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