$\mathcal P_1$-covers over commutative rings

Silvana Bazzoni; Giovanna Le Gros

doi:10.4171/rsmup/54

𝒫_{1}

-covers over commutative rings

Silvana Bazzoni ; Giovanna Le Gros

Rendiconti del Seminario Matematico della Università di Padova, Tome 144 (2020), pp. 27-43.

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Résumé

In this paper we consider the class $𝒫_{1} (R)$ of modules of projective dimension at most one over a commutative ring $R$ and we investigate when $𝒫_{1} (R)$ is a covering class. More precisely, we investigate Enochs' Conjecture, that is the question of whether $𝒫_{1} (R)$ is covering necessarily implies that $𝒫_{1} (R)$ is closed under direct limits. We answer the question affirmatively in the case of a commutative semihereditary ring $R$ . This gives an example of a cotorsion pair $(𝒫_{1} (R), 𝒫_{1} {(R)}^{⊥})$ which is not necessarily of finite type such that $𝒫_{1} (R)$ satisfies Enochs' Conjecture. Moreover, we describe the class $lim_{\to} 𝒫_{1} (R)$ over (not necessarily commutative) rings which admit a classical ring of quotients.

Publié le : 2020-12-10

DOI : 10.4171/rsmup/54

Classification : 13, 18

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     author = {Silvana Bazzoni and Giovanna Le Gros},
     title = {$\mathcal P_1$-covers over commutative rings},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {27--43},
     volume = {144},
     year = {2020},
     doi = {10.4171/rsmup/54},
     mrnumber = {4186444},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/54/}
}

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Silvana Bazzoni; Giovanna Le Gros. $\mathcal P_1$-covers over commutative rings. Rendiconti del Seminario Matematico della Università di Padova, Tome 144 (2020), pp. 27-43. doi : 10.4171/rsmup/54. http://www.numdam.org/articles/10.4171/rsmup/54/

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