# Explicit cogenerators for the homotopy category of projective modules over a ring

Annales scientifiques de l'École Normale Supérieure (2011)

- Volume: 44, Issue: 4, page 607-629
- ISSN: 0012-9593

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topNeeman, Amnon. "Explicit cogenerators for the homotopy category of projective modules over a ring." Annales scientifiques de l'École Normale Supérieure 44.4 (2011): 607-629. <http://eudml.org/doc/272245>.

@article{Neeman2011,

abstract = {Let $R$ be a ring. In two previous articles [12, 14] we studied the homotopy category $\mathbf \{K\}(R\text\{-\}\mathrm \{Proj\})$ of projective $R$-modules. We produced a set of generators for this category, proved that the category is $\aleph _1$-compactly generated for any ring $R$, and showed that it need not always be compactly generated, but is for sufficiently nice $R$. We furthermore analyzed the inclusion $j_!^\{\}:\mathbf \{K\}(R\text\{-\}\mathrm \{Proj\})\rightarrow \mathbf \{K\}(R\text\{-\}\mathrm \{Flat\})$ and the orthogonal subcategory $\mathcal \{S\}=\{\mathbf \{K\}(R\text\{-\}\mathrm \{Proj\})\}^\perp $. And we even showed that the inclusion $\mathcal \{S\}\rightarrow \mathbf \{K\}(R\text\{-\}\mathrm \{Flat\})$ has a right adjoint; this forces some natural map to be an equivalence $\mathbf \{K\}(R\text\{-\}\mathrm \{Proj\})\rightarrow \mathcal \{S\}^\perp $.
In this article we produce a set of cogenerators for $\mathbf \{K\}(R\text\{-\}\mathrm \{Proj\})$. More accurately, this set of cogenerators naturally lies in the equivalent $\mathcal \{S\}^\perp \cong \mathbf \{K\}(R\text\{-\}\mathrm \{Proj\})$; it can be used to give yet another proof of the fact that the inclusion $\mathcal \{S\}\rightarrow \mathbf \{K\}(R\text\{-\}\mathrm \{Flat\})$ has a right adjoint. But by now several proofs of this fact already exist.},

author = {Neeman, Amnon},

journal = {Annales scientifiques de l'École Normale Supérieure},

keywords = {triangulated categories; generators; cogenerators; flat modules; projective modules},

language = {eng},

number = {4},

pages = {607-629},

publisher = {Société mathématique de France},

title = {Explicit cogenerators for the homotopy category of projective modules over a ring},

url = {http://eudml.org/doc/272245},

volume = {44},

year = {2011},

}

TY - JOUR

AU - Neeman, Amnon

TI - Explicit cogenerators for the homotopy category of projective modules over a ring

JO - Annales scientifiques de l'École Normale Supérieure

PY - 2011

PB - Société mathématique de France

VL - 44

IS - 4

SP - 607

EP - 629

AB - Let $R$ be a ring. In two previous articles [12, 14] we studied the homotopy category $\mathbf {K}(R\text{-}\mathrm {Proj})$ of projective $R$-modules. We produced a set of generators for this category, proved that the category is $\aleph _1$-compactly generated for any ring $R$, and showed that it need not always be compactly generated, but is for sufficiently nice $R$. We furthermore analyzed the inclusion $j_!^{}:\mathbf {K}(R\text{-}\mathrm {Proj})\rightarrow \mathbf {K}(R\text{-}\mathrm {Flat})$ and the orthogonal subcategory $\mathcal {S}={\mathbf {K}(R\text{-}\mathrm {Proj})}^\perp $. And we even showed that the inclusion $\mathcal {S}\rightarrow \mathbf {K}(R\text{-}\mathrm {Flat})$ has a right adjoint; this forces some natural map to be an equivalence $\mathbf {K}(R\text{-}\mathrm {Proj})\rightarrow \mathcal {S}^\perp $.
In this article we produce a set of cogenerators for $\mathbf {K}(R\text{-}\mathrm {Proj})$. More accurately, this set of cogenerators naturally lies in the equivalent $\mathcal {S}^\perp \cong \mathbf {K}(R\text{-}\mathrm {Proj})$; it can be used to give yet another proof of the fact that the inclusion $\mathcal {S}\rightarrow \mathbf {K}(R\text{-}\mathrm {Flat})$ has a right adjoint. But by now several proofs of this fact already exist.

LA - eng

KW - triangulated categories; generators; cogenerators; flat modules; projective modules

UR - http://eudml.org/doc/272245

ER -

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