symmetric monoidal (∞,1)-category of spectra
Deformation theory studies problems of extending structures to extensions of their domains. Formal deformation theory, is the part of the deformation theory where the extensions are infinitesimal.
A typical problem in formal deformation theory has the structure that
and infinitesimal thickenings and of and are prescribed, with injection morphisms and
and asks whether a bottom horizontal morphism in the diagram
may be found. This morphism would be called an infinitesimal deformation of .
In other words:
Formal deformation theory studies the obstruction theory of extensions to infinitesimal thickenings.
A typical example of an infinitesimal thickening is a square-0-extension of a ring:
Then consider the new ring, whose underlying group is the direct sum , equipped with the product structure
This is the square 0-extension of by . It should be thought of as the algebra of functions that consists of elements of and , where the elements in are thought of as functions with values in infinitesimal quantities, so that their would-be product “” vanishes.
So the ring may be thought of as the ring of functions on the infinitesimal extension of , which is the space obtained by adding to all the vectors of infinitesimal length in the vector bundle over .
There is a canonical ring homomorphism that is the identity on and on . This is to be thought of as the pullback of functions on spaces along the inclusion of spaces (which in turn may be thought of as the 0-section of the vector bundle on ).
Similarly, let be another ring with module and square-0 extension , thought of, respectively, as the ring of functions on a space , the module of sections of a vector bundle on and the ring of functions on the space of infinitesimal vectors of this vector bundle.
In terms of these function rings, a morphism of spaces corresponds to a ring homomorphism . Hence we have a situation
The obvious obstruction problem now is whether we can deform to a morphism of rings, such that we get a commuting diagram
This is the archetypical problem that deformation theory deals with. As always, after studying this a bit it turns out that in order to obtain a good theory, one needs to adopt the nPOV. Problems as above may be stated in the category Ring of rings, but they may have good answers only in categorifications of this for instance to the (∞,1)-category of E-∞-rings.
In order to better see the structure of the above archetypical problem of deformation theory, we describe some aspects of the canonical bifibration of ring modules in a way that nicely organizes all the concepts module, derivation, Kähler differential in a single picture that lends itself to vertical categorification. (Following DefTheory.)
But there is also another functor of interest: for any -module, we may form the ring called the square 0-extension of , in which multiplication is given by
This may be organized into a single functor
Using the module of Kähler differentials is not appropriate in general; instead we need to take its derived version. To talk about the nonabelian derived functors, Quillen introduced a model category structure on the category of simplicial commutative rings. Given a morphism of rings, which makes an -algebra, the category of abelian group objects in the slice category - of -algebras over is equivalent both to the category of -modules and the trivial (= square zero) extensions of by -modules. In particular we can consider the forgetful functor which has a left adjoint . All said is true for simplicial commutative rings as well. Now the relative cotangent complex is the value on of the left derived functor . Regarding that the left adjoint at the nonderived level (and for usual rings) can be expressed via Kähler differentials, this explains the phrase “derived version of the module Kähler differentials”.
The above situation generalizes from the category Ring to an arbitrary presentable (∞,1)-category by replacing the bifibration by the stabilization of the codomain fibration of : the tangent (∞,1)-category of .
The projection still has a left adjoint
for which a representative which is also a section (in a strict sense) of may be taken; any such representative is called the cotangent complex functor for . The special property section property, like in the motivating example above, says that the composition
is the identity (∞,1)-functor.
Over a field of characteristic zero, there is an approach to deformation theory via differential graded Lie algebras (or more generally L-infinity algebras). One can find some exposition about this approach in the Kontsevich and Lurie references below. See also discussion at MathOverflow: def theory and dgla-s.
In this approach, one begins with an object (for example a scheme, or a complex manifold, or a vector bundle, or an associative algebra, or a dg category, or …) that one would like to deform. Then the general principle is that there exists a dgLa with the property that the functor , which sends a local Artin algebra to the set of Maurer-Cartan solutions in modulo the gauge action of , is isomorphic to the functor which sends a local Artin algebra to the set of isomorphism classes of deformations of over . Note the similarity with Schlessinger’s theory: both here and in Schlessinger’s work, we deal with functors from Artin algebras to sets.
In the case of a compact complex manifold, the dgLa in question is given by the so-called Kodaira-Spencer? dgLa: holomorphic vector fields tensor -forms (this is just the Dolbeault resolution of the sheaf of holomorphic vector fields). In the case of an associative algebra (or a dg algebra, or an A-infinity algebra, or a dg category, or an A-infinity category), the appropriate dgLa is the Hochschild complex with the Hochschild differential and the Gerstenhaber bracket.
