22.6 Cones
We introduce cones for the category of differential graded modules.
Definition 22.6.1. Let $(A, \text{d})$ be a differential graded algebra. Let $f : K \to L$ be a homomorphism of differential graded $A$-modules. The cone of $f$ is the differential graded $A$-module $C(f)$ given by $C(f) = L \oplus K$ with grading $C(f)^ n = L^ n \oplus K^{n + 1}$ and differential
\[ d_{C(f)} = \left( \begin{matrix} \text{d}_ L
& f
\\ 0
& -\text{d}_ K
\end{matrix} \right) \]
It comes equipped with canonical morphisms of complexes $i : L \to C(f)$ and $p : C(f) \to K[1]$ induced by the obvious maps $L \to C(f)$ and $C(f) \to K$.
The formation of the cone triangle is functorial in the following sense.
Lemma 22.6.2. Let $(A, \text{d})$ be a differential graded algebra. Suppose that
\[ \xymatrix{ K_1 \ar[r]_{f_1} \ar[d]_ a & L_1 \ar[d]^ b \\ K_2 \ar[r]^{f_2} & L_2 } \]
is a diagram of homomorphisms of differential graded $A$-modules which is commutative up to homotopy. Then there exists a morphism $c : C(f_1) \to C(f_2)$ which gives rise to a morphism of triangles
\[ (a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) \to (K_1, L_1, C(f_1), f_2, i_2, p_2) \]
in $K(\text{Mod}_{(A, \text{d})})$.
Proof.
Let $h : K_1 \to L_2$ be a homotopy between $f_2 \circ a$ and $b \circ f_1$. Define $c$ by the matrix
\[ c = \left( \begin{matrix} b
& h
\\ 0
& a
\end{matrix} \right) : L_1 \oplus K_1 \to L_2 \oplus K_2 \]
A matrix computation show that $c$ is a morphism of differential graded modules. It is trivial that $c \circ i_1 = i_2 \circ b$, and it is trivial also to check that $p_2 \circ c = a \circ p_1$.
$\square$
Comments (2)
Comment #676 by Martin Olsson on
Comment #698 by Johan on