nLab
deformation theory

Idea

Deformation theory studies problems of extending structures to extensions of their domains. Formal deformation theory, is the part where the extensions are infinitesimal.

A typical problem in deformation theory has the structure that

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

• and infinitesimal thickenings $\tilde{X}$ and $\tilde{Y}$ of $X$ and $Y$ are prescribed, with injection morphisms $X\to \tilde{X}$ and $Y\to \tilde{Y}$

and asks whether a bottom horizontal morphism $\tilde{f}$ in the diagram

$$\begin{array}{ccc}X& \stackrel{f}{\to }& Y\\ \downarrow & & \downarrow \\ \tilde{X}& \stackrel{\tilde{f}}{\to }& \tilde{Y}\end{array}$$\array{ X &\stackrel{f}{\to}& Y \\ \downarrow && \downarrow \\ \tilde X &\stackrel{\tilde f}{\to}& \tilde Y }

may be found. This morphism $\tilde{f}$ would be called an infinitesimal deformation of $f$.

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 $R$ be a ring, to be thought of as the ring of functions on the space $X$ in the above diagram. Let furthermore $N$ be an $R$-module, to be thought of as the $R$-modules of sections of a vector bundle over $X$.

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

$$(r_1,n_1)\cdot (r_2,n_2)=(r_1{r}_2,r_1{n}_2+n_2{r}_1).$$(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 $R$ by $N$. It should be thought of as the algebra of functions that consists of elements of $R$ and $N$, where the elements in $N$ are thought of as functions with values in infinitesimal quantities, so that their would-be product ”$n_1\cdot n_2$” vanishes.

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

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

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

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

$$\begin{array}{ccc}{R}_1& \stackrel{f^{*}}{\leftarrow }& R_2\\ \uparrow & & \uparrow \\ R_1\oplus N_1& & R_2\oplus N_2\end{array}.$$\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 $f^{*}$ to a morphism $R_1\oplus N_1\leftarrow R_2\oplus N_2:{\tilde{f}}^{*}$ of rings, such that we get a commuting diagram

$$\begin{array}{ccc}{R}_1& \stackrel{f^{*}}{\leftarrow }& R_2\\ \uparrow & & \uparrow \\ R_1\oplus N_1& \stackrel{{\tilde{f}}^{*}}{\leftarrow }& R_2\oplus N_2\end{array}.$$\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 $R_1$.

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

$$p:\mathrm{Mod}\to \mathrm{Ring}$$p : Mod \to Ring

for the bifibration of modules over rings: objects of $\mathrm{Mod}$ are pairs consisting of a ring $R$ an an $R$-module $N$, and morphism $(R_1,N_1)\to (R_2,N_2)$ are pairs consisting of a ring homomorphism $f:R_1\to R_2$ and a morphism $F:N_1\to N_2{\otimes }_f{R}_2$ of $R_2$-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 $G:\mathrm{Mod}\to \mathrm{Ring}$ of interest: for $N$ any $R$-module, we may form the ring $G(N):=R\oplus N$ called the square 0-extension of $R$, in which multiplication is given by

$$(r_1,n_1)\cdot (r_2,n_2):=(r_1{r}_2,n_1{r}_2+n_2{r}_1).$$(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 $G(N)\to R$ given by sending $(r,n)\mapsto r$. A section $v:R\to G(n)$ of this morphism is precisely a derivation of $R$ with values in the module $N$.

This may be organized into a single functor

$$\mathrm{Mod}\to [I,\mathrm{Ring}]$$Mod \to [I,Ring]

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

$$\begin{array}{ccccc}\mathrm{Mod}& & \stackrel{p}{\to }& & \mathrm{Ring}\\ & \searrow & & {\nearrow }_{d_1}\\ & & [I,\mathrm{Ring}]\end{array}.$$\array{ Mod &&\stackrel{p}{\to}&& Ring \\ & \searrow && \nearrow_{\mathrlap{d_1}} \\ && [I,Ring] } \,.