In this language, the Tian-Todorov theorem? on the unobstructedness of deformations of Calabi-Yau manifolds translates to the statement that the Kodaira-Spencer dgLa of a Calabi-Yau manifold is homotopy abelian — that is, it is quasi-isomorphic to an abelian dg Lie algebra. Barannikov-Kontsevich proved more generally that the dgLa given by holomorphic polyvector fields tensor -forms on a Calabi-Yau manifold is homotopy abelian. The deformation-theoretic consequence is that the “extended deformations” of Calabi-Yau manifolds are unobstructed. These “extended deformations” should be realized by certain deformations of (a dg enhancement of) the derived category of coherent sheaves on the Calabi-Yau.
The following paper is a good introduction to these ideas:
The Kontsevich and Soibelman references below are also good.
C. Doran, Deformation Theory: An Historical Annotated Bibliography (pdf)
M. Doubek, M. Markl, P. Zima, Deformation theory (lecture notes), Archivum mathematicum 43 (5), 2007, 333–371, arXiv:0705.3719
M. Markl, Deformation theory of algebras and their diagrams, 129 pp, CBMS 116, AMS 2012, book page
E. Sernesi, An overview of classical deformation theory, pdf
Alexander Grothendieck, Catégories cofibrées additives et complexe cotangent relatif, Lecture Notes in Mathematics 79
Luc Illusie, Complexe cotangent et déformations I, Lec. Notes Math. 239, Springer 1971, xv+355 pp.; II, LNM 283, Springer 1972. vii+304 xv+355 pp.
Maxim Kontsevich, Yan Soibelman, Deformation theory I (ps); Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I, math.AG/0606241 – two parts of large unfinished books on the subject; Yan Soibelman, Lectures on deformation theory and mirror symmetry (ps)
Jacob Lurie, Deformation Theory (arXiv:0709.3091) – describes a very setup for deformation theory over any (∞,1)-category is described. Then as an application the deformation theory of E-∞-rings is developed. An application: J. Lurie, Moduli problems for ring spectra, moduli.pdf.
E. Sernesi, Deformations of algebraic schemes (monograph) Grundlehren der Math. Wiss. 334, Springer 2006. xii+339 pp. MR2008e:14011
Martin C. Olsson, Deformation theory of representable morphisms of algebraic stacks, Mathematische Zeitschrift 253, n. 1, 25–62 (2006) doi; Tangent spaces and obstructon theories, lectures, MSRISummer07.pdf
B. Fantechi, M. Manetti, Obstruction calculus for functors of Artin rings I, J. Algebra 202 (1998), no. 2, 541–576, MR99f:14004.
Domenico Fiorenza, Marco Manetti, Elena Martinengo, Semicosimplicial DGLAs in deformation theory, arxiv/0803.0399
S. Merkulov, B. Vallette, Deformation theory of properads, arXiv:0707.0889
Vladimir Hinich, Deformations of homotopy algebras, Communication in Algebra, 32 (2004), 473-494, MR2005m:18009, doi, pdf; Deformations of sheaves of algebras, Adv. Math. 195 (2005), no. 1, 102–164, MR2007d:13021, doi, pdf; DG coalgebras as formal stacks, J. Pure Appl. Algebra 162 (2001), no. 2-3, 209–250 (doi), math.AG/9812034; Formal deformations of sheaves of algebras, video of a talk at MSRI 2002, link
V. Hinich, V. Schechtman, Deformation theory and Lie algebra homology I. Algebra Colloq. 4 (1997), no. 2, 213–240, MR2000c:14006a; part II., Algebra Colloq. 4 (1997), no. 3, 291–316, MR2000c:14006b; preprint version alg-geom/9405013;_Homotopy Lie algebras_, I.M. Gelfand Seminar, Adv. in Sov. Math., 16 (1993), Part 2, 1-28.
W. Lowen, M. Van den Bergh, Deformation theory of abelian categories, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5441–5483; arXiv:math.CT/0405226.
M. Van den Bergh, Notes on formal deformations of abelian categories, arXiv:1002.0259
B. Mazur, Perturbations, deformations, and variations (and “Near-misses”) in Geometry, Physics, and Number Theory, BAMS 41(3), 307-336
M. Artin, Deformations of singularities, TATA Lecture Notes vol. 54.
M. Artin, Versal deformations and algebraic stacks, Invent. Math. 1974
K. Kodaira, L. Nirenberg, D. C. Spencer, On the existence of deformation of complex analytic structures, Ann. Math. 68, 450-459 (1958).
K. Kodaira, D. C. Spencer, On deformation of complex analytic structures, I II, Ann. Math. 67, 328-466 (1958).
M. Schlessinger, Functors of Artin rings, Trans. AMS 130, 208-222 (1968) – this was a groundbreaking article at the time, still much cited.
B. Osserman, Deformation theory and moduli in algebraic geometry, pdf
homepage of Ravi Vakil’s graduate Stanford class on deformation theory and moduli spaces