nLab spectral algebraic geometry



Higher geometry

Higher algebra



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.


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

See also


The foundations of the theory are developed in

A textbook account is developing in

See also

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 May 2, 2023 at 09:13:47. See the history of this page for a list of all contributions to it.