Deformation Theory
Khai-Hoan Nguyen-Dang
December 2022
Contents
1 Introduction
2
2 Tentative Schedule
4
Abstract
This note is a proposal for the study group on deformation theory.
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1
Introduction
Deformation theory is a study of the way algebraic structures vary locally, i.e
concerning families of objects X (which can be simply schemes, or complicated
structures, like sheaves, bundles, etc) over a scheme S, restricting to a given
object X0 over some point s0 ∈ S.
X0
X
s0
S
For example, let A be a local Artin ring and k be its residue field. Suppose
that X is a parameter space (e.g moduli space), then the maps Spec A → X
correspond to families of objects over Spec A. If we fix a map s0 : Spec k → A
and restrict our attention to A with residue field k, we can consider elements
of the family restricting to k under the natural map Spec k → Spec A, and this
corresponds to studying families of objects over Spec A which restrict to the
fixed object Spec k the reduced point; such families are called (infinitesimal)
deformations of s0 over Spec A. In other words, deformation theory provides a
method that one can go from positive characteristic to characteristic 0.
Initially, Kodaira and Spencer laid a foundational study of the deformation of
complex varieties in the 50s. Their results connected deformation theory with
cohomological algebra. After that, Grothendieck and his school generalized
these ideas to scheme theory in 60s. His approach to this problem was to
formalize the method of Kodaira and Spencer via formal schemes, which consist
of a formal construction followed by a proof of convergence.
X0
Xn
Spec(k)
Spec(A/mn )
From there, deformation theory has been developed and played big roles in
many aspects of mathematics such as Deligne proof on Hodge to absolute Hodge
for abelian varieties, Serre-Tate for ordinary abelian varieties, Mori’s bend and
break techniques, Deformation of Galois representations,... The main references
for classical deformation in algebraic geometry are books by Hartshorne [Har10]
and Sernesi [Ser06].
One vein of our seminar is to study deformation of Galois representations.
Mazur’s theory gives one a universal deformation ring which can be considered
a parameter space for all lifts of a given residual representation (up to conjugation). More precisely, given a continuous 22-dimensional mod pp-representation
ρ : GQ,S → GL2 (Fp )
where GQ,S is the Galois group of the maximal algebraic extension of Q unramified outside the finite set S of primes of Q, if ρ is absolutely irreducible there
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is a complete Noetherian local ring R with residue field Fp and a continuous
homomorphism
ρ : GQ,S → GL2 (R)
which is universal in the natural sense. The ring depends on the residual representation and on supplementary conditions one imposes on the lifts. If the
residual representation is modular and if the deformation conditions are such
that the p-adic lifts satisfy needs that hold for modular Galois representations,
then one expects in many cases that the natural homomorphism R → T from
the universal ring R to a suitably defined Hecke algebra T is an isomorphism.
The proof of such isomorphisms called R = T theorems or modularity theorems
is at the heart of the proof of Fermat’s Last Theorem. It expresses that all p-adic
Galois representations of the type described by R are modular and in particular that they arise from geometry. We will follow lecture notes by [Gou01].
Mazur’s original paper [Maz89] is the main reference. For more explanation
and motivation, look at his introductory paper [Maz97].
Another part of the seminar is to investigate the application of deformation
to abelian schemes. Remarkably, the Serre-Tate theorem tells us that deformations of abelian varieties are basically deformations of its pp-divisible groups.
One of the main points in Serre–Tate theory is the statement that, if AA is
ordinary, its formal deformation space has a canonical structure of a formal
torus over the ring of Witt vectors W (k). In particular, this gives rise to a
canonical lifting over W (k), corresponding to the identity section of the formal
group. The theorem provides us coordinates of local deformations in terms of
Tate modules. To conclude the big picture, the theory of Grothendieck-Messing
will reduce deformations of pp-divisible groups to some linear algebra via the
theory of crystals. We use Shatz’s exposition paper [Sha86] for the first three
talks and also look at Tate’s papers [Tat67], [Tat97]. For formal schemes, a
good source is Illusie’s paper [Ill05]. The main reference for Serre-Tate theory
is Katz’s exposition paper [Kat81] and Chai-Oort’s lecture notes [CO09]. Other
materials will be discussed and added during the seminar.
