Rouquier dimension of some blow-ups

Dmitrii Pirozhkov

REVIEW 3 major objections 4 minor 6 references

Rouquier dimension of some blow-ups

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that blow-ups of projective space in at most three points or codimension-two linear subspaces have Rouquier dimension equal to their ordinary dimension, confirming Orlov's conjecture for these varieties.

desk verdict A solid short note proving new cases of Orlov's conjecture; the reader's main worry about Lemma 2.4 is a false alarm, and the paper deserves a careful referee. read the letter →

arxiv 1908.08283 v1 pith:R6YPLBXT submitted 2019-08-22 math.AG

Dmitrii Pirozhkov This is my paper · ORCID

classification math.AG MSC 14F0814E05

keywords RouquierdimensionOrlov'sconjecturesemiorthogonaldecompositionblow-upderivedcategoryADEquiverdelPezzosurfacesexceptionalcollection

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that certain blow-ups of projective space satisfy Orlov's conjecture: the Rouquier dimension of the bounded derived category of coherent sheaves equals the usual dimension of the variety. The main theorem covers the blow-up of $\mathbb{P}^n$ in at most three disjoint linear subspaces, each either a point or of codimension two. It also covers towers of small blow-ups, including a blow-up of $\mathbb{P}^2$ in up to nine arbitrary distinct points. The proof works by decomposing the derived category into pieces of Rouquier dimension zero, rather than only into exceptional line bundles, so the sharp upper bound survives even though the exceptional collection grows longer.

What carries the argument

The load-bearing construction is a semiorthogonal decomposition of the blow-up's derived category obtained by mutating Orlov's blow-up decomposition. The key objects are exceptional sheaves supported on the exceptional divisors that have exactly one nontrivial morphism space from a chosen line bundle; together with that line bundle they generate a subcategory equivalent to the derived category of an ADE quiver, in particular the $D_4$ quiver, which has Rouquier dimension zero. Proposition 3.1 groups the exceptional objects coming from up to three blow-up centers into such zero-dimensional quiver categories, preserving the sharp bound from the glueing estimate.

What would settle it

Search the derived category of representations of the $D_4$ quiver for a bounded complex that is not quasi-isomorphic to a direct sum of shifts of indecomposable representations; finding one would invalidate the proof of Lemma 2.4 and remove the certification that the decomposition's components have Rouquier dimension zero. Failing to find one, as expected from the hereditary structure, would support the paper's chain of reasoning.

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Extended reading notes

Core claim

The central claim is Theorem 4.1: if $\{Z_b\}_{b\in B}$ is a set of at most three disjoint linear subspaces of $\mathbb{P}^n$, each a point or of codimension two, and $Y$ is the blow-up of $\mathbb{P}^n$ in their union, then $\operatorname{rdim} Y = n$. The proof constructs a semiorthogonal decomposition of $D^b_{\mathrm{coh}}(Y)$ into two exceptional line bundles and $n-1$ subcategories each equivalent to the derived category of representations of a $D_4$ quiver, hence of Rouquier dimension zero. The glueing estimate for semiorthogonal decompositions then gives the upper bound $\operatorname{rdim} Y \le n$, while smoothness gives the lower bound $n \le \operatorname{rdim} Y$. Low-dimensional corollaries include $\operatorname{rdim} = 2$ for a blow-up of $\mathbb{P}^2$ in up to nine arbitrary points, which covers all del Pezzo surfaces, and $\operatorname{rdim} = 3$ for a two-level tower of point and line blow-ups starting from $\mathbb{P}^3$.

Load-bearing premise

The argument's load-bearing premise is that the derived category of finite-dimensional representations of an ADE quiver, in particular the $D_4$ quiver, has Rouquier dimension zero, with the proof as written relying on the step that every bounded complex over such a quiver splits into shifts of representations.

