${𝒫}_1$-covers over commutative rings

Bazzoni, Silvana; Le Gros, Giovanna

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

In this paper we consider the class ${𝒫}_1\left(R\right)$ of modules of projective dimension at most one over a commutative ring $R$ and we investigate when ${𝒫}_1\left(R\right)$ is a covering class. More precisely, we investigate Enochs' Conjecture, that is the question of whether ${𝒫}_1\left(R\right)$ is covering necessarily implies that ${𝒫}_1\left(R\right)$ 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 $({𝒫}_1\left(R\right),{𝒫}_1{\left(R\right)}^{\perp })$ which is not necessarily of finite type such that ${𝒫}_1\left(R\right)$ satisfies Enochs' Conjecture. Moreover, we describe the class $\underrightarrow{lim}{𝒫}_1\left(R\right)$ over (not necessarily commutative) rings which admit a classical ring of quotients.

Classification : 13, 18