nLab prime spectrum of a symmetric monoidal stable (∞,1)-category

Idea

The categorification of the concept of (commutative) ring, hence the concept of (commutative) 2-ring in homotopy theory/(∞,1)-category theory may be taken to be that of (symmetric) monoidal stable (∞,1)-category.

In analogy to the prime spectrum of a commutative ring, the prime spectrum of a symmetric monoidal stable $(\infty,1)$-category is the collection of thick subcategories which are prime ideals, in the evident higher sense, under the given tensor product. On the level of homotopy categories this concept was introduced in (Balmer 04) as the spectrum of a tensor triangulated category.

Examples

The prime spectrum of the category of spectra

The prime spectrum of the (∞,1)-category of spectra (p-local and finite) is labeled by the Morava K-theories (Balmer 10, prop. 9.4). This follows from the the thick subcategory theorem.

This is the topic of chromatic homotopy theory, see at Spec(S).

Generally, the kernel of any generalized homology theory here is a thick ideal, but not generally a prime ideal.

Related concepts

• spectrum of a tensor triangulated category

• 2-algebraic geometry

References

The concept was introduced at the level of triangulated categories in

• Paul Balmer, The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math., 588:149–168, 2005 (arXiv:math/0409360)

• Paul Balmer, Spectra, spectra, spectra—tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebr. Geom. Topol., 10(3):1521–1563, 2010 (pdf)

Review includes

• Greg Stevenson, Tensor actions and locally complete intersection PhD thesis 2011 (pdf)

Exposition with an eye towards application in chromatic homotopy theory in general and tmf in particular is in

• Aaron Mazel-Gee, You could’ve invented $tmf$, April 2013 (pdf slides)

Further discussion is in

• Beren Sanders, Higher comparison maps for the spectrum of a tensor triangulated category (arXiv:1302.4521)

See also

• Jack Morava, Toward a fundamental groupoid for the stable homotopy category (arXiv:math/0509001)