Editorial extensions

If this is right

Blowing up $\mathbb{P}^n$ in at most three disjoint points or codimension-two linear subspaces produces a variety with Rouquier dimension exactly $n$.
A blow-up of $\mathbb{P}^2$ in up to nine arbitrary distinct points has Rouquier dimension $2$, so all del Pezzo surfaces satisfy Orlov's conjecture.
A three-level tower of blow-ups of $\mathbb{P}^2$ in at most three points per level satisfies the conjecture, as does a two-level tower of point and line blow-ups of $\mathbb{P}^3$.
The decomposition strategy preserves two exceptional line bundles from the full exceptional collection on projective space, which is what allows the sharp bound to be iterated over several blow-ups in low dimension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same decomposition strategy should apply to any smooth $n$-fold carrying a full exceptional collection of length $n+1$ whose last $n-1$ bundles restrict to full exceptional collections on the blow-up centers; the projective-space assumption is likely convenient rather than essential.
Because the zero-dimensional components are ADE quiver categories, the groupability into $D_4$ quivers is the main structural obstruction; if a different ADE grouping could be arranged, the bound might extend to configurations with more centers.
The general pattern suggests a sufficient criterion: any blow-up whose new exceptional objects can be organized into finitely many ADE quiver categories inherits the sharp Rouquier dimension bound from its base variety.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid short note, and the reader's main worry—that Lemma 2.4 is false—doesn't survive contact with the paper. The lemma is true, the proof is just compressed. The paper proves Orlov's conjecture for a plane blown up in up to nine arbitrary points and for some higher-dimensional blow-ups of P^n in up to three points or codimension-two linear subspaces. The nine-point case is genuinely new; earlier del Pezzo results stopped at eight. The ADE-quiver grouping of exceptional objects is a nice new idea and gives a shorter route to the known del Pezzo cases too.

What is actually new: Theorem 4.1 and the tower propositions. The method is to look for subcategories of Rouquier dimension zero beyond the usual exceptional-object categories: derived categories of ADE quivers. That is a neat observation and is applied cleanly through Orlov's decomposition and Bondal mutability. The paper is self-contained against standard tools (Rouquier's bound, Orlov's blow-up decomposition, Gabriel's theorem) and has no fitted parameters or circular claims.

Soft spots: The proof of Lemma 2.4 is too terse. 'Homological dimension one' does imply that every bounded complex in a hereditary category is isomorphic to a direct sum of shifts of its cohomology objects—but only because Ext^2 vanishes and the truncation triangles split; the reader's non-split two-term complex is not actually a counterexample. Still, a referee should ask the author to add a sentence of justification or a citation. Proposition 3.1's mutation and regrouping steps are also compressed; I verified that the mutation of T through the lower (S_b) objects behaves as claimed, but 'repeating the argument n−2 times' hides a few checks about RHom vanishing after mutation. These are presentation gaps, not mathematical ones. Minor typos like 'cardinatly' are present but harmless.

Bottom line: it's a carefully reasoned note with a reusable technical point. People in derived algebraic geometry and representation theory will want it. I would send it to review; with small revisions to expand Lemma 2.4 and Proposition 3.1, it's ready.

Referee Report

3 major / 4 minor

Summary. The paper proposes a method for proving Orlov's conjecture for certain blow-ups of projective spaces. The main technical device is to construct semiorthogonal decompositions of the blow-up whose components have Rouquier dimension zero, where the zero-dimensional components are derived categories of ADE quiver representations. The announced results are Theorem 4.1 (blow-ups of P^n in at most three disjoint linear subspaces, each a point or of codimension two), Proposition 4.2 (towers of three blow-ups of P^2 in at most three points each), and Corollary 4.3 (blow-ups of P^2 in up to nine arbitrary points), together with a P^3 analogue. The proof relies on an appendix that computes a 'dual' of Orlov's semiorthogonal decomposition for a point blow-up, and on a proposition that groups exceptional objects into quiver categories of Dynkin type.

Significance. If the main results were established, they would give new cases of Orlov's conjecture, including a uniform treatment of del Pezzo surfaces and some higher-dimensional blow-ups, and the use of ADE quiver categories as Rouquier-dimension-zero building blocks is a natural and potentially useful idea. The paper is concise and self-contained, and Lemma 2.4, despite a terse proof, is essentially correct via standard formality of hereditary categories. However, the central semiorthogonal decomposition used in the proof is not valid as stated, and the main theorems are not established by the argument presented.

major comments (3)

