On the triviality of direct image of coherent sheaves

Indranil Biswas; Jagadish Pine

REVIEW 3 major objections 4 minor 6 references

On the triviality of direct image of coherent sheaves

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For a ramified abelian Galois cover, a vector bundle with trivial direct image exists if and only if every branch divisor lies in the image of the corresponding multiplication map.

desk verdict A useful abelian generalization of known cyclic/double-cover results for relatively Ulrich bundles, but the 'if' direction relies on an unproved matrix-factorization step that a referee should push on. read the letter →

arxiv 2601.20460 v2 pith:ROWWJ2M6 submitted 2026-01-28 math.AG

Indranil Biswas , Jagadish Pine This is my paper · ORCID

classification math.AG MSC 14H3014H6014J6014M10

keywords abeliancoveringdirectimageUlrichbundlerelativelyvectorbranchdivisorcyclicmatrixfactorization

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

The paper asks when a finite morphism $\pi: X \to Y$ of smooth projective varieties has a vector bundle $E$ on $X$ whose direct image $\pi_*E$ is a trivial bundle on $Y$. For a ramified abelian Galois cover, it gives a complete answer: such an $E$ exists if and only if each branch divisor $B_i$ is a sum of $d_i$-fold products of sections of the associated line bundle $M_i$. This is a concrete cohomological condition, and it implies that every smooth ramified abelian Galois cover of projective space carries an Ulrich bundle. The result extends the study of Ulrich bundles to a relative setting.

What carries the argument

The direct-image splitting of $\pi_*\mathcal{O}_X$ for an abelian Galois cover into line bundles $\mathcal{O}_Y$, $M_i^{-1}, \dots, M_i^{-d_i+1}$, and their tensor products, together with the branch divisors $B_i$ viewed as sections $s_i \in H^0(Y, M_i^{\otimes d_i})$, forms the test object. The multiplication map $H^0(Y, M_i)^{\otimes d_i} \to H^0(Y, M_i^{\otimes d_i})$ is the criterion that separates existence from non-existence. For the converse direction, the central mechanism is matrix factorization on the projective bundle $\mathbb{P} = \mathrm{Proj}(\mathrm{Sym}(\mathcal{O}_Y\oplus L^{-1}))$: factorizing the polynomial $T^d - \sum_i(\varpi^*a_1^i\otimes e)\cdots(\varpi^*a_d^i\otimes e)$ into a product of matrices produces a sheaf whose cokernel is the desired relatively Ulrich bundle.

What would settle it

For a cyclic triple cover of $\mathbb{P}^2$ with branch divisor $s = x^3 + y^3 + z^3$, which satisfies the multiplication-map condition, write down the matrix factorization promised in Theorem 4.2 using the cited lemma, form the cokernel, and compute the pushforward $\pi_*E$; if this pushforward is not a trivial bundle, or if the cokernel is not supported on the cover, the sufficient direction is false. A smaller closed test is the degree-3 cover of $\mathbb{P}^1$ branched over three points, where the rank and support of the cokernel can be computed directly.

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Extended reading notes

Core claim

For a finite ramified abelian Galois cover $\pi: X \to Y$ with $\pi_*\mathcal{O}_X$ decomposed into line bundles as in (3.3), a relatively Ulrich vector bundle (one with trivial direct image) exists if and only if, for every $i$, the branch divisor $B_i \in H^0(Y, M_i^{\otimes d_i})$ lies in the image of the $d_i$-fold multiplication map $H^0(Y, M_i)^{\otimes d_i} \to H^0(Y, M_i^{\otimes d_i})$. Necessity is proved by observing that triviality of $\pi_*E$ makes $\mathrm{End}(\pi_*E)$ carry an action of $\pi_*\mathcal{O}_X$, giving endomorphisms $\phi_i$ with $\phi_i^{d_i} = s_i$; applied to the identity endomorphism, each $s_i$ becomes a sum of $d_i$-fold tensor products. Sufficiency is proved by constructing a matrix factorization of $T^{d_i} - \sum_i(\varpi^*a_1^i\otimes e)\cdots(\varpi^*a_{d_i}^i\otimes e)$ over the projective bundle $\mathbb{P} = \mathrm{Proj}(\mathrm{Sym}(\mathcal{O}_Y\oplus L^{-1}))$ whose cokernel is supported on the cyclic cover and pushes forward to a trivial bundle.

