nLab triangulated category of singularities




A derived category of coherent sheaves on singularities in a variety.

For a smooth variety XX then the full bounded derived category D b(X)=D b(CohX)D^b(X) = D^b(Coh X) coincides with that of perfect complexes, while for a singular variety there are objects not represented by perfect complexes. Hence these may be attributed to be due to the contribution of the singularities. The quotient D b(X)/Perf(X)D^b(X)/Perf(X) hence serves as the derived category of the singularities themselves.

×\mathbb{C}^\times-equivariant singularity category is in good cases equivalent to the category of matrix factorizations, which is the category of B-branes in a Landau-Ginzburg theory.

One sometimes considers a dg-enhancement (which is unique by Orlov-Lunts) of the derived category instead, the dg-category of singularities.


The concept is due to

  • Dmitri Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. 2004, no. 3 (246), 227–248 (arXiv:math/0302304)

  • Dmitri Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, Arithmetic, and Geometry: In Honor of Yu.I. Manin, vol. II, Progr. Math. 270, Birkhäuser 2009, pp. 503-531 (arXiv:0503632)

  • Dmitri O. Orlov, Triangulated categories of singularities and equivalences between Landau-Ginzburg models, Sbornik: Mathematics 197:12 (2006) 1827 doi

Idempotent completions of triangulated categories of singularities of two schemes are equivalent if the formal completions of these schemes along singularities are isomorphic:

  • Dmitri Orlov, Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math. 226:1 (2011) 206-217 doi

The algebraic K-theory of these categories of singularities is considered in

If a variety YY over a field kk is the zero scheme of a section of a vector bundle on a smooth kk-variety, then there is a construction of a singular variety ZZ so that k ×k^\times-equivariant/graded singularity category of ZZ is equivalent (as an enhanced triangulated category) to the bounded derived category of coherent sheaves on YY:

  • M. Umut Isik, Equivalence of the derived category of a variety with a singularity category, International Mathematics Research Notices 2013, no. 12, pp. 2787-2808, doi arXiv:1011.1484

There is also a relative singularity category (usually defined in terms of some ring and module theoretic data) related to matrix factorizations

Last revised on July 28, 2023 at 16:12:58. See the history of this page for a list of all contributions to it.