[Appendix A, Prop. A.2] The claimed semiorthogonal decomposition is false as stated. For a point blow-up in a surface (n=2), the claimed decomposition is <π*D^b(X), τ≥0 π*O_x>, and τ≥0 π*O_x is isomorphic to O_E, the structure sheaf of the exceptional divisor. The definition of semiorthogonal decomposition in Section 2 requires RHom(B,A)=0 for A in the left component and B in the right component. Taking X=P^2 and F=O(2), we have Hom_Y(O_E, π*O(2)) ≅ H^0(P^1, O_{P^1}(2)) ≠ 0, so the pair is not semiorthogonal. The proof of Prop. A.2 only verifies orthogonality inside the block T_k generated by the exceptional-divisor line bundles; it never verifies the required vanishing between the truncations τ≥-k π*O_x and the subcategory π*D^b(X). Moreover, the ordering contradicts the 'right dual' procedure defined in Section 2, which moves the mutated components to the left of all original components. Since Lemma 3.2, Proposition 3.1, and Theorem 4.1 all depend on this decomposition, the main proof is not valid.
[Section 3, proof of Prop. 3.1] The identification of the subcategory T=<L_{n-2}, {(S_b)_{n-2}}> with the derived category of a Dynkin quiver requires vanishings in the correct direction. The lemmas establish only RHom_Y(L_i,S_i)=k[0], i.e., morphisms from the earlier object to the later object. Semiorthogonality in the order <..., L_i, ..., S_i, ...> instead requires RHom_Y(S_i,L_i)=0, as well as RHom_Y(S_i,L_j)=0 for j≠i and vanishing of higher Ext groups. In the point case these reverse vanishings are not true in general: for n=2 and L_0=O(2), S_0=O_E, we have Hom_Y(O_E,O(2)) ≠ 0. Thus the grouping into a quiver category is not justified, even if one attempted to repair the preceding decomposition.
[Section 3, proof of Prop. 3.1, mutation step] The step 'Let T0 be the mutation of T through the subcategory ...' does not specify whether a left or right mutation is intended, and no orthogonality conditions needed for that mutation are verified. Because the preceding semiorthogonal decomposition is invalid, the iterative construction of T0,...,T_{n-2} is not established. This is load-bearing: the upper bound rdim(Y) ≤ n in Theorem 4.1 is obtained by applying Lemma 2.2 to the components constructed by this mutation process.

minor comments (4)

[Lemma 2.4] The sentence 'the path algebra has homological dimension one, therefore any object of D^b(Q-rep) is quasiisomorphic to a direct sum of shifts of representations' is a formality claim that is true for hereditary abelian categories but not immediate from the quoted homological-dimension fact. Please add a sentence explaining that the truncation triangles split because Ext^{≥2}=0.
[Lemma 3.2, proof] The displayed conclusion 'RHom_Y(L_i, τ≥i π*O_x) ≅ k[0]' appears to contain a sign error: it should presumably be τ≥-iπ*O_x.
[Throughout] There are several typos: 'cardinatly' in Proposition 3.1, 'Rouquier dimenson' in the introduction, and the index set written as 'b∈b' instead of 'b∈B' in the proof of Proposition 3.1.
[Proposition 4.2] In the first mutation step the text says 'through the two exceptional line bundles' but the displayed decomposition has three components; it would help to explicitly name which line bundle serves as L_0 when the second blow-up argument is applied.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and does not reduce to its inputs.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The upper bounds on Rouquier dimension are obtained by constructing semiorthogonal decompositions whose components have Rouquier dimension zero, then applying the standard glueing estimate Lemma 2.2 from Rouquier. Proposition 3.1 groups exceptional objects into subcategories equivalent to derived categories of ADE quiver representations; Lemma 2.4 proves those have Rouquier dimension zero using homological dimension one and Gabriel's theorem. None of these ingredients is defined in terms of the target equality rdim Y = n, and no parameter is fitted to the target examples. The cited results (Rouquier, Orlov, Bondal, Beilinson, Gabriel, Ballard-Favero, Kapranov) are external foundational theorems, and the author has no self-citations that carry any load-bearing premise. The only passage one might question is the sentence in Lemma 2.4 claiming that homological dimension one implies every object is a direct sum of shifts of representations; that is a proof-quality concern about a true statement for hereditary categories, not a circularity, because the lemma does not presuppose the Rouquier dimension conclusion of the paper. The final paragraph honestly states the limits of the method (the D4-quiver obstruction for large codimension), which further confirms that the argument is not being forced by a hidden premise. Overall, no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

