Lemma 22.7.3. Let $(A, \text{d})$ be a differential graded algebra. Let
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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)