# Rigidity, Residues and Duality: Overview and Recent Progress

@inproceedings{Yekutieli2021RigidityRA, title={Rigidity, Residues and Duality: Overview and Recent Progress}, author={Amnon Yekutieli}, year={2021} }

In this article we explain the theory of rigid residue complexes in commutative algebra and algebraic geometry, summarizing the background, recent results and anticipated future results. Unlike all previous approaches to Grothendieck Duality, the rigid approach concentrates on the construction of rigid residue complexes over rings, and their intricate yet robust properties. The geometrization, i.e. the passage to rigid residue complexes on schemes and Deligne-Mumford (DM) stacks, by gluing, is…

## References

SHOWING 1-10 OF 29 REFERENCES

Rigid Dualizing Complexes Over Commutative Rings

- Mathematics
- 2006

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the…

Grothendieck Duality for Deligne-Mumford Stacks

- Mathematics
- 2008

We prove the existence of the dualizing functor for a separated morphism of algebraic stacks with affine diagonal; then we explicitly develop duality for compact Deligne-Mumford stacks focusing in…

Rigid complexes via DG algebras

- Mathematics
- 2006

Let A be a commutative ring, B a commutative A-algebra and M a complex of B-modules. We begin by constructing the square Sq B/A M, which is also a complex of B-modules. The squaring operation is a…

PERFECT COMPLEXES ON ALGEBRAIC STACKS

- 2017

We develop a theory of unbounded derived categories of quasicoherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks…

∞-Categories for the Working Mathematician

- 2018

homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition,…

Dualizing complexes over noncommutative graded algebras

- Mathematics
- 1992

The motivation for the work presented in this paper was two questions posed by M. Artin. Let A be a d dimensional regular graded algebra over the field k, with augmentation ideal m (see Def. 4.13).…

Central Extensions of Gerbes

We introduce the notion of central extension of gerbes on a topo-logical space X. We show that there are obstruction classes to lifting objects and isomorphisms in a central extension. We also…

The Squaring Operation for Commutative DG Rings

- Mathematics
- 2014

Let A -> B be a homomorphism of commutative rings. The squaring operation is a functor Sq_{B/A} from the derived category D(B) of complexes B-modules into itself. The squaring operation is needed for…

An improvement on the base-change theorem and the functor $f^!$

- Mathematics
- 2014

The major improvement in this paper is that we can extend the functor $(-)^!$ of Grothendieck duality to the unbounded derived category of sufficiently nice algebraic stacks. The original motivation…