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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

It might be nice here to discuss the universal property of the cone, as in the usual case of complexes in an additive category (Verdier's thesis, Chapter I, 3.1.3).

Comment #698 by on

Somebody borrowed my copy of Verdier's thesis. Please return it whoever you are!

Anyway, I looked it up on the web and yes we should add this some time. In fact this should be added to Section 13.9. To anybody who reads this: feel free to write it up (should only be 1 or 2 pages I think) and submit it. Thanks!

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