Categorical resolution of singularities

By V. Lunts. 2009

104 citations recorded by the source corpus.

Abstract Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We then propose the definition of a categorical resolution of singularities. Our main example is the derived category D ( X ) of quasi-coherent sheaves on a scheme X. We prove that D ( X ) has a canonical categorical resolution if the base field is perfect and X is a separated scheme of finite type with a dualizing complex.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/fc345731128a2eb5457bb2ff8660dcfb0e147db9

How researchers see this paper

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In Geometry and Topology

1. L. Katzarkov
2. D. Auroux
3. Dmitri Orlov
4. Alexander I. Efimov
5. Nick Sheridan
6. M. Abouzaid
7. I. Smith
8. P. Seidel

In Statistical and Nonlinear Physics

1. L. Katzarkov
2. M. Abouzaid
3. Dmitri Orlov
4. D. Auroux
5. Alexander I. Efimov
6. Alexander Kuznetsov
7. L. Positselski
8. V. Lunts

Citations in the corpus

• Coherent analogues of matrix factorizations and relative singularity categories (2011, result): “ed than the similar category of locally free matrix factorizations of finite rank. Other (and in some way related) arguments in support of the same conclusion are provided by the results of the papers [27, 12] showing that the DG-category corresponding to the absolute derived category of coherent mat”
• Categorical resolutions of irrational singularities (2012, background): “Finally, in papers [17, 18] the notion of a categorical resolution in the context of big categories was considered.”
• SMOOTH AND PROPER NONCOMMUTATIVE SCHEMES AND GLUING OF DG CATEGORIES (2014, background): “odule over A˝ bA. 21 This property depends on the base field k. For example, a finite inseparable extension F Ą k is not smooth over k and it is smooth over itself. The following statement is proved in [Lu] see Lemmas 3.5. and 3.6. Proposition 3.24. If a small DG category A is smooth, then it is regul”
• Homological mirror symmetry for punctured spheres (2011, background): “This Fukaya category, whose objects also include non-compact Lagrangian submanifolds, more accurately reflects the symplectic geometry of open symplectic manifolds, and by recent work [2, 18], is known in some generality to be homologically smooth in the sense of Kontsevich [28] (homological smoothn”
• Non-commutative resolutions of quotient singularities for reductive groups (2015, unknown): “Then the functor D(S)→ D( ) : N → M L ⊗S N is fully faithful and hence yields a categorical resolution of singularities of S in the sense of [33,35,42].”