Formal completions and idempotent completions of triangulated categories of singularities

By Dmitri Orlov. 2009

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Abstract The main goal of this paper is to prove that the idempotent completions of triangulated categories of singularities of two schemes are equivalent if the formal completions of these schemes along singularities are isomorphic. We also discuss Thomason's theorem on dense subcategories and a relation to the negative K-theory.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/01852c26c63db185ce8d71bab0afc8efdd2d5131

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A geometric approach to Orlov’s theorem (2010, background): “Orlov [Orl09b] has obtained a similar theorem in the case of categories of singularities.”
Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras (2011, unknown): “gard Dsg(X) as a full subcategory of D SG(X). The aim of this subsection is to prove the following result, which plays a crucial role in this paper. Note that part (2) also follows from work of Orlov [O09], and is well-known to experts [BK]. Theorem 3.2. Suppose that X is a Gorenstein scheme of dime”
Singularity categories of gentle algebras (2012, background): “2) and Orlov’s localization theorem [20], the singularity category of Xn may be described as follows:”
Homological mirror symmetry for the genus two curve (2008, background): “More recently, Katzarkov [28, 33] has proposed a further extension of Homological Mirror Symmetry, encompassing some varieties of general type.”
Matrix factorizations for nonaffine LG–models (2011, background): “In [11] we show that for any two schemes X and X ′ satisfying (ELF), whose formal completions X and X along the singularities are isomorphic, we have that the idempotent completions of the triangulated categories of singularities DSg(X) and DSg(X ′) are equivalent.”
Flops and clusters in the homological minimal model programme (2014, background): “[O09] D. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, Adv.”
Compact generators in categories of matrix factorizations (2009, methodology): “that Corollary 4.3 was ﬁrst proven by Schoutens in [Sch03]. An independent proof by Murfet will be contained as an appendix in [KVdB08]. It is also possible to deduce the statement using results from [Orl09] as explained in [Sei08, 11.1]. However, Theorem 3.1 is stronger since it also implies that t”
Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (2010, background): “,w) can be viewed as a full triangulated subcategory of the perfect derived category of matrix factorizations Per(MF(R,w)) and is isomorphic to it when Ris regular and complete by [10, Thm. 5.7] (see [38] for a more general statement). For a pair of elements a,b∈ Rdenote by {a,b} the matrix factoriz”