Pseudo-dualizing complexes of torsion modules and semi-infinite MGM duality

TL;DR

This paper develops a duality theory for torsion modules over commutative rings using pseudo-dualizing complexes, establishing triangulated equivalences in a generalized MGM framework, extending previous results to non-Noetherian settings.

Contribution

It introduces pseudo-dualizing complexes for torsion modules and constructs new triangulated equivalences in the MGM duality setting, generalizing prior work to broader classes of rings.

Findings

Constructed triangulated equivalences for $J$-torsion modules and $J$-contramodules.

Extended MGM duality to non-Noetherian rings with weakly proregular ideals.

Defined pseudo-dualizing complexes as tensor products involving dual Koszul complexes.

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 $J\subset S$.