Cut cotorsion pairs

We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

THE FINITISTIC DIMENSION AND CHAIN CONDITIONS ON IDEALS

Let Λ be an artin algebra and $0=I_{0}\subseteq I_{1} \subseteq I_{2}\subseteq\cdots \subseteq I_{n}$ a chain of ideals of Λ such that $(I_{i+1}/I_{i})\rad(\Lambda/I_{i})=0$ for any $0\leq i\leq n-1$ and $\Lambda/I_{n}$ is semisimple. If either none or the direct sum of exactly two consecutive ideals has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. As a consequence, we have that if either none or the direct sum of exactly two consecutive terms in the radical series of Λ has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. Some known results are obtained as corollaries.

Finitistic dimension and endomorphism algebras of 𝒲-Gorenstein modules

Let [Formula: see text] be an artin algebra, [Formula: see text] be a [Formula: see text]-Gorenstein [Formula: see text]-module and [Formula: see text], then [Formula: see text] is a [Formula: see text]-[Formula: see text]-bimodule. We use the restricted flat dimension of [Formula: see text] and the finitistic [Formula: see text]-dimension of [Formula: see text] to characterize the finitistic dimension of [Formula: see text], and obtain the following main result: if [Formula: see text] is [Formula: see text]-finite with [Formula: see text], then we have: (1) If [Formula: see text] or [Formula: see text], then [Formula: see text] (2) If [Formula: see text], then [Formula: see text]