2
$\begingroup$

This is a sort of follow up to this MO question.

Let $R$ be a ring (eventually with good properties) and $\mathrm{Ch}(R)$ be the category of chain complexes of $R$-modules (eventually bounded).

Following the hint by F. Muro given in his answer, I would like to know if the following two classes $$ \begin{align*} \mathcal E &= \left\{ \left. f:\begin{array}{c} X\\ \downarrow \\ Y\end{array}\right| \begin{array}{ccc} Z_0 X &\to & Z_0Y \\ \downarrow && \downarrow \\ X_0 & \underset{f_0}\to & Y_0 \end{array} \text{ is a pushout, } f_n \text{ iso }\forall (n<0) \right\}\\ \mathcal M &= \left\{ \left. g:\begin{array}{c} A\\ \downarrow \\ B\end{array}\right| \begin{array}{ccc} Z_0 A &\to & Z_0B \\ \downarrow && \downarrow \\ A_0 & = & A_0 \end{array} \text{ is a pushout, } g_n \text{ iso }\forall (n>0) \right\} \end{align*} $$ form a factorization system on $\mathrm{Ch}(R)$, where $Z_0(-)\to (-)_0$ is the inclusion of the 0-cycles.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.