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Shlomi A
Shlomi A
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Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it induces isomorphisms $H_kM\to H_kN$ on all homologies, and a cofibration if each $f_k:M_k\to N_k$ is a monomorphism, with a projective R-module as its cokernel.

Let A be a projective R-module. The chain complex $(*)$ $0\to A\to A\to0$ is a projective object in the world of chain complexes, $Ch_R$. So is any direct sum of such chain complexes. It is a known result (see [DS95, 7.10] for instance) that any acyclic object in $Ch_R$ which is level-wise projective is is isomorphic to a direct sum of chains complexes as in $(*)$, with $A$'s projective.

We have here two notions, of (i) a chain complex which is level-wise project, and of (ii) a projective object in $Ch_R$. My gut's feeling is that (i) doesn't imply (ii). In the other way around I'm not sure. I would be glad to have an answer in both directions. :-)

[DS95] = Dwyer & Spalinski, Homotopy theories and model categories.

Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it induces isomorphisms $H_kM\to H_kN$ on all homologies, and a cofibration if each $f_k:M_k\to N_k$ is a monomorphism, with a projective R-module as its cokernel.

Let A be a projective R-module. The chain complex $(*)$ $0\to A\to A\to0$ is a projective object in the world of chain complexes, $Ch_R$. So is any direct sum of such chain complexes. It is a known result [DS95, 7.10] that any acyclic object in $Ch_R$ which is level-wise projective is isomorphic to a direct sum of chains complexes as in $(*)$, with $A$'s projective.

We have here two notions, of (i) a chain complex which is level-wise project, and of (ii) a projective object in $Ch_R$. My gut's feeling is that (i) doesn't imply (ii). In the other way around I'm not sure. I would be glad to have an answer in both directions. :-)

[DS95] = Dwyer & Spalinski, Homotopy theories and model categories.

Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it induces isomorphisms $H_kM\to H_kN$ on all homologies, and a cofibration if each $f_k:M_k\to N_k$ is a monomorphism, with a projective R-module as its cokernel.

Let A be a projective R-module. The chain complex $(*)$ $0\to A\to A\to0$ is a projective object in the world of chain complexes, $Ch_R$. So is any direct sum of such chain complexes. It is a known result (see [DS95, 7.10] for instance) that any acyclic object in $Ch_R$ which is level-wise projective is isomorphic to a direct sum of chains complexes as in $(*)$, with $A$'s projective.

We have here two notions, of (i) a chain complex which is level-wise project, and of (ii) a projective object in $Ch_R$. My gut's feeling is that (i) doesn't imply (ii). In the other way around I'm not sure. I would be glad to have an answer in both directions. :-)

[DS95] = Dwyer & Spalinski, Homotopy theories and model categories.

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Shlomi A
Shlomi A
  • 583
  • 5
  • 15

On the difference between a projective chain complex and a level-wise projective chain complex

Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it induces isomorphisms $H_kM\to H_kN$ on all homologies, and a cofibration if each $f_k:M_k\to N_k$ is a monomorphism, with a projective R-module as its cokernel.

Let A be a projective R-module. The chain complex $(*)$ $0\to A\to A\to0$ is a projective object in the world of chain complexes, $Ch_R$. So is any direct sum of such chain complexes. It is a known result [DS95, 7.10] that any acyclic object in $Ch_R$ which is level-wise projective is isomorphic to a direct sum of chains complexes as in $(*)$, with $A$'s projective.

We have here two notions, of (i) a chain complex which is level-wise project, and of (ii) a projective object in $Ch_R$. My gut's feeling is that (i) doesn't imply (ii). In the other way around I'm not sure. I would be glad to have an answer in both directions. :-)

[DS95] = Dwyer & Spalinski, Homotopy theories and model categories.