Dimensions of triangulated categories

By Raphaël Rouquier. 2003

403 citations recorded by the source corpus.

We define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/566313fc117a20c7babbf1f7eac7728f5f993856

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Annihilation of cohomology and strong generation of module categories (2014, background): “The part of this result dealing with rings essentially of finite type over a field was proved by Rouquier [29, Theorem 7.38] when the field is perfect, and extended to the general case by Keller and Van den Bergh [19, Proposition 5.1.2].”
Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I (2006, result): “ion of Xover CP1. This result can be generalized to other Calabi-Yau manifolds. In Z/2-graded case examples of homologically smooth compact A∞-algebras come from Landau-Ginzburg categories (see [Or], [R]) and from Fukaya categories for Fano varieties. Remark 8.2.6 Formal deformation theory of smooth”
Handbook of Tilting Theory: Fourier-Mukai transforms (2004, background): “. Let us now discuss a similar but related problem. For a given scheme X one may want to ﬁnd algebras A derived equivalent to X. One has the following very general result. Theorem 7.5 ([12], see also [66]). Assume that X is separated. Then there exists a perfect complex E such that D(Qcoh(X)) is equ”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, methodology): “es of quasi-coherent sheaves, in the non-equivariant setting. This could be extended to handle our situation. However, this is not the approach we choose. Instead, we follow the method of Rouquier in [Rou08] and focus on Db(Qcoh GX), the bounded derived category of all quasi-coherent G-equivariant s”
Homology of perfect complexes (2006, methodology): “2.1 one gets thickn T(C) op = thickn Top(C op) 2.2.4. The thickenings of C provide a natural ﬁltration of its thick closure in T: {0}= thick0 T(C) ⊆thick 1 T(C) ⊆···⊆ [ n∈Z thickn T(C) = thick (C) In [11, 33] the ﬁltration above is used to deﬁne the dimension of T as the inﬁmum of the integers d ≥0 ”
Derived equivalences and finite dimensional algebras (2006, background): “Let us explain how to associate a dimension to a triangulated category T (cf [88]).”
On triangulated orbit categories (2005, background): “ sometimes says that T is Calabi-Yau of fractional dimension d/e. Note that d ∈ Z is only determined up to a multiple of the order of S. It would be interesting to link the CY-dimension to Rouquier’s [51] notion of dimension of a triangulated category. The terminology has its origin in the following”
Categorical resolution of singularities (2009, background): “e ﬁeld k is algebraically closed and has characteristic p. Then again the Koszul functor KA is a categorical resolution of D(A). 6. Categoricalresolutionforschemes The following theorem was proved in [Rou]. Theorem 6.1. Let X be a separated scheme of ﬁnite type over a perfect ﬁeld. Then there exists”