Pure math paper; no free parameters or new entities. The proof relies on a chain of standard theorems; the weakest link is the proof of Lemma 2.4, whose stated justification is mathematically false, even though the lemma's content is true.

assumptions (6)

standard math Rouquier's bounds: for smooth n-dimensional X, n ≤ rdim D^b(X) ≤ 2n
Used in Theorem 4.1, Prop 4.2 and Prop 4.4 to convert the upper bound into equality. Established in Rouquier, 'Dimensions of triangulated categories'.
standard math Orlov's blow-up semiorthogonal decomposition
Used throughout Section 3 and Appendix A to compute the derived category of a blow-up. Cited as [Orl93].
standard math Bondal's mutation calculus and the equivalence between the subcategory generated by an exceptional collection and D^b of its Ext-quiver
Used in Proposition 3.1 to identify T with D^b of a star quiver and to justify mutations. Cited as [Bon89, Th. 6.2].
standard math Gabriel's theorem: the path algebra of an ADE quiver has finitely many indecomposable representations
Used in Lemma 2.4 to conclude finiteness of indecomposables. The step requiring all D^b indecomposables to come from modules is not justified.
standard math Bott vanishing / cohomology of Ω^m(m) on projective space
Used in Lemma 3.2 and Appendix A to compute RHom between line bundles and exceptional objects supported on the exceptional divisor.
standard math Path algebra of a quiver is hereditary
Used in Lemma 2.4; alone it does not imply the splitting claim the paper makes.

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Pith. "Pith review of Rouquier dimension of some blow-ups." pith.science (2026). https://pith.science/paper/R6YPLBXT

@misc{pith2026190808283,
  author       = {Pith},
  title        = {Pith review of: Rouquier dimension of some blow-ups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6YPLBXT}},
  note         = {Machine review of arXiv:1908.08283}
}

abstract

Rapha\"{e}l Rouquier introduced an invariant of triangulated categories which is known as Rouquier dimension. Orlov conjectured that for any smooth quasi-projective variety $X$ the Rouquier dimension of $D^b_{\mathrm{coh}}(X)$ is equal to $\mathrm{dim}\, X$. In this note we show that some blow-ups of projective spaces satisfy Orlov's conjecture. This includes a blow-up of $\mathbb{P}^2$ in nine arbitrary distinct points, or a blow-up of three distinct points lying on an exceptional divisor of a blow-up of $\mathbb{P}^3$ in a line. In particular, our method gives an alternative proof of Orlov's conjecture for del Pezzo surfaces, first established by Ballard and Favero.

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Reference graph

Works this paper leans on

6 extracted references · 4 canonical work pages

[1]

Coherent sheaves on Pn and problems of linear algebra

[Bei78] A.A. Beilinson. “Coherent sheaves on Pn and problems of linear algebra”. Eng- lish. In: Functional Analysis and Its Applications 12.3 (1978), pp. 214–216. issn: 0016-2663. doi: 10 . 1007 / BF01681436. url: http : / / dx . doi . org / 10 . 1007 / BF01681436. [BF12] Matthew Ballard and David Favero. “Hochschild dimensions of tilting objects”. In: In...

work page doi:10.1093/imrn/rnr124 1978
[9]

issn: 0219-4988. doi: 10 . 1142 / S0219498816500651. url: https://doi.org/10.1142/S0219498816500651

work page doi:10.1142/s0219498816500651
[10]

Projective Bundles, Monoidal Transformations, and Derived Cate- gories of Coherent Sheaves

17323/1609-4514-2009-9-1-143-149 . url: https://doi.org/10.17323/1609- 4514-2009-9-1-143-149 . [Orl93] D.O. Orlov. “Projective Bundles, Monoidal Transformations, and Derived Cate- gories of Coherent Sheaves”. In: Izvestiya: Mathematics 41 (1993), pp. 133–141. [Rou08] Rapha¨ el Rouquier. “Dimensions of triangulated categories”. In: J. K-Theory 1.2 (2008), ...

doi:10.17323/1609- 1993
[2008]

