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deformation theory

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Higher algebra

Formal geometry

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Idea

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

  • a morphism f:XYf : X \to Y of certain spaces is given,

  • and infinitesimal thickenings X˜\tilde X and Y˜\tilde Y of XX and YY are prescribed, with injection morphisms XX˜X \to \tilde X and YY˜Y \to \tilde Y

and asks whether a bottom horizontal morphism f˜\tilde f in the diagram

X f Y X˜ f˜ Y˜ \array{ X &\stackrel{f}{\to}& Y \\ \downarrow && \downarrow \\ \tilde X &\stackrel{\tilde f}{\to}& \tilde Y }

may be found. This morphism f˜\tilde f would be called an infinitesimal deformation of ff.

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:

let RR be a ring, to be thought of as the ring of functions on the space XX in the above diagram. Let furthermore NN be an RR-module, to be thought of as the RR-module of sections of a vector bundle over XX.

Then consider the new ring, whose underlying group is the direct sum RNR \oplus N, equipped with the product structure

(r 1,n 1)(r 2,n 2)=(r 1r 2,r 1n 2+n 2r 1). (r_1, n_1) \cdot (r_2, n_2) = (r_1 r_2, r_1 n_2 + n_2 r_1) \,.

This is the square 0-extension of RR by NN. It should be thought of as the algebra of functions that consists of elements of RR and NN, where the elements in NN are thought of as functions with values in infinitesimal quantities, so that their would-be product ”n 1n 2n_1 \cdot n_2” vanishes.

So the ring RNR \oplus N may be thought of as the ring of functions on the infinitesimal extension X˜\tilde X of XX, which is the space obtained by adding to XX all the vectors of infinitesimal length in the vector bundle over XX.

There is a canonical ring homomorphism RNRR\oplus N \to R that is the identity on RR and 00 on NN. This is to be thought of as the pullback of functions on spaces along the inclusion of spaces XX˜X \to \tilde X (which in turn may be thought of as the 0-section of the vector bundle on XX).

Similarly, let R 2R_2 be another ring with module N 2N_2 and square-0 extension R 2N 2R_2 \oplus N_2, thought of, respectively, as the ring of functions on a space YY, the module of sections of a vector bundle on YY and the ring of functions on the space of infinitesimal vectors of this vector bundle.

In terms of these function rings, a morphism f:XYf : X \to Y of spaces corresponds to a ring homomorphism R 1R 2:f *R_1 \leftarrow R_2 : f^*. Hence we have a situation

R 1 f * R 2 R 1N 1 R 2N 2. \array{ R_1 &\stackrel{f^*}{\leftarrow}& R_2 \\ \uparrow && \uparrow \\ R_1 \oplus N_1 && R_2 \oplus N_2 } \,.

The obvious obstruction problem now is whether we can deform f *f^* to a morphism R 1N 1R 2N 2:f˜ *R_1 \oplus N_1 \leftarrow R_2 \oplus N_2 : \tilde f^* of rings, such that we get a commuting diagram

R 1 f * R 2 R 1N 1 f˜ * R 2N 2. \array{ R_1 &\stackrel{f^*}{\leftarrow}& R_2 \\ \uparrow && \uparrow \\ R_1 \oplus N_1 &\stackrel{\tilde f^*}{\leftarrow}& R_2 \oplus N_2 } \,.

The obstruction to the existence of such lifts is measured by cohomology with coefficients in the cotangent complex of R 1R_1.

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.

Modules, derivations and Kähler differentials

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.)

With Ring denoting the category of (commutative, unital) rings, write

p:ModRing p : Mod \to Ring

for the bifibration of modules over rings: objects of ModMod are pairs consisting of a ring RR an an RR-module NN, and morphism (R 1,N 1)(R 2,N 2)(R_1,N_1) \to (R_2, N_2) are pairs consisting of a ring homomorphism f:R 1R 2f : R_1 \to R_2 and a morphism F:N 1N 2 fR 2F : N_1 \to N_2 \otimes_f R_2 of R 2R_2-modules.

(Recall for instance from the discussion at Sweedler coring) that this bifibration is a way to think of the stack of algebraic vector bundles.)

But there is also another functor G:ModRingG : Mod \to Ring of interest: for NN any RR-module, we may form the ring G(N):=RNG(N) := R \oplus N called the square 0-extension of RR, in which multiplication is given by

(r 1,n 1)(r 2,n 2):=(r 1r 2,n 1r 2+n 2r 1). (r_1,n_1) \cdot (r_2, n_2) := (r_1 r_2, n_1 r_2 + n_2 r_1) \,.

