nLab spectral algebraic geometry

Idea

Spectral algebraic geometry (or maybe E-∞ geometry) is the theory of homotopical algebraic geometry specialized to the (infinity,1)-category of spectra. Hence it is a generalization of ordinary algebraic geometry where instead of commutative rings, spectral schemes are locally modelled on commutative ring spectra.

Applications

Elliptic cohomology

Historically, the first application of spectral algebraic geometry was in the study of elliptic cohomology and topological modular forms. In particular it allowed the construction and study of the tmf spectrum as a certain derived moduli stack of derived elliptic curves. This construction is based on the Artin-Lurie representability theorem. See

• Jacob Lurie, A Survey of Elliptic Cohomology
• Jacob Lurie, Elliptic Cohomology I

See also

• derived algebraic geometry
• brave new algebra
• E-infinity ring

References

The foundations of the theory are developed in

• Jacob Lurie, DAG VII: Spectral schemes, pdf.

• Jacob Lurie, DAG VIII: Quasi-coherent sheaves and Tannaka duality theorems, pdf.

• Jacob Lurie, DAG IX: Closed immersions, pdf.

• Jacob Lurie, DAG XI: Descent theorems, pdf.

• Jacob Lurie, DAG XII: Proper morphisms, completions, and the Grothendieck existence theorem, pdf.

• Jacob Lurie, DAG XIV: Representability theorems, pdf.

A textbook account is developing in

• Jacob Lurie, Spectral Algebraic Geometry (pdf)

See also

• Clark Barwick, Applications of derived algebraic geometry to homotopy theory, lecture notes, mini-course in Salamanca, 2009, pdf.

• Paul Goerss, Topological Algebraic Geometry - A Workshop

For the 2014 installment of UOregon’s Moursund Lecture Series, Jacob Lurie gave three (video recorded) lectures on spectral algebraic geometry

• Jacob Lurie, Theory of ‘Spectral’ Algebraic Geometry

• Jacob Lurie, Ambidexterity

• Jacob Lurie, Roots of Unity in Stable Homotopy Theory

More review:

• Charles Rezk, Spectral algebraic geometry (pdf) in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds.), Stable categories and structured ring spectra, MSRI Book Series, Cambridge University Press