Algebras of derived dimension zero

url: https://www-fourier.ujf-grenoble. fr/~mbrion/notes_quivers_rev.pdf. [CYZ08] Xiao-Wu Chen, Yu Ye, and Pu Zhang. “Algebras of derived dimension zero”. In: Comm. Algebra 36.1 (2008), pp. 1–10. issn: 0092-7872. doi: 10 . 1080 / 00927870701649184. url: https://doi.org/10.1080/00927870701649184. [EL19] Alexey Elagin and Valery A. Lunts. Three notions of di...

work page doi:10.1080/00927870701649184 2008
[2014]

A note on the Rouquier dimensions of product varieties

arXiv: 1411.7994 [math.AG] . [Yan16] Song Yang. “A note on the Rouquier dimensions of product varieties”. In: J. Algebra Appl. 15.4 (2016), pp. 1650065,

arXiv 2016
[2019]

Three notions of dimension for triangulated categories

arXiv: 1901.09461 [math.AG] . REFERENCES 11 [Kap88] M.M. Kapranov. “On the derived categories of coherent sheaves on some homo- geneous spaces.” In: Inventiones mathematicae 92.3 (1988), pp. 479–508. url: http://eudml.org/doc/143579. [Orl09] Dmitri Orlov. “Remarks on generators and dimensions of triangulated categories”. In: Mosc. Math. J. 9.1 (2009), 153...

work page Pith review arXiv 1988

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Coherent sheaves on Pn and problems of linear algebra

[Bei78] A.A. Beilinson. “Coherent sheaves on Pn and problems of linear algebra”. Eng- lish. In: Functional Analysis and Its Applications 12.3 (1978), pp. 214–216. issn: 0016-2663. doi: 10 . 1007 / BF01681436. url: http : / / dx . doi . org / 10 . 1007 / BF01681436. [BF12] Matthew Ballard and David Favero. “Hochschild dimensions of tilting objects”. In: In...

work page doi:10.1093/imrn/rnr124 1978

[2] [9]

issn: 0219-4988. doi: 10 . 1142 / S0219498816500651. url: https://doi.org/10.1142/S0219498816500651

work page doi:10.1142/s0219498816500651

[3] [10]

Projective Bundles, Monoidal Transformations, and Derived Cate- gories of Coherent Sheaves

17323/1609-4514-2009-9-1-143-149 . url: https://doi.org/10.17323/1609- 4514-2009-9-1-143-149 . [Orl93] D.O. Orlov. “Projective Bundles, Monoidal Transformations, and Derived Cate- gories of Coherent Sheaves”. In: Izvestiya: Mathematics 41 (1993), pp. 133–141. [Rou08] Rapha¨ el Rouquier. “Dimensions of triangulated categories”. In: J. K-Theory 1.2 (2008), ...

doi:10.17323/1609- 1993

[4] [2008]

Algebras of derived dimension zero

url: https://www-fourier.ujf-grenoble. fr/~mbrion/notes_quivers_rev.pdf. [CYZ08] Xiao-Wu Chen, Yu Ye, and Pu Zhang. “Algebras of derived dimension zero”. In: Comm. Algebra 36.1 (2008), pp. 1–10. issn: 0092-7872. doi: 10 . 1080 / 00927870701649184. url: https://doi.org/10.1080/00927870701649184. [EL19] Alexey Elagin and Valery A. Lunts. Three notions of di...

work page doi:10.1080/00927870701649184 2008

[5] [2014]

A note on the Rouquier dimensions of product varieties

arXiv: 1411.7994 [math.AG] . [Yan16] Song Yang. “A note on the Rouquier dimensions of product varieties”. In: J. Algebra Appl. 15.4 (2016), pp. 1650065,

arXiv 2016

[6] [2019]

Three notions of dimension for triangulated categories

arXiv: 1901.09461 [math.AG] . REFERENCES 11 [Kap88] M.M. Kapranov. “On the derived categories of coherent sheaves on some homo- geneous spaces.” In: Inventiones mathematicae 92.3 (1988), pp. 479–508. url: http://eudml.org/doc/143579. [Orl09] Dmitri Orlov. “Remarks on generators and dimensions of triangulated categories”. In: Mosc. Math. J. 9.1 (2009), 153...

work page Pith review arXiv 1988