Deformation theory

Course Notes Fall 1996

DEFORMATION THEORY

Sections

Heuristic computation of first order deformations of an affine curve.

Algebras finite dimensional over an algebraically closed field.

A second heuristic computation of first order deformations.

Moduli.

First order deformations as tangents to the moduli space.

Flatness.

The general set-up for studying infinitesimal deformations.

First order deformations of a commutative algebra.

First order noncommutative deformations; Noncommutative differentiation.

Grobner bases.

First order deformations via Grébner bases.

Commutative Grobner bases and commutative deformations.

Hochschild Cohomology.

An example of an obstructed deformation.

The obstruction i in Hochschild cohomology.

The abstract approach, and why first order deformations are linear.

Universal and versal objects.

A sample computation of a versal deformation.

Application to deformations; Smooth maps in commutative algebra.

Parametrizing finite dimensional algebras.

Groupoids.

The groupoid associated to a family of algebras.

The Amitsur complex.

Descent via a faithfully flat ring homomorphism.

Descent when the tensor products S tensor... S are rings.

Interpretation of descent for extensions of commutative rings.

Forms of a structure.

Sheaves and cohomology.

Azumaya algebras.

Noncommutative deformations of commutative algebras.

Solving polynomial equations.

Rigidity of etale maps.

Noncommutative deformations of commutative polynomial rings.

Deforming smooth algebras.

Flatness of the completion.

Deformations of a commutative power series ring.

Deforming smooth schemes.

Proofs.