A higher algebraic approach to liftings of modules over derived quotients
Ryo Ishizuka
TL;DR
This work extends the Auslander--Reiten lifting framework from regular and complete-intersection quotients to derived quotients using higher algebra and animated rings. The core contribution is a derived-version of the ADS lifting lemma: if a finitely generated module M over a derived quotient A/^L f satisfies $Ext^2_{A/^L f}(M,M)=0$, then M lifts to A, expressed as $M \cong L ⊗^L_A A/^L f$ for some finitely generated A-module L. The paper also builds a fiber-sequence apparatus that translates Ext-vanishing into concrete liftability criteria and provides an obstruction in $Ext^2_{A_1}(M,M)$ whose vanishing guarantees liftings; together with Auslander's zero-divisor theorem, the derived-quotient Ext-vanishing criterion is equivalent to the regularity of the sequence, i.e., local complete intersection. This yields a conceptual, higher-algebra proof of the classical circle connecting liftability and regular sequences, and clarifies how derived quotients capture additional torsion data absent in ordinary quotients. The results unify and extend prior DG-based approaches, offering a robust framework for analyzing liftings across derived quotients with potential applications to the Auslander--Reiten conjecture and singularity theory.
Abstract
We show a certain existence of a lifting of modules under the self-$\mathrm{Ext}^2$-vanishing condition over the "derived quotient" by using the notion of higher algebra. This refines a work of Auslander-Ding-Solberg's solution of the Auslander-Reiten conjecture for complete interesctions. Together with Auslander's zero-divisor theorem, we show that the existence of such $\mathrm{Ext}$-vanishing module over derived quotients is equivalent to being local complete intersections.