higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
symmetric monoidal (∞,1)-category of spectra
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.
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
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
See also
Clark Barwick, Applications of derived algebraic geometry to homotopy theory, lecture notes, mini-course in Salamanca, 2009, pdf.
For the 2014 installment of UOregon’s Moursund Lecture Series, Jacob Lurie gave three (video recorded) lectures on spectral algebraic geometry
More review:
Last revised on July 23, 2021 at 06:45:23. See the history of this page for a list of all contributions to it.