Pseudo-dualizing complexes of torsion modules and semi-infinite MGM duality
Leonid Positselski
TL;DR
The paper develops a relative MGM duality framework for torsion modules and contramodules over a pair of rings linked by a morphism of ring-ideal pairs with weakly proregular ideals. It introduces pseudo-dualizing and relative dualizing complexes to build a network of Bass/Auslander classes and ordinary, pseudo-, and semi-derived categories, establishing a web of triangulated equivalences that generalize covariant Serre–Grothendieck duality and the co-contra correspondence to the relative, non-Noetherian setting. Core contributions include explicit constructions of (lower and upper) pseudo-derived categories, a comparison of dedualizing and dualizing notions, and robust base-change results via quotflat morphisms and relative U-objects, culminating in semiderived equivalences that interpolate between classical derived categories and coderived/contraderived categories. The results unify duality phenomena in MGM theory under weak proregularity and adic coherence hypotheses, with detailed finiteness and coherence conditions enabling several equivalences and adjunctions across torsion and contramodule worlds. Collectively, the work provides a comprehensive, relative framework for co-contra correspondences and MGM dualities applicable to a broad class of non-Noetherian, adically complete settings, including relative base change and semiderived contexts.
Abstract
This paper is an MGM version of arXiv.org:1703.04266 and arXiv:1907.03364, and a follow-up to Section 5 of arXiv:1503.05523. In the setting of a commutative ring $S$ with a weakly proregular finitely generated ideal $J\subset S$, we consider the maximal, abstract, and minimal corresponding classes of $J$-torsion $S$-modules and $J$-contramodule $S$-modules with respect to a given pseudo-dualizing complex of $J$-torsion $S$-modules $L^\bullet$, and construct the related triangulated equivalences. As a special case, we obtain an equivalence of the semiderived categories for an $I$-adically coherent commutative ring $R$ with a weakly proregular ideal $I\subset R$, a dualizing complex of $I$-torsion $R$-modules $D^\bullet$, and a ring homomorphism $f\colon R\rightarrow S$ such that $f(I)\subset J$ and $S$ is a flat $R$-module. (If the ring $S$ is not Noetherian, then a certain further assumption, which we call quotflatness of the morphism of pairs $f\colon (R,I)\rightarrow(S,J)$, needs to be imposed.) In that case, the pseudo-dualizing complex $L^\bullet$ is constructed as a complex of $J$-torsion $S$-modules quasi-isomorphic to the tensor product of $D^\bullet$ with the infinite dual Koszul complex for some set of generators of the ideal $J\subset S$.