# The derived deformation theory of a point

@article{Booth2021TheDD, title={The derived deformation theory of a point}, author={Matt Booth}, journal={Mathematische Zeitschrift}, year={2021} }

We provide a prorepresenting object for the noncommutative derived deformation problem of deforming a module $X$ over a differential graded algebra. Roughly, we show that the corresponding deformation functor is homotopy prorepresented by the dual bar construction on the derived endomorphism algebra of $X$. We specialise to the case when $X$ is one-dimensional over the base field, and introduce the notion of framed deformations, which rigidify the problem slightly and allow us to obtain derived… Expand

#### One Citation

Singularity categories via the derived quotient

- Mathematics
- 2021

Abstract Given a noncommutative partial resolution A = End R ( R ⊕ M ) of a Gorenstein singularity R, we show that the relative singularity category Δ R ( A ) of Kalck–Yang is controlled by a certain… Expand

#### References

SHOWING 1-10 OF 140 REFERENCES

Noncommutative deformations and flops

- Mathematics
- 2016

We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33-fold is representable, and hence, we associate to every such curve a… Expand

Noncommutative deformations of modules

- Mathematics
- 2002

The classical deformation theory for modules on a k-algebra, where k is a Þeld, is generalized. For any k-algebra, and for any Þnite family of r modules, we consider a deformation functor deÞned on… Expand

The A∞ Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus

- Mathematics
- 2008

Abstract Let A be an augmented algebra over a semi-simple algebra S . We show that the Ext algebra of S as an A -module, enriched with its natural A -infinity structure, can be used to reconstruct… Expand

Derived functors of I-adic completion and local homology

- Mathematics
- 1992

In recent topological work [2], we were forced to consider the left derived functors of the I-adic completion functor, where I is a finitely generated ideal in a commutative ring A. While our concern… Expand

The derived contraction algebra

- Mathematics
- 2019

Using Braun-Chuang-Lazarev's derived quotient, we enhance the contraction algebra of Donovan-Wemyss to an invariant valued in differential graded algebras. Given an isolated contraction $X \to… Expand

An introduction to noncommutative deformations of modules

- Mathematics
- 2003

This paper gives an elementary introduction to noncommutative deformations of modules. The main results of this deformation theory are due to Laudal.
Let k be an algebraically closed (commutative)… Expand

Unifying derived deformation theories

- Mathematics
- 2007

Abstract We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In… Expand

The homotopy theory of dg-categories and derived Morita theory

- Mathematics
- 2004

The main purpose of this work is to study the homotopy theory of dg-categories up to quasi-equivalences. Our main result is a description of the mapping spaces between two dg-categories C and D in… Expand

Deformation theory of objects in homotopy and derived categories I: General theory

- Mathematics
- 2009

Abstract This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories. Namely, for a DG module E over a DG category we… Expand

DG coalgebras as formal stacks

- Mathematics
- 1998

Abstract The category of unital (unbounded) dg cocommutative coalgebras over a field of characteristic zero is provided with a structure of simplicial closed model category. This generalizes the… Expand