nLab spectrum of a tensor triangulated category

Idea

The concept of tensor triangulated category may be thought of as a categorification of the concept of a ring in that the monoidal category structure refines the monoid structure of a ring, and the fact that it is a triangulated category and hence stable makes it the homotopy theory analog of a monoid in abelian groups, hence of a ring.

Accordingly, one may consider the prime spectrum of a tensor triangulated category in direct analogy: the prime ideals are taken to be the thick subcategories which behave as prime ideals under the tensor product, in the obvious way. (Thomason 97, Balmer 02).

In terms of modern stable (∞,1)-category theory this may be understood as being the shadow on homotopy categories of the prime spectrum of a symmetric monoidal stable (∞,1)-category.

A typical application of the spectrum construction for tensor triangulated categories is to the derived categories of quasicoherent sheaves of algebraic varieties. If one restricts to perfect complexes then often the variety may be reconstructed from the spectrum of its category of coherent sheaves. More generally this reconstruction theory applies to geometric stacks, see at Tannaka duality for geometric stacks.

In some good cases such reconstruction works even when forgetting the monoidal structure (Bondal-Orlov 03): The reconstruction theorems of nice classes of schemes from the abelian categories of (quasi)coherent sheaves have versions from weaker data of the corresponding derived categories, viewed as triangulated or enhanced triangulated categories (see triangulated categories of sheaves). This is important for study of deformations, homological mirror symmetry and noncommutative generalizations.

Related concepts

• spectrum of an abelian category

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

References

An important theorem of reconstruction using the triangulated category (but not the monoidal structure) is the Bondal-Orlov reconstruction theorem:

• Aleksei Bondal, Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Mathematica 125 (03), 327-344 doi, pdf

Several spectra for triangulated categories were systematically studied by Rosenberg:

• A. L. Rosenberg, Geometry of Noncommutative ‘Spaces’ and Schemes, pdf

Along the lines of the general spectral cookbook, these spectra are obtained considering certain preorders on objects of the triangulated category. The preorders can be easily modified to respect additional structure, say a monoidal product, if there.

In this vein, Thomason, Balmer, Garkusha and others considered instead tensor triangulated categories and corresponding spectral theory, as well as stronger reconstruction theorems of schemes from derived categories of coherent sheaves (or, in some cases, perfect complexes) taking into account the monoidal structures:

• R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105(1):1–27, 1997 MR98b:18017 doi

• Paul Balmer, Presheaves of triangulated categories and reconstruction of schemes, Mathematische Annalen 324:3 (2002), 557-580 dvi, pdf ps; The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588:149–168, 2005 pdf ps; Spectra, spectra, spectra - Tensor triangular spectra versus Zariski spectra of endomorphism rings, Alg. and Geom. Topology 10:3 (2010) 1521-1563 dvi pdf ps

• Grigory Garkusha, M. Prest, Reconstructing projective schemes from Serre subcategories, J. Algebra 319(3) (2008), 1132-1153, pdf; Triangulated categories and the Ziegler spectrum, Algebras Repr. Theory 8 (2005), 499-523 pdf

• 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)

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

This line has been enhanced by exploring the theory of locales (for the case of quasicompact quasiseparated (i.e. coherent) schemes and derived category of perfect complexes) in

• Joachim Kock, Wolfgang Pitsch, Hochster duality in derived categories and point-free reconstruction of schemes, arxiv/1305.1503

Further discussion in the context of 2-algebraic geometry is in

• Martin Brandenburg, Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)

For the equivariant stable homotopy category:

• Paul Balmer, Beren Sanders, The spectrum of the equivariant stable homotopy category of a finite group (arXiv:1508.03969)

In relation to Drinfeld centers:

• Kent B. Vashaw, Balmer spectra and Drinfeld centers (arXiv:2010.11287)

category: geometry, noncommutative geometry, triangulated categories