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

Amnon Neeman

Annales scientifiques de l'École Normale Supérieure (2011)

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

Abstract

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Let R be a ring. In two previous articles [12, 14] we studied the homotopy category 𝐊 ( R - Proj ) of projective R -modules. We produced a set of generators for this category, proved that the category is ℵ 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 ! : 𝐊 ( R - Proj ) → 𝐊 ( R - Flat ) and the orthogonal subcategory 𝒮 = 𝐊 ( R - Proj ) ⊥ . And we even showed that the inclusion 𝒮 → 𝐊 ( R - Flat ) has a right adjoint; this forces some natural map to be an equivalence 𝐊 ( R - Proj ) → 𝒮 ⊥ . In this article we produce a set of cogenerators for 𝐊 ( R - Proj ) . More accurately, this set of cogenerators naturally lies in the equivalent 𝒮 ⊥ ≅ 𝐊 ( R - Proj ) ; it can be used to give yet another proof of the fact that the inclusion 𝒮 → 𝐊 ( R - Flat ) has a right adjoint. But by now several proofs of this fact already exist.

How to cite

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Neeman, 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 -

References

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