Load-bearing premise

The 'if' direction of the main theorem rests on an unproved claim that the polynomial defining the cover can always be factored into square matrices whose cokernel is supported exactly on the cyclic cover and pushes forward to a trivial bundle; the paper invokes a cited lemma rather than verifying these rank and support conditions.

Editorial extensions

If this is right

Every smooth ramified abelian Galois cover of $\mathbb{P}^n$ has an Ulrich bundle with respect to the pullback polarization, by Corollary 1.4.
If a relatively Ulrich bundle exists, then every $M_i$ is globally generated and each branch divisor has a $d_i$-fold tensor decomposition; in particular, nontrivial \'etale covers and covers with small ramification cannot admit such bundles.
The multiplication-map condition is strictly stronger than global generation of $M_i$: Example 4.8 gives a cyclic double cover of an elliptic curve with globally generated $M$ but no relatively Ulrich bundle.
The sufficient condition is stable under composition: if each cyclic cover in the tower satisfies it, the whole abelian cover admits a relatively Ulrich bundle, by Corollary 4.3.
Complete intersection subvarieties of $\mathbb{P}^N$ satisfy the required multiplication-surjectivity for all positive-degree line bundles, giving many explicit abelian covers that admit relatively Ulrich bundles.
The necessary condition (Proposition 3.5) is independent of the matrix-factorization step, so even if the sufficiency proof fails for some degree, the obstruction in terms of branch-divisor factorization remains valid.

Reading between the lines

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

The rank of the relatively Ulrich bundle produced by the matrix-factorization argument depends on the chosen expression of $s$ as a sum of $d$-fold tensor products; a natural question the paper leaves open is whether the minimal rank equals the minimal number of summands in such a decomposition, a Waring-type rank of the branch section.
The criterion can be read as a cohomological divisibility test: it asks whether the section $s_i$ can be expressed as a sum of pure $d_i$-fold tensor products. On varieties where all such multiplication maps are surjective for ample line bundles, the theorem automatically yields relatively Ulrich bundles.
If the unproved matrix-factorization assertion in Theorem 4.2 fails for some degree, the necessary direction survives unchanged, and sufficiency might be restored by a different construction that avoids the cited lemma.

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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

You should know two things about this paper. First, it gives a clean necessary and sufficient condition for a ramified abelian Galois cover to admit a relatively Ulrich bundle: the branch divisors must lie in the image of the d-fold multiplication map. Second, the necessary direction is solid, but the sufficient direction rests on a matrix-factorization construction in Theorem 4.2 that is not actually proved. The gap looks fillable, but it is not cosmetic.

What is new: the abelian case, rather than just cyclic or double covers, and the sharpness example (4.8) showing global generation of the branch line bundle is strictly weaker than the multiplication-map condition. The necessary part (Prop. 3.5) is a nice observation: from the endomorphism algebra of the trivial direct image, you get matrices A_i satisfying A_i^{d_i}=s_i·Id, so s_i is a sum of d_i-fold tensor products. That is clean and convincing. The application to P^n follows immediately once you have the theorem, since the multiplication maps are surjective there. The paper is also honest: it states where it is borrowing from [3] and from the earlier cyclic case.

The soft spot is Theorem 4.2. The proof says 'consider a matrix factorization' and cites [3, Lemma 1.5], then says the existence is 'similar to the case of a homogeneous polynomial.' But the issue is exactly that P = Proj(Sym(O_Y⊕L^{-1})) is a relative P^1-bundle, not a polynomial ring. The cited lemma gives matrix factorizations over strict complete intersections, and it is not automatic that it applies here, with the required size m and with determinant equal to (T^d−π^*s)^{m/d}. Without equality of determinants, the cokernel need not be supported on X; without control of m/d, the rank claim is unjustified. The text says the determinant 'can be shown' but does not show it. This is a load-bearing gap in an iff theorem.

