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Elliptic Cohomology I

Contents

Contents

This entry is about the text

  • Jacob Lurie, Elliptic Cohomology I: Spectral Abelian Varieties, (pdf)

It is described as the first in a series of papers whose ultimate aim is to carry out the details of the program outlined in A Survey of Elliptic Cohomology.

This first paper provides a discussion of abelian varieties in the setting of spectral algebraic geometry. It sets out to answer the question:

What does it mean to give an elliptic curve (or, more generally, an abelian variety) over a cohomology theory AA? To what extent do such objects behave like their counterparts in classical algebraic geometry?

It is followed up by the further parts

  • Jacob Lurie, Elliptic Cohomology II: Orientations, 2018. 288pp (pdf)

  • Jacob Lurie, Elliptic Cohomology III: Tempered Cohomology, 2019. 286pp (pdf)

  • Jacob Lurie, Elliptic Cohomology IV: Equivariant elliptic cohomology, to appear.

Contents

1 Abelian Varieties in Spectral Algebraic Geometry

  • 1.1 Varieties over E E_{\infty}-Rings
  • 1.2 Abelian Group Objects of \infty-Categories
  • 1.3 Commutative Monoid Objects of \infty-Categories
  • 1.4 Abelian Varieties
  • 1.5 Strict Abelian Varieties

2 Moduli of Elliptic Curves

  • 2.1 Comparing Abelian Varieties with Strict Abelian Varieties
  • 2.2 Deformation Theory of Abelian Varieties
  • 2.3 Deformation Theory of Strict Abelian Varieties
  • 2.4 The Moduli Stack of Elliptic Curves

3 Cartier Duality

  • 3.1 Coalgebra Objects of \infty-Categories
  • 3.2 Duality between Algebra and Coalgebra Objects
  • 3.3 Bialgebra Objects of \infty-Categories
  • 3.4 The Spectrum of a Bialgebra
  • 3.5 The Affine Line
  • 3.6 Smash Products of E E_{\infty}-Spaces
  • 3.7 The Cartier Dual of a Functor
  • 3.8 Duality for Bialgebras
  • 3.9 Duality for Hopf Algebras

4 Biextensions and the Fourier-Mukai Transform

  • 4.1 Line Bundles and Invertible Sheaves
  • 4.2 Biextensions of Abelian Varieties
  • 4.3 Digression: Tannaka Duality
  • 4.4 Biextensions: Tannakian Perspective
  • 4.5 Categorical Digression
  • 4.6 The Convolution Product
  • 4.7 The Fourier-Mukai Transform

5 Duality Theory for Abelian Varieties

  • 5.1 Perfect Biextensions
  • 5.2 Dualizing Sheaves
  • 5.3 Multiplicative Line Bundles
  • 5.4 The Functor 𝒫ic X 0\mathcal{P}ic^0_X
  • 5.5 Representability of the Functor 𝒫ic X m\mathcal{P}ic^m_X
  • 5.6 Existence of the Dual Abelian Variety

6 pp-Divisible Groups

  • 6.1 Finite Flat Group Schemes
  • 6.2 Epimorphisms and Monomorphisms
  • 6.3 Cartier Duality for Finite Flat Group Schemes
  • 6.4 pp-Torsion Objects of \infty-Categories
  • 6.5 pp-Divisible Groups
  • 6.6 Cartier Duality for pp-Divisible Groups
  • 6.7 The pp-Divisible Group of a Strict Abelian Variety
  • 6.8 Comparison of Duality Theories

7 The Serre-Tate Theorem

  • 7.1 Deformation Theory of the Functor RBT h(R)R \mapsto BT_h(R)
  • 7.2 The Case of a Trivial Square-Zero Extension
  • 7.3 Proof of the Serre-Tate Theorem
  • 7.4 Application: Lifting Abelian Varieties from Classical to Spectral Algebraic Geometry
category: reference

Last revised on April 23, 2019 at 05:34:31. See the history of this page for a list of all contributions to it.