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2
Tentative Schedule
PART I: Galois Theory
Talk I
• Galois Representations:
Galois groups of infinite algebraic extension. Galois group of Q and their
representations, give some examples.
Speaker:
Talk II
• Deformation of Representations:
Introduction to Witt vectors. Deformation functor and universal deformation.
Speaker:
Talk III
• Cohomological Interpretation:
Representable funtor, fiber product and tangent space.
Speaker:
Talk IV
• Universal Deformation Ring:
Schlessinger’s criteria and the existence of universal deformation rings.
Speaker:
Talk V
• Universal Deformation Ring:
Functorial, cohomological properties. Obstructed and unobstructed problems.
Speaker:
Talk VI
• Explicit Deformations:
Give some computation, and focus on tame cases.
Speaker:
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PART II: Algebraic Geometry
Talk I
• Finite Flat Group Schemes:
Introduction to finite group schemes with examples, highlight on N -torsion
points of abelian varieties. Their basic properties and the connected-étale sequence.
Speaker:
Talk II
• p-divisible Groups:
Introduction to p-divisible group, examples, their basic properties and the
connected-étale sequence. Tate module of p-divisible groups and their properties.
Speaker:
Talk III
• Formal Schemes:
Give definition, examples and basic properties, especially in the affine case.
Formal completion. Formal groups.
Speaker:
Talk IV
• Fppf Sheaf:
View these objects above as fppf sheaves. Cartier duality.
Speaker:
Talk V
• Drinfeld’s Rigidity Lemma:
Present lemma 1.1.3 in Katz’s paper.
Speaker:
Talk VI
• Serre-Tate Deformation Theorem and Local Coordinates:
Theorem 1.2.1 and theorem 2.1 in Katz’s paper
Speaker:
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References
[CO09] Ching-Li Chai and Frans Oort. Moduli of abelian varieties and pdivisible groups. In Arithmetic geometry, volume 8 of Clay Math. Proc.,
pages 441–536. Amer. Math. Soc., Providence, RI, 2009.
[Gou01] Fernando Q. Gouvêa. Deformations of Galois representations. In Arithmetic algebraic geometry (Park City, UT, 1999), volume 9 of IAS/Park
City Math. Ser., pages 233–406. Amer. Math. Soc., Providence, RI,
2001. Appendix 1 by Mark Dickinson, Appendix 2 by Tom Weston
and Appendix 3 by Matthew Emerton.
[Har10] Robin Hartshorne. Deformation theory, volume 257 of Graduate Texts
in Mathematics. Springer, New York, 2010.
[Ill05]
Luc Illusie. Grothendieck’s existence theorem in formal geometry.
In Fundamental algebraic geometry, volume 123 of Math. Surveys
Monogr., pages 179–233. Amer. Math. Soc., Providence, RI, 2005. With
a letter (in French) of Jean-Pierre Serre.
[Kat81] N. Katz. Serre-Tate local moduli. In Algebraic surfaces (Orsay, 1976–
78), volume 868 of Lecture Notes in Math., pages 138–202. Springer,
Berlin-New York, 1981.
[Maz89] B. Mazur. Deforming Galois representations. In Galois groups over Q
(Berkeley, CA, 1987), volume 16 of Math. Sci. Res. Inst. Publ., pages
385–437. Springer, New York, 1989.
[Maz97] Barry Mazur. An introduction to the deformation theory of Galois
representations. In Modular forms and Fermat’s last theorem (Boston,
MA, 1995), pages 243–311. Springer, New York, 1997.
[Ser06] Edoardo Sernesi. Deformations of algebraic schemes, volume 334 of
Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006.
[Sha86] Stephen S. Shatz. Group schemes, formal groups, and p-divisible
groups. In Arithmetic geometry (Storrs, Conn., 1984), pages 29–78.
Springer, New York, 1986.
[Tat67] J. T. Tate. p-divisible groups. In Proc. Conf. Local Fields (Driebergen,
1966), pages 158–183. Springer, Berlin, 1967.
[Tat97] John Tate. Finite flat group schemes. In Modular forms and Fermat’s
last theorem (Boston, MA, 1995), pages 121–154. Springer, New York,
1997.
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