Singular Hochschild complex and Cartan matrix

Xiaozhuan Liang; Yu Wang

For symmetric algebras, Hochschild homology of the dg-enhanced singularity category equals singular Hochschild homology.

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load-bearing objection Paper gives two precise equivalences for symmetric and basic algebras plus counterexamples showing failure for general Frobenius algebras.

arxiv 2606.11641 v1 pith:LNPOZQJZ submitted 2026-06-10 math.RT math.CTmath.KTmath.RA

Singular Hochschild complex and Cartan matrix

Yu Wang , Xiaozhuan Liang This is my paper

classification math.RT math.CTmath.KTmath.RA

keywords Hochschild homologysingularity categoryCartan matrixsymmetric algebraFrobenius algebramixed complexdg enhancement

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves that when an algebra A is symmetric, the Hochschild homology obtained from the dg enhancement of its singularity category coincides exactly with the singular Hochschild homology of A. It further shows that, for basic finite-dimensional algebras over a field, the Cartan matrix of A is symmetric if and only if the k-dual of the mixed complex attached to that dg enhancement is isomorphic to its own shift by minus one. The authors supply two explicit counterexamples demonstrating that both statements fail when the algebra is only Frobenius rather than symmetric. These identifications link a classical matrix invariant of the algebra to homological data carried by its singularity category.

Core claim

If A is a symmetric algebra, then Hochschild homology of the dg enhancement of the singularity category of A agrees with singular Hochschild homology of A. For a basic finite dimensional k algebra A, the Cartan matrix of A is symmetric if and only if the k dual of the mixed complex of the dg enhancement of its singularity category is isomorphic to its shift by -1.

What carries the argument

The dg enhancement of the singularity category of A together with its mixed complex.

Load-bearing premise

The standard definitions and constructions of the dg enhancement of the singularity category and the associated mixed complex behave as expected for the algebras considered.

What would settle it

An explicit symmetric algebra for which the two homologies differ, or a basic finite-dimensional algebra whose Cartan matrix is symmetric yet the dual mixed complex fails to be isomorphic to its negative shift.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Paper gives two precise equivalences for symmetric and basic algebras plus counterexamples showing failure for general Frobenius algebras.

read the letter

The main things here are the two conditional results: for symmetric algebras the Hochschild homology of the dg-enhanced singularity category matches singular Hochschild homology, and for basic finite-dimensional k-algebras the Cartan matrix is symmetric precisely when the dual of the mixed complex of that enhancement is isomorphic to its shift by -1. They back this with two explicit counterexamples that fail for general Frobenius algebras.

The paper does well by drawing the boundaries clearly and supplying concrete counterexamples instead of stopping at the positive statements. That makes the claims easier to evaluate and shows the authors checked where the statements stop holding. The work sits squarely in representation theory of finite-dimensional algebras and links singularity categories to classical invariants like the Cartan matrix.

The soft spots are minor and technical rather than foundational. Everything rests on the usual definitions of dg enhancements and mixed complexes; if those are handled correctly in the proofs the results look clean. No circularity shows up in the abstract, and the counterexamples reduce the risk that the claims are overfitted. The restriction to basic algebras in the second statement is stated up front, which is honest.

This is for people already working with singularity categories, Hochschild invariants, and finite-dimensional algebras over a field. A reader in that niche gets direct value from the equivalences and the negative results. It deserves a serious referee because the statements are specific, the counterexamples add credibility, and the topic connects standard tools without obvious gaps.

Referee Report

0 major / 3 minor

Summary. The paper proves that if A is a symmetric algebra then the Hochschild homology of the dg enhancement of its singularity category coincides with the singular Hochschild homology of A. It further shows that, for a basic finite-dimensional k-algebra A, the Cartan matrix of A is symmetric if and only if the k-dual of the mixed complex of the dg enhancement of the singularity category is isomorphic to its shift by -1. Two explicit counterexamples demonstrate that neither statement holds for general Frobenius algebras.

