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

Abstract

For any abelian category $\mathcal{A}$, Auslander constructed a localisation $w:\mathrm{fp}(\mathcal{A}^{\mathrm{op}},\mathrm{Ab})\to \mathcal{A}$ called the defect, which is the left adjoint to the Yoneda embedding $Y:\mathcal{A}\to\mathrm{fp}(\mathcal{A}^{\mathrm{op}},\mathrm{Ab})$. If $\mathcal{A}$ 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 $\mathcal{A}$ is hereditary. We also discuss several subcategories of $\mathrm{fp}(\mathcal{A}^{\mathrm{op}},\mathrm{Ab})$ 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 $\psi\leqslant\rho\leqslant\phi$ which agrees with $\phi$ on injectives, and that there is a largest pp formula $\mu$ such that $\psi\leqslant \mu\leqslant \phi$ and $\psi R=\mu R$. When $R$ is left coherent, we show that there is a largest pp formula $\sigma$ such that $\psi\leqslant\sigma\leqslant \phi$ 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 $\psi\leqslant \nu\leqslant\phi$ and $\phi R=\nu R$. We also show that, for any pp-pair $\phi/\psi$, $w(\phi/\psi)\cong (D\psi)R/(D\phi)R$, where $D$ is the elementary duality of pp formulas. We also give an expression for $w(\phi/\psi)$ in terms of the free realisation of $\phi$ and $\psi$.