Cotorsion pairs generated by modules of bounded projective dimension

Abstract

We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations.We also prove that if the ring is an order in an ℵ0-noetherian ring Q of small finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and Cohen Macaulay noetherian commutative rings.

Tilting for regular rings of Krull dimension two

“…For a domain R, let L δ = ⊥ ∞ (Add δ), and recall [7] that the tilting cotorsion pair induced by δ is (P 1 , D). If R is Gorenstein, we have:…”

DIVISIBLE ENVELOPES, $\mathcal{P}_1$-COVERS AND WEAK-INJECTIVE MODULES

“…In [6], Herbera and the author proved that, for a large class of rings including all commutative domains, the right Ext-orthogonal to P 1 is the class of divisible modules. From this fact it follows that every module over an almost perfect domain has a divisible envelope and in [21], Salce conjectured that the commutative domains for which every module admits a divisible envelope are exactly the almost perfect domains.…”

Sigma-cotorsion modules over valuation domains

“…Every countably presented flat module F is a countable direct limit of finitely generated projective (or even free) modules (see [23]); hence F has projective dimension at most one, by [19]. Now, by [4,Corollary 8.2], for any commutative domain R, the class of divisible R-modules is the right Ext-orthogonal to the class of modules of projective dimension at most one, that is Ext Recall that a commutative domain R is called a valuation domain if its ideals form a chain with respect to inclusion. For proofs of the following facts, unexplained terminology, and other results on modules over valuation domains we refer to [12,XIII].…”