Significance. If the proofs are correct, the results give a precise dictionary between classical invariants (Cartan matrix symmetry) and homological data attached to the dg enhancement of the singularity category, together with a clear demarcation of the symmetric versus Frobenius setting. The explicit counterexamples are a positive feature that makes the scope of the claims falsifiable and concrete.

minor comments (3)

The abstract and introduction should explicitly state the base field k and the standing assumptions on A (finite-dimensional, basic) at the first appearance of each theorem, rather than deferring them to later sections.
Notation for the mixed complex and its dual should be introduced once in a dedicated subsection or table, with consistent symbols across the two main theorems.
The counterexamples in the final section would benefit from a short table summarizing the algebras, their Cartan matrices, and the computed homology groups to make the failure of the statements immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the accurate summary of our results and the positive assessment of their significance. The recommendation for minor revision is noted, but no specific issues or comments were raised in the major comments section.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states two conditional theorems (agreement of homologies for symmetric algebras; Cartan matrix symmetry iff dual mixed complex isomorphism for basic finite-dimensional algebras) and supplies explicit counterexamples showing failure for general Frobenius algebras. These rest on standard definitions of dg enhancements of singularity categories, mixed complexes, and singular Hochschild homology, with no reduction of any claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation is self-contained against external benchmarks and does not invoke uniqueness theorems or ansatzes from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard definitions from homological algebra and representation theory for symmetric algebras, Frobenius algebras, singularity categories, and mixed complexes; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)

domain assumption Standard properties of symmetric and Frobenius algebras over a field k
The statements are conditioned on these classes of algebras as defined in the literature.
domain assumption Existence and basic properties of the dg enhancement of the singularity category and its mixed complex
The claims invoke these constructions without re-deriving them.

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Works this paper leans on

11 extracted references · 1 canonical work pages

[1]

Jorgensen,Tate-Hochschild homology and cohomology of Frobenius algebras, J

Petter Andreas Bergh and David A. Jorgensen,Tate-Hochschild homology and cohomology of Frobenius algebras, J. Noncommut. Geom.7(2013), no. 4, 907–937

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Ragnar-Olaf Buchweitz,Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings,http://hdl.handle.net/1807/16682(1986), 155 pp

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[3]

Xiao-Wu Chen and Zhengfang Wang,Differential graded enhancements of singularity categories, arXiv:2312.12138v1[math.RT]

work page arXiv
[4]

Algebra272(2004), no

Vladimir Drinfeld,DG quotients of DG categories, J. Algebra272(2004), no. 2, 643–691

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Algebra321(2009), no

Ching-Hwa Eu and Travis Schedler,Calabi-Yau Frobenius algebras, J. Algebra321(2009), no. 3, 774–815

2009
[6]

Pure Appl

Bernhard Keller,Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123(1998), no. 1-3, 223–273

1998
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Pure Appl

,On the cyclic homology of exact categories, J. Pure Appl. Algebra136(1999), no. 1, 1–56

1999
[8]

301, Springer-Verlag, Berlin, 1998, Appendix E by María O

Jean-LouisLoday,Cyclic homology, seconded., GrundlehrenderMathematischenWissenschaften[Fun- damental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili

1998
[9]

Dmitri Orlov,Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova246(2004), 240–262

2004
[10]

Noncommut

Manuel Rivera and Zhengfang Wang,Singular Hochschild cohomology and algebraic string operations, J. Noncommut. Geom.13(2019), no. 1, 297–361

2019
[11]

Math.360(2022), 491-496

Yu Wang, Umamaheswaran Arunachalam, and Bernhard Keller,On the Hochschild homology of sin- gularity categories, C.R. Math.360(2022), 491-496. School of Mathematics and Statistics, Taiyuan Normal University, Shanxi 030619, P. R. China Email address:yu.wang@tynu.edu.cn School of Mathematics and Statistics, Taiyuan Normal University, Shanxi 030619, P. R. Chi...

2022

[1] [1]

Jorgensen,Tate-Hochschild homology and cohomology of Frobenius algebras, J

Petter Andreas Bergh and David A. Jorgensen,Tate-Hochschild homology and cohomology of Frobenius algebras, J. Noncommut. Geom.7(2013), no. 4, 907–937

2013

[2] [2]

Ragnar-Olaf Buchweitz,Maximal Cohen