Finally notice that the functor $G$ has a left adjoint functor

$$\Omega :\mathrm{Ring}\to \mathrm{Mod}$$\Omega : Ring \to Mod

that sends a ring $R$ to the $R$-module ${\Omega }_K(R)$ 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 $f:A\to B$ of rings, which makes $B$ an $A$-algebra, the category $\mathrm{AbGr}(A-\mathrm{Alg}/B)$ of abelian group objects in the slice category $A$-$\mathrm{Alg}/B$ of $A$-algebras over $B$ is equivalent both to the category of $B$-modules and the trivial (= square zero) extensions of $A$ by $B$-modules. In particular we can consider the forgetful functor $\mathrm{AbGr}(A-\mathrm{Alg}/B)\to A-\mathrm{Alg}/B$ which has a left adjoint ${\mathrm{Ab}}_{B/A}:A-\mathrm{Alg}/B\to \mathrm{AbGr}(A-\mathrm{Alg}/B)\cong {}_B\mathrm{Mod}$. All said is true for simplicial commutative rings as well. Now the relative cotangent complex $L_{B/A}$ is the value on $B$ of the left derived functor $𝕃{\mathrm{Ab}}_{B/A}(B)$. 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 $C$ by replacing the bifibration $\mathrm{Mod}\to \mathrm{Ring}$ by the stabilization $T_C\to C$ of the codomain fibration of $C$: the tangent (∞,1)-category of $C$.

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

$$\Omega :C\to T_C$$\Omega : C \to T_C

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

$$C\stackrel{\Omega }{\to }{T}_C\stackrel{p}{\to }C$$C \stackrel{\Omega}{\to} T_C \stackrel{p}{\to} C

is the identity (∞,1)-functor.

Further categorification

…

Related entries

Related $n$lab entries include cotangent complex, Maurer-Cartan equation, derived algebraic geometry, formal scheme, formal smoothness. Deformation problems are often phrased in terms of differential graded Lie algebras, and, more generally, L-infinity algebras. See also discussion at MathOverflow: def theory and dgla-s.

References

• M. Doubek, M. Markl, P. Zima, Deformation theory (lecture notes), Archivum mathematicum 43 (5), 2007, 333–371, arXiv:0705.3719

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

• E. Sernesi, An overview of classical deformation theory, pdf

In

• Jacob Lurie, Deformation Theory (arXiv:0709.3091)

a very setup for deformation theory over any (∞,1)-category is described. Then as an application the deformation theory of E-∞-rings is developed.

• Yan Soibelman, Lectures on deformation theory and mirror symmetry (ps)

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

• 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

• L. 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.

• Alexander Grothendieck, Catégories cofibrées additives et complexe cotangent relatif, Lecture Notes in Mathematics 79

• E. Sernesi, Deformations of algebraic schemes (monograph) Grundlehren der Math. Wiss. 334, Springer 2006. xii+339 pp. MR2247603 (2008e:14011)

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

  • I: general theory arXiv:math/0702838;
  • II: pro-representability of the deformation functor arXiv:math/0702839;
  • III: abelian categories arXiv:math/0702840

• Wikipedia: deformation theory, cotangent complex

• C. Doran, Deformation theory: a historical annotated bibliography

• Kai Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88, MR1437495 (98e:14022) arXiv:alg-geom/9601010

• B. Fantechi, M. Manetti, Obstruction calculus for functors of Artin rings I, J. Algebra 202 (1998), no. 2, 541–576, MR1617687 (99f:14004).

• Martin C. Olsson, Deformation theory of representable morphisms of algebraic stacks, Mathematische Zeitschrift 253, n. 1, 25–62 (2006) doi:10.1007/s00209-005-0875-9

• S. Merkulov, B. Vallette, Deformation theory of properads, arXiv:0707.0889

• V. Hinich, DG coalgebras as formal stacks, J. Pure Appl. Algebra 162 (2001), no. 2-3, 209–250 (doi), math.AG/9812034.

• Vladimir Hinich, Formal deformations of sheaves of algebras, video of a talk at MSRI 2002, link

• 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.

• 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

• Kodaira K., Nirenberg L., Spencer D.C. On the existence of deformation of complex analytic structures, Ann. Math. 68, 450-459 (1958).

• Kodaira K., Spencer D.C. 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