nLab triangulated category of singularities

Idea

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

For a smooth variety $X$ then the full bounded derived category $D^b(X) = D^b(Coh X)$ 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 $D^b(X)/Perf(X)$ 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.

References

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

• Goncalo Tabuada, Algebraic K-theory with coefficients of cyclic quotient singularities, Comptes Rendus Mathematique 354:5 (2016) 449-452 (arXiv:1512.01205 doi)

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

• 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

• Alexander I. Efimov, Leonid Positselski, Coherent analogues of matrix factorizations and relative singularity categories, Algebra and Number Theory, 9(5):1159-1292 (2015) arXiv:1102.0261
• Massimo Pippi, On the structure of dg categories of relative singularities, High. Struct. 6:1 (2022) 375–402 arXiv:1911.01332