Moreover, there is a natural morphism of rings G(N)RG(N) \to R given by sending (r,n)r(r,n) \mapsto r. A section v:RG(n)v : R \to G(n) of this morphism is precisely a derivation of RR with values in the module NN.

This may be organized into a single functor

Mod[I,Ring] Mod \to [I,Ring]

into the arrow category of Ring, that sends to the RR-module NN to the morphism G(N)RG(N) \to R. The original bifibration factors through this morphism by the right endpoint evaluation

Mod p Ring d 1 [I,Ring]. \array{ Mod &&\stackrel{p}{\to}&& Ring \\ & \searrow && \nearrow_{\mathrlap{d_1}} \\ && [I,Ring] } \,.

Finally notice that the functor GG has a left adjoint functor

Ω:RingMod \Omega : Ring \to Mod

that sends a ring RR to the RR-module Ω K(R)\Omega_K(R) of Kähler differentials, i.e. to the module that encodes the cotangent bundle.

Cotangent complex

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 f:ABf: A\to B of rings, which makes BB an AA-algebra, the category AbGr(AAlg/B)AbGr(A-Alg/B) of abelian group objects in the slice category AA-Alg/BAlg/B of AA-algebras over BB is equivalent both to the category of BB-modules and the trivial (= square zero) extensions of AA by BB-modules. In particular we can consider the forgetful functor AbGr(AAlg/B)AAlg/BAbGr(A-Alg/B)\to A-Alg/B which has a left adjoint Ab B/A:AAlg/BAbGr(AAlg/B) BModAb_{B/A} : A-Alg/B\to AbGr(A-Alg/B)\cong {}_B Mod. All said is true for simplicial commutative rings as well. Now the relative cotangent complex L B/AL_{B/A} is the value on BB of the left derived functor 𝕃Ab B/A(B)\mathbb{L} Ab_{B/A}(B). 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 CC by replacing the bifibration ModRingMod \to Ring by the stabilization T CCT_C \to C of the codomain fibration of CC: the tangent (∞,1)-category of CC.

The projection p:T CCp : T_C \to C still has a left adjoint

Ω:CT C \Omega : C \to T_C

for which a representative which is also a section (in a strict sense) of pp may be taken; any such representative is called the cotangent complex functor for CC. The special property section property, like in the motivating example above, says that the composition

CΩT CpC C \stackrel{\Omega}{\to} T_C \stackrel{p}{\to} C

is the identity (∞,1)-functor.

Further categorification

Deformation theory via differential graded Lie algebras

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 XX (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 L XL_X with the property that the functor Def L X:ArtSetDef_{L_X} : Art \to Set, which sends a local Artin algebra (A,m)(A,m) to the set of Maurer-Cartan solutions in (L Xm) 1(L_X \otimes m)^1 modulo the gauge action of (L Xm) 0(L_X \otimes m)^0, is isomorphic to the functor which sends a local Artin algebra (A,m)(A,m) to the set of isomorphism classes of deformations of XX over SpecA\operatorname{Spec} A. 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 (0,q)(0,q)-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 (0,q)(0,q)-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 A A_\infty 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:

  • Marco Manetti, Deformation theory via differential graded Lie algebras , °(arXiv:0507284)

The Kontsevich and Soibelman references below are also good.

derived algebraic geometry, formal scheme, formal smoothness

References

  • 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

  • Wikipedia: deformation theory, cotangent complex

  • 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)

  • Maxim Kontsevich, Topics in deformation theory (A rough write up of a Berkeley course, early 90-s), 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

  • Alexander I. Efimov, Valery A. Lunts, Dmitri O. Orlov, Deformation theory of objects in homotopy and derived categories

  • 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

  • M. Talpo, A. Vistoli, Deformation theory from the point of view of fibered categories, arxiv/1006.0497

  • 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

  • Robin Hartshorne, Deformation theory, Grad. texts in math. Springer 2010, viii+234 pp. (draft of the chap. I-IV: pdf), gBooks, doi

  • homepage of Ravi Vakil’s graduate Stanford class on deformation theory and moduli spaces

  • Kai Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88, MR98e:14022 arXiv:alg-geom/9601010

Revised on April 7, 2014 08:01:01 by Urs Schreiber (88.128.80.110)