Deformation theory (original) (raw)

Derived deformation theory of algebraic structures

arXiv (Cornell University), 2019

The main purpose of this article is to develop an explicit derived deformation theory of algebraic structures at a high level of generality, encompassing in a common framework various kinds of algebras (associative, commutative, Poisson...) or bialgebras (associative and coassociative, Lie, Frobenius...), that is algebraic structures parametrized by props. A central aspect is that we define and study moduli spaces of deformations of algebraic structures up to quasi-isomorphisms (and not only up to isomorphims or ∞-isotopies). To do so, we implement methods coming from derived algebraic geometry, by encapsulating these deformation theories as classifying (pre)stacks with good infinitesimal properties and derived formal groups. In particular, we prove that the Lie algebra describing the deformation theory of an object in a given ∞-category of dg algebras can be obtained equivalently as the tangent complex of loops on a derived quotient of this moduli space by the homotopy automorphims of this object. Moreover, we provide explicit formulae for such derived deformation problems of algebraic structures up to quasi-isomorphisms and relate them in a precise way to other standard deformation problems of algebraic structures. This relation is given by a fiber sequence of the associated dg-Lie algebras of their deformation complexes. Our results provide simultaneously a vast generalization of standard deformation theory of algebraic structures which is suitable (and needed) to set up algebraic deformation theory both at the ∞categorical level and at a higher level of generality than algebras over operads. In addition, we study a general criterion to compare formal moduli problems of different algebraic structures and apply our formalism to En-algebras and bialgebras.

The Moduli Space and Versal Deformationsof Algebraic Structures

In this talk I consider deformations of algebraic structures. The notion of 1-parameter deformation is due to Gerstenhaber. Here I give a generalization of the classical notion by considering deformations with a commutative algebra base, and define the miniversal formal deformation. This notion is necessary to describe non-equivalent deformations with the same infinitesimal part, and to find singular nontrivial deformations with zero infinitesimal part. I use the example of a vector field Lie algebra to demonstrate the computation. Another example which underlines the importance of such general deformations is to consider moduli spaces of Lie algebras. This I also demonstrate on an example.

Deformation theory from the point of view of fibered categories

2010

Abstract: We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We include proofs of two key results: a versione of Schlessinger's Theorem in this context, and the Ran--Kawamata vanishing theorem for obstructions.

Deformations and Contractions of Algebraic Structures

Труды Математического института им. Стеклова, 2014

We describe the basic notions of versal deformation theory of algebraic structures and compare it with the analytic theory. As a special case, we consider the notion of versal deformation used by Arnold. With the help of versal deformation we get a stratification of the moduli space into projective orbifolds. We compare this with Arnold's stratification in the case of similarity of matrices. The other notion we discuss is the opposite notion of contraction.

On the tangent space of the deformation functor of curves with automorphisms

Algebra & Number Theory, 2007

We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, p-cyclic covers of the affine line and to Lehr-Matignon curves.

A Superficial Working Guide to Deformations and Moduli

This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results, regarding the question of local homeomorphism between Kuranishi and Teichmueller space, and a survey of new results with Ingrid Bauer, concerning the discrepancy between the deformation of the action of a group G on a minimal models S, respectively the deformation of the action of G on the canonical model X. Here Def(S) maps properly onto Def(X), but the same does not hold for pairs: Def(S,G) does not map properly onto Def(X,G). Indeed the connected components of Def(S), in the case of tertiary Burniat surfaces, only map to locally closed sets. The last section contains anew result on some surfaces whise Albanese map has generic degree equal to 2.

Making lifting obstructions explicit

Proceedings of the London Mathematical Society, 2013

If P → X is a topological principal K-bundle and K a central extension of K by Z, then there is a natural obstruction class δ1(P ) ∈Ȟ 2 (X, Z) in sheaf cohomology whose vanishing is equivalent to the existence of a K-bundle P over X with P ∼ = P /Z. In this paper we establish a link between homotopy theoretic data and the obstruction class δ1(P ) which in many cases can be used to calculate this class in explicit terms. Writing ∂ P d : π d (X) → π d−1 (K) for the connecting maps in the long exact homotopy sequence, two of our main results can be formulated as follows. If Z is a quotient of a contractible group by the discrete group Γ, then the homomorphism π3(X) → Γ induced by δ1(P ) ∈Ȟ 2 (X, Z) ∼ = H 3 sing (X, Γ) coincides with ∂ K 2 •∂ P 3 and if Z is discrete, then δ1(P ) ∈Ȟ 2 (X, Z) induces the homomorphism −∂ K 1 •∂ P

Modules Which Are Lifting Relative To Module Classes

Kyungpook mathematical journal, 2008

In this paper, we study a module which is lifting and supplemented relative to a module class. Let R be a ring, and let X be a class of R-modules. We will define X-lifting modules and X-supplemented modules. Several properties of these modules are proved. We also obtain results for the case of specific classes of modules.