}(s)$. Then we obtain a morphism
\[ \delta = \pi \circ \text{d}_ L \circ s : M \to K[1] \]
of $\text{Mod}_{(A, \text{d})}$ which induces the boundary maps in the long exact sequence of cohomology (22.4.2.1).
Proof.
The map $\pi \circ \text{d}_ L \circ s$ is compatible with the $A$-module structure and the gradings by construction. It is compatible with differentials by Homology, Lemmas 12.14.10. Let $R$ be the ring that $A$ is a differential graded algebra over. The equality of maps is a statement about $R$-modules. Hence this follows from Homology, Lemmas 12.14.10 and 12.14.11.
$\square$
Lemma 22.7.3. Let $(A, \text{d})$ be a differential graded algebra. Let
\[ \xymatrix{ K \ar[r]_ f \ar[d]_ a & L \ar[d]^ b \\ M \ar[r]^ g & N } \]
be a diagram of homomorphisms of differential graded $A$-modules commuting up to homotopy.
If $f$ is an admissible monomorphism, then $b$ is homotopic to a homomorphism which makes the diagram commute.
If $g$ is an admissible epimorphism, then $a$ is homotopic to a morphism which makes the diagram commute.
Proof.
Let $h : K \to N$ be a homotopy between $bf$ and $ga$, i.e., $bf - ga = \text{d}h + h\text{d}$. Suppose that $\pi : L \to K$ is a graded $A$-module map left inverse to $f$. Take $b' = b - \text{d}h\pi - h\pi \text{d}$. Suppose $s : N \to M$ is a graded $A$-module map right inverse to $g$. Take $a' = a + \text{d}sh + sh\text{d}$. Computations omitted.
$\square$
Lemma 22.7.4. Let $(A, \text{d})$ be a differential graded algebra. Let $\alpha : K \to L$ be a homomorphism of differential graded $A$-modules. There exists a factorization
\[ \xymatrix{ K \ar[r]^{\tilde\alpha } \ar@/_1pc/[rr]_\alpha & \tilde L \ar[r]^\pi & L } \]
in $\text{Mod}_{(A, \text{d})}$ such that
$\tilde\alpha $ is an admissible monomorphism (see Definition 22.7.1),
there is a morphism $s : L \to \tilde L$ such that $\pi \circ s = \text{id}_ L$ and such that $s \circ \pi $ is homotopic to $\text{id}_{\tilde L}$.
Proof.
The proof is identical to the proof of Derived Categories, Lemma 13.9.6. Namely, we set $\tilde L = L \oplus C(1_ K)$ and we use elementary properties of the cone construction.
$\square$
Lemma 22.7.5. Let $(A, \text{d})$ be a differential graded algebra. Let $L_1 \to L_2 \to \ldots \to L_ n$ be a sequence of composable homomorphisms of differential graded $A$-modules. There exists a commutative diagram
\[ \xymatrix{ L_1 \ar[r] & L_2 \ar[r] & \ldots \ar[r] & L_ n \\ M_1 \ar[r] \ar[u] & M_2 \ar[r] \ar[u] & \ldots \ar[r] & M_ n \ar[u] } \]
in $\text{Mod}_{(A, \text{d})}$ such that each $M_ i \to M_{i + 1}$ is an admissible monomorphism and each $M_ i \to L_ i$ is a homotopy equivalence.
Proof.
The case $n = 1$ is without content. Lemma 22.7.4 is the case $n = 2$. Suppose we have constructed the diagram except for $M_ n$. Apply Lemma 22.7.4 to the composition $M_{n - 1} \to L_{n - 1} \to L_ n$. The result is a factorization $M_{n - 1} \to M_ n \to L_ n$ as desired.
$\square$
Lemma 22.7.6. Let $(A, \text{d})$ be a differential graded algebra. Let $0 \to K_ i \to L_ i \to M_ i \to 0$, $i = 1, 2, 3$ be admissible short exact sequence of differential graded $A$-modules. Let $b : L_1 \to L_2$ and $b' : L_2 \to L_3$ be homomorphisms of differential graded modules such that
\[ \vcenter { \xymatrix{ K_1 \ar[d]_0 \ar[r] & L_1 \ar[r] \ar[d]_ b & M_1 \ar[d]_0 \\ K_2 \ar[r] & L_2 \ar[r] & M_2 } } \quad \text{and}\quad \vcenter { \xymatrix{ K_2 \ar[d]^0 \ar[r] & L_2 \ar[r] \ar[d]^{b'} & M_2 \ar[d]^0 \\ K_3 \ar[r] & L_3 \ar[r] & M_3 } } \]
commute up to homotopy. Then $b' \circ b$ is homotopic to $0$.
Proof.
By Lemma 22.7.3 we can replace $b$ and $b'$ by homotopic maps such that the right square of the left diagram commutes and the left square of the right diagram commutes. In other words, we have $\mathop{\mathrm{Im}}(b) \subset \mathop{\mathrm{Im}}(K_2 \to L_2)$ and $\mathop{\mathrm{Ker}}((b')^ n) \supset \mathop{\mathrm{Im}}(K_2 \to L_2)$. Then $b \circ b' = 0$ as a map of modules.
$\square$
Comments (0)