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

Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). One has the natural endofunctors $\tau_{\leq n}$ and $\tau_{\geq n}$ associated with the standard t-structure on $\mathrm{Chains}(R)$.

What I'm wondering is the following. Is it true that for every morphism $f\colon A\to B$ of chain complexes there exist morphisms $h:A\to C$ and $g:C\to B$ with $\tau_{\leq -1}(h)\colon \tau_{\leq -1}(A)\to \tau_{\leq -1}(C)$ a weak equivalence (in the standard model structure on chain complexes), $\tau_{\geq 0}(g)\colon \tau_{\geq 0}(C)\to \tau_{\geq 0}(B)$ a weak equivalence, and with $f$ homotopy equivalent to $g\circ h$?

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

Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). One has the natural endofunctors $\tau_{\leq n}$ and $\tau_{\geq n}$ associated with the standard t-structure on $\mathrm{Chains}(R)$.

What I'm wondering is the following. Is it true that for every morphism $f\colon A\to B$ of chain complexes there exist morphisms $h:A\to C$ and $g:C\to B$ with $\tau_{\leq -1}(h)\colon \tau_{\leq -1}(A)\to \tau_{\leq -1}(C)$ a weak equivalence (in the standard model structure on chain complexes), $\tau_{\geq 0}(g)\colon \tau_{\geq 0}(C)\to \tau_{\geq 0}(B)$ a weak equivalence, and with $f$ homotopy equivalent to $g\circ h$?

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

Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). One has the natural endofunctors $\tau_{\leq n}$ and $\tau_{\geq n}$ associated with the standard t-structure on $\mathrm{Chains}(R)$.

What I'm wondering is the following. Is it true that for every morphism $f\colon A\to B$ of chain complexes there exist morphisms $h:A\to C$ and $g:C\to B$ with $\tau_{\leq -1}(h)\colon \tau_{\leq -1}(A)\to \tau_{\leq -1}(C)$ a weak equivalence (in the standard model structure on chain complexes), $\tau_{\geq 0}(g)\colon \tau_{\geq 0}(C)\to \tau_{\geq 0}(B)$ a weak equivalence, and with $f$ homotopy equivalent to $g\circ h$?

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edit approved May 31, 2014 at 15:26
user62675
edit approved May 31, 2014 at 15:26
user62675

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

Let $R$ be a ring (eventually with good properties) and $Chains(R)$$\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). One has the natural endofunctors $\tau_{\leq n}$ and $\tau_{\geq n}$ associated with the standard t-structure on $Chains(R)$$\mathrm{Chains}(R)$.

What I'm wondering is the following. Is it true that for every morphism $f\colon A\to B$ of chain complexes there exist morphisms $h:A\to C$ and $g:C\to B$ with $\tau_{\leq -1}(h)\colon \tau_{\leq -1}(A)\to \tau_{\leq -1}(C)$ a weak equivalence (in the standard model structure on chain complexes), $\tau_{\geq 0}(g)\colon \tau_{\geq 0}(C)\to \tau_{\geq 0}(B)$ a weak equivalence, and with $f$ homotopy equivalent to $g\circ h$?

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

Let $R$ be a ring (eventually with good properties) and $Chains(R)$ be the category of chain complexes of $R$-modules (eventually bounded). One has the natural endofunctors $\tau_{\leq n}$ and $\tau_{\geq n}$ associated with the standard t-structure on $Chains(R)$.

What I'm wondering is the following. Is it true that for every morphism $f\colon A\to B$ of chain complexes there exist morphisms $h:A\to C$ and $g:C\to B$ with $\tau_{\leq -1}(h)\colon \tau_{\leq -1}(A)\to \tau_{\leq -1}(C)$ a weak equivalence (in the standard model structure on chain complexes), $\tau_{\geq 0}(g)\colon \tau_{\geq 0}(C)\to \tau_{\geq 0}(B)$ a weak equivalence, and with $f$ homotopy equivalent to $g\circ h$?

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

Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). One has the natural endofunctors $\tau_{\leq n}$ and $\tau_{\geq n}$ associated with the standard t-structure on $\mathrm{Chains}(R)$.

What I'm wondering is the following. Is it true that for every morphism $f\colon A\to B$ of chain complexes there exist morphisms $h:A\to C$ and $g:C\to B$ with $\tau_{\leq -1}(h)\colon \tau_{\leq -1}(A)\to \tau_{\leq -1}(C)$ a weak equivalence (in the standard model structure on chain complexes), $\tau_{\geq 0}(g)\colon \tau_{\geq 0}(C)\to \tau_{\geq 0}(B)$ a weak equivalence, and with $f$ homotopy equivalent to $g\circ h$?

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domenico fiorenza
domenico fiorenza
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Homotopy factorization of morphisms of chain complexes

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

Let $R$ be a ring (eventually with good properties) and $Chains(R)$ be the category of chain complexes of $R$-modules (eventually bounded). One has the natural endofunctors $\tau_{\leq n}$ and $\tau_{\geq n}$ associated with the standard t-structure on $Chains(R)$.

What I'm wondering is the following. Is it true that for every morphism $f\colon A\to B$ of chain complexes there exist morphisms $h:A\to C$ and $g:C\to B$ with $\tau_{\leq -1}(h)\colon \tau_{\leq -1}(A)\to \tau_{\leq -1}(C)$ a weak equivalence (in the standard model structure on chain complexes), $\tau_{\geq 0}(g)\colon \tau_{\geq 0}(C)\to \tau_{\geq 0}(B)$ a weak equivalence, and with $f$ homotopy equivalent to $g\circ h$?