n-Projective Model Structures on Chain Complexes

Details n-Projective Model Structures on Chain Complexes n-Projective Model Structures on Chain Complexes

2012-02-12

arxiv.org/abs/1202.2481v2

Mathematics

Category Theory

Second version, 49 pages. arXiv admin note: text overlap with arXiv:math/0607769 by other authors

Scientific paper

We study completeness of the cotorsion pairs $(\widetilde{\mathcal{P}_n}, dg\widetilde{\mathcal{P}^{\perp}_n})$ and $(dg\widetilde{\mathcal{P}_n}, \widetilde{\mathcal{P}^{\perp}_n})$ in ${\rm Ch}(R)$ induced by the cotorsion pair $(\mathcal{P}_n, \mathcal{P}_n^{\perp})$ in $R -Mod$, where $\mathcal{P}_n$ is the class of left $R$-modules with projective dimension at most $n$. The completeness of the pair $(dg\widetilde{\mathcal{P}_n}, \widetilde{\mathcal{P}^{\perp}_n})$ is a consequence of a result proved by J. Gillespie in \cite{Gillespie}. We use a generalization of the zig-zag argument to show that $(\widetilde{\mathcal{P}_n}, dg\widetilde{\mathcal{P}^{\perp}_n})$ is also complete. This gives rise to a new model structure on ${\rm Ch}(R)$.

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