The defect recollement, the MacPherson-Vilonen construction, and pp formulas

JOURNAL OF ALGEBRA

卷:532

Article

关键词: Finitely presented functors;    Coherent functors;    Recollements of abelian categories;    pp formula;    Model theory of modules;   

DOI  :  10.1016/j.jalgebra.2019.05.015

来源: Elsevier

【 摘 要 】

For any abelian category A, Auslander constructed a localisation w : fp(A(op), Ab) -> A called the defect, which is the left adjoint to the Yoneda embedding Y : A -> fp(A(op), Ab). If A has enough projectives, then this localisation is part of a recollement called the defect recollement. We show that this recollement is an instance of the MacPherson-Vilonen construction if and only if A is hereditary. We also discuss several subcategories of fp(A(op), Ab) which arise as canonical features of the defect recollement, and characterise them by properties of their projective presentations and their orthogonality with other subcategories. We apply some parts of the defect recollement to the model theory of modules. Let R be a ring and let phi/psi be a pp-pair. When R is an artin algebra, we show that there is a smallest pp formula rho such that psi <= rho <= phi which agrees with phi on injectives, and that there is a largest pp formula mu such that psi <= mu <= phi and psi R = mu R. When R is left coherent, we show that there is a largest pp formula sigma such that psi <= sigma <= phi which agrees with psi on injectives, and that the pp-pair psi/phi is isomorphic to a pp formula if and only if psi = sigma, and that there is a smallest pp formula nu such that psi <= nu <= phi and phi R = nu R. We also show that, for any pp-pair phi/psi, w(phi/psi) congruent to (D psi)R/(D phi)R, where D is the elementary duality of pp formulas. We also give expressions for w(phi/psi) in terms of free realisation of phi and psi. (C) 2019 Published by Elsevier Inc.

【 授权许可】

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