Uniqueness of enhancement for triangulated categories

By V. Lunts, Dmitri Orlov. 2009

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The paper contains general results on the uniqueness of a DG enhancement for trian- gulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded de- rived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de- rived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a42446cacf0b09fbf86fdc8ecfb848f7dc87a0ad

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, background): “ ×···×Xt), which by our Theorem 7.3 and Lemma 7.4 is a component in the semi-orthogonal decomposition of Db(cohZ), in the mirror, this category corresponds to the deep singular ﬁber at the origin (by [LO10], we need not be careful about the choice of dg-structure on Db(cohX i)). We arrive at the fol”
ON EQUIVARIANT TRIANGULATED CATEGORIES (2014, background): “od geometric situation: for smooth projective algebraic variety Xand an autoequivalence of Db(coh(X)) given by a kernel on X× X, it is not clear how to do it. It is known (see Lunts and Orlov’s paper [15]) that for a projective variety X, DG-enhancement of Db(coh(X)) is strongly unique. But this uni”
Chow motives versus non-commutative motives (2011, background): “The gap between algebraic varieties and dg categories can be bridged by associating to every smooth projective variety X its dg category D perf(X) of perfect complexes of OX -modules; see Lunts-Orlov [31] or [13, Example 4.”
On the Fukaya category of a Fano hypersurface in projective space (2013, result): “(equivalence of the homotopy categories is proven in [28], and quasi-equivalence as DG categories then follows from the results of [23]).”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “DOI 10.1063/1.2768185 [5] Ballard, M., Favero, D., Katzarkov, L.: Orlov spectra: bounds and gaps.”
A Guided Tour Through the Garden of Noncommutative Motives (2011, background): “1Consult Lunts-Orlov [35] for the uniqueness of this enhancement.”
Spherical DG-functors (2013, methodology): “ries Aand Band the (non-triangulated) category ExFun(A;B) of exact functors between them, one should work with enhancements Aand Bof Aand Bin Ho(DG-Cat) (which are often unique up to isomorphism, cf. [LO10]), the enhanced triangulated category RHom(A;B) and the functor H0(RHom(A;B)) ! ExFun(A;B). Fo”