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Orlov's conjecture on Rouquier dimension

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The Rouquier dimension of a triangulated category is the minimum number of successive cone operations needed to generate it from a single object.

Conjecture. For every smooth quasi-projective variety XX,

Rdim⁡DbCoh⁡(X)=dim⁡X.\operatorname{Rdim}D^b\operatorname{Coh}(X)=\dim X.

The lower bound Rdim⁡Db(X)≥dim⁡X\operatorname{Rdim}D^b(X)\geq\dim X is known quite generally. The difficult direction is to construct a generator whose generation time is at most dim⁡X\dim X.

Thus the conjecture asks whether ordinary geometric dimension can be recovered purely from the triangulated category. It is known for regular quasi-affine schemes, normal toric varieties, curves, and several other classes, but remains open for general smooth projective varieties.

Equivalent formulations 2Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Rouquier's dimension conjecture for smooth quasi-projective schemes

    Let XX be a smooth quasi-projective scheme of dimension nn. Write Dbcoh⁡(X)D^b\operatorname{coh}(X) for the bounded derived category of coherent sheaves on XX, and let dim⁡Dbcoh⁡(X)\dim D^b\operatorname{coh}(X) denote its triangulated dimension. Rouquier's dimension conjecture.

    dim⁡Dbcoh⁡(X)=n.\dim D^b\operatorname{coh}(X)=n.

    This conjecture proposes that the triangulated dimension of the derived category of coherent sheaves on a smooth quasi-projective scheme equals the scheme's geometric dimension. The preceding result establishes the claim for smooth projective curves, while the general quasi-projective case remains open in the supplied text.

    source: Dmitri Orlov, “Remarks on generators and dimensions of triangulated categories”, arXiv:0804.1163 (2008).

  2. Orlov's conjecture on Rouquier dimension

    Let XX be a smooth projective variety over a field. Its Rouquier dimension is the minimal generation time among all strong generators of Dcoh⁡b(X)D^b_{\operatorname{coh}}(X), and its Krull dimension is denoted dim⁡X\dim X. Orlov's conjecture. The Rouquier dimension of Dcoh⁡b(X)D^b_{\operatorname{coh}}(X) equals the Krull dimension of XX:

    dim⁡Dcoh⁡b(X)=dim⁡X.\dim D^b_{\operatorname{coh}}(X)=\dim X.

    The conjecture relates the categorical complexity of the bounded derived category to the geometric dimension of the variety; its resolution status is not specified in the supplied text.

    source: Pat Lank, “Descent conditions for generation in derived categories”, arXiv:2308.08080 (2024).

algebraic-geometryderived-categoriesrouquier-dimensiontriangulated-categories
Posted by Vihaan Dheer Posted

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