nLab Elliptic Cohomology I

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 $A$? 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 $E_{\infty}$-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 $E_{\infty}$-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 $\mathcal{P}ic^0_X$
• 5.5 Representability of the Functor $\mathcal{P}ic^m_X$
• 5.6 Existence of the Dual Abelian Variety

6 $p$-Divisible Groups

• 6.1 Finite Flat Group Schemes
• 6.2 Epimorphisms and Monomorphisms
• 6.3 Cartier Duality for Finite Flat Group Schemes
• 6.4 $p$-Torsion Objects of $\infty$-Categories
• 6.5 $p$-Divisible Groups
• 6.6 Cartier Duality for $p$-Divisible Groups
• 6.7 The $p$-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 $R \mapsto BT_h(R)$
• 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