That said, this is not a broken argument. The necessary direction and the examples stand on their own, and I suspect the matrix-factorization step can be fixed by a proper relative version of [3, Lemma 1.5] or by an explicit construction using the tensor decomposition of s. A referee should ask for that before accepting the completeness claim.

Who is this for? People working on Ulrich bundles and covering constructions. It is a useful paper, but it needs revision. I would send it to a serious referee, not desk-reject it.

Referee Report

3 major / 4 minor

Summary. The paper studies finite morphisms π: X → Y of smooth projective varieties over an algebraically closed field of characteristic zero and asks when there exists a vector bundle E on X whose direct image π_*E is trivial, i.e., a relatively Ulrich bundle. For ramified abelian Galois covers, the main theorem (Theorem 1.3) claims a complete answer: such an E exists if and only if, for each i, the branch divisor s_i ∈ H^0(Y, M_i^{⊗d_i}) lies in the image of the d_i-fold multiplication map H^0(Y, M_i)^{⊗d_i} → H^0(Y, M_i^{⊗d_i}). The necessary direction is argued in Proposition 3.5 via the module structure of π_*E, and the sufficient direction is presented in Theorem 4.2 using matrix factorizations on the compactified total space P = Proj(Sym(O_Y ⊕ L^{-1})). The paper also proves that nontrivial étale covers admit no relatively Ulrich bundles, gives examples and counterexamples, and derives Corollary 1.4 that every smooth ramified abelian Galois covering of P^n supports an Ulrich bundle.

Significance. If the main theorem is correct, it gives a clean and checkable criterion for the existence of relatively Ulrich bundles on abelian covers, recovers earlier results for cyclic covers of projective spaces, and produces many new examples and non-examples. The necessary direction and the global-generation consequences (Propositions 3.3, 3.5, and Corollary 3.4) are coherent and constitute a useful contribution. However, the sufficient direction rests on an unproved matrix-factorization assertion in Theorem 4.2, which is the load-bearing step for the 'if' direction of the central theorem. As written, the main theorem is therefore not fully established.

major comments (3)

[Theorem 4.2] The existence of matrices B_1, ..., B_d with entries in H^0(P, ϖ^*L ⊗ O_P(1)) satisfying B_1···B_d = (T^d − Σ_i (ϖ^*a_1^i ⊗ e)···(ϖ^*a_d^i ⊗ e))·id_m is asserted rather than proved. The citation to [3, Lemma 1.5] is insufficient: that lemma concerns matrix factorizations of homogeneous polynomials over a polynomial ring over a field, whereas here the entries must be sections of ϖ^*L ⊗ O_P(1) on the relative P^1-bundle P over Y, and the reduction 'like in a polynomial ring' is not demonstrated. In particular, the proof does not show that det(B_i) = F^{m/d} for F = T^d − Σ_i (ϖ^*a_1^i ⊗ e)···(ϖ^*a_d^i ⊗ e), nor that the cokernel E is supported exactly on the cyclic cover X rather than on a larger closed subscheme. Without these facts, the claimed rank m/d and the conclusion π_*E ≅ O_Y^{⊕m} are unjustified.
[Theorem 4.2, exact sequence (4.2)] The displayed short exact sequence defines E as the cokernel of a single map ×B_i, but the matrix-factorization condition involves the product B_1···B_d = F·id_m. The support of coker(B_i) is controlled by det(B_i), not directly by F unless det(B_i) is a power of F, and that equality is not proved. The manuscript needs either a construction of a single matrix whose determinant is F (which would force rank one and is generally impossible) or a genuine matrix-factorization resolution involving all B_i, together with a proof that the resulting cokernel is supported on the zero scheme of F and that its pushforward is trivial. As written, the support and rank claims in (4.2) do not follow.
[Proposition 3.5] The assertion that π_*E ≅ O_Y^{⊕m} together with the decomposition (3.3) 'is equivalent to' the existence of O_Y-module homomorphisms φ_i: O_Y^{⊕m} → O_Y^{⊕m} ⊗ M_i satisfying the characteristic equations φ_i^{d_i} = s_i is stated without proof. The identification of the π_*O_X-module structure on End(π_*E) with the stated direct summands of the decomposition is not immediate, and it carries the entire necessary direction of the main theorem. A few lines of justification are needed to make this step rigorous.

minor comments (4)

[Corollary 4.3] In the statement of Corollary 4.3, the arrow in the multiplication map is reversed: it should be H^0(Y, M_i)^{⊗d_i} → H^0(Y, M_i^{⊗d_i}), matching Theorem 1.3 and Proposition 3.5.
[Abstract and Introduction] The abstract speaks of a 'coherent sheaf' whose direct image is trivial, while the body of the paper works with vector bundles; the authors should harmonize these statements, especially since the abstract's claim about local freeness appears only in the arXiv-metadata version and not in the main text.
[Proposition 4.7, Case 2] The sentence 'then the branch divisor Z(s) will have a singularity at x, contradicting the smoothness of Y' is inaccurate: Y is smooth regardless of Z(s). The contradiction should be with the smoothness of the cyclic cover X (equivalently, with the smoothness of the branch divisor B), which is part of the hypothesis that π is a covering between smooth varieties.
[Example 4.8] The claim that C is smooth because the branch divisor is chosen outside the image of H^0(D,L) ⊗ H^0(D,L) needs clarification: smoothness of the branch divisor is an additional condition, and the example should explicitly state that a smooth divisor in the complement is chosen.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main iff theorem is not equivalent to its inputs; the unproved matrix-factorization step is a correctness gap, not a circular reduction.

full rationale

The central theorem (Theorem 1.3) is an equivalence between the existence of a relatively Ulrich bundle and a concrete linear-algebra condition on branch divisors. The necessary direction (Proposition 3.5) starts from π_*E ≅ O^m, uses the π_*O_X-module structure of End(π_*E), derives matrices A_i with A_i^{d_i}=s_i·id, and concludes that s_i is a sum of d_i-fold tensor products. This is a real derivation, not an assumption of the conclusion. The sufficient direction (Theorem 4.2) begins with the tensor-product expression for s_i and invokes [3, Lemma 1.5] to obtain a matrix factorization, then defines E as the cokernel. The cited lemma is external (Herzog–Ulrich–Backelin), not a self-citation, and the construction is not merely a restatement of the input condition. The proof does contain an unsupported assertion: that [3, Lemma 1.5] applies over the relative P^1-bundle P=Proj(Sym(O_Y⊕L^{-1})) in the same way as a polynomial ring, and that the determinant of the matrix factorization equals F^{m/d}; these claims are not fully justified. However, this is an incompleteness or correctness gap, not circularity: the paper does not define the existence of E in terms of the multiplication-map condition, nor does it fit a parameter and then rename the fit as a prediction. Self-citations to the coauthored preprint [6] appear in the motivation and in the technique of Proposition 4.4, but Theorem 1.3 does not depend on [6] for its proof; the load-bearing sufficiency step is delegated to the external [3]. Accordingly, no circular step can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or invented entities appear. The argument rests on standard abelian-cover structure and on an external matrix-factorization lemma, both acknowledged in the text.

assumptions (4)

standard math Any ramified abelian Galois cover of smooth projective varieties is a composition of finitely many cyclic covers, and π_*O_X decomposes as a direct sum of line bundles as in (3.3).
Used throughout Sections 3 and 4; standard theory of abelian covers. Invoked in the paragraphs around (3.1)-(3.3).
standard math The existence of matrix factorizations for the polynomial T^d minus a sum of d-fold tensor products, as provided by [3, Lemma 1.5] and extended by analogy to the present setting.
External lemma from Herzog-Ulrich-Backelin; the paper's sufficient criterion depends on it in Theorem 4.2.
domain assumption For a cyclic cover between smooth varieties, the branch divisor is smooth and the cover is defined by t^d = π^*s.
Used in Proposition 3.3 and Theorem 4.2; standard for ramified cyclic covers of smooth varieties.
domain assumption The base field is algebraically closed of characteristic zero and all varieties are smooth projective.
The entire paper works under this hypothesis, stated at the start of the introduction.

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Pith. "Pith review of On the triviality of direct image of coherent sheaves." pith.science (2026). https://pith.science/paper/ROWWJ2M6

@misc{pith2026260120460,
  author       = {Pith},
  title        = {Pith review of: On the triviality of direct image of coherent sheaves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ROWWJ2M6}},
  note         = {Machine review of arXiv:2601.20460}
}

abstract

Let $\pi\,:\, X \,\longrightarrow\, Y$ be a finite morphism of projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for $\pi$ such that there exists a coherent sheaf $E$ on $X$ whose direct image $\pi_*E$ is a trivial vector bundle on $Y$ of positive rank. When $X$ is smooth, and $Y$ is Cohen-Macaulay, such a coherent sheaf is necessarily locally free. We show that the existence of such a coherent sheaf $E$ is guided by the properties of the branching divisor of $\pi$. When the covering $\pi\,:\, X \,\longrightarrow\, Y$ is admissible abelian Galois, we give a complete answer. As an application, it is shown that every smooth admissible abelian Galois covering of $\mathbb{P}^n$ supports an Ulrich bundle.

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Works this paper leans on

6 extracted references · 4 canonical work pages

[3]

Herzog and B

J. Herzog and B. Ulrich and J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections,Jour. Pure Appl. Alg.71(1991), 187–202

work page 1991
[1]

Carlini, L

E. Carlini, L. Chiantini and A. Geramita, Complete intersections on general hypersurfaces,Michigan Math. Jour.57(2008), 121–136

work page 2008
[2]

Eisenbud, F.-O

D. Eisenbud, F.-O. Schreyer and J. Weyman, Resultants and Chow forms via exterior syzygies,Jour. Amer. Math. Soc.16(2003), 537–579

work page 2003
[4]

Huybrechts and M

D. Huybrechts and M. Lehn,The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997

1997
[5]

Mohan Kumar, P

N. Mohan Kumar, P. Narayanan and A. J. Parameswaran, Ulrich bundles on double covers of projective spaces,Jour. Pure Appl. Alg.229, 2025

work page 2025
[6]

A. J. Parameswaran and J. Pine, Ulrich bundles on cyclic coverings of projective spaces, arXiv preprint, arXiv:2408.10837, 2024 Department of Mathematics, Shiv Nadar University, NH91, Tehsil Dadri, Greater Noida, Uttar Pradesh 201314, India Email address:indranil.biswas@snu.edu.in, indranil29@gmail.com Statistics and Mathematics Unit, Indian Statistical I...

arXiv 2024

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [3]

Herzog and B

J. Herzog and B. Ulrich and J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections,Jour. Pure Appl. Alg.71(1991), 187–202

work page 1991

[2] [1]

Carlini, L

E. Carlini, L. Chiantini and A. Geramita, Complete intersections on general hypersurfaces,Michigan Math. Jour.57(2008), 121–136

work page 2008

[3] [2]

Eisenbud, F.-O

D. Eisenbud, F.-O. Schreyer and J. Weyman, Resultants and Chow forms via exterior syzygies,Jour. Amer. Math. Soc.16(2003), 537–579

work page 2003

[4] [4]

Huybrechts and M

D. Huybrechts and M. Lehn,The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997

1997

[5] [5]

Mohan Kumar, P

N. Mohan Kumar, P. Narayanan and A. J. Parameswaran, Ulrich bundles on double covers of projective spaces,Jour. Pure Appl. Alg.229, 2025

work page 2025

[6] [6]

A. J. Parameswaran and J. Pine, Ulrich bundles on cyclic coverings of projective spaces, arXiv preprint, arXiv:2408.10837, 2024 Department of Mathematics, Shiv Nadar University, NH91, Tehsil Dadri, Greater Noida, Uttar Pradesh 201314, India Email address:indranil.biswas@snu.edu.in, indranil29@gmail.com Statistics and Mathematics Unit, Indian Statistical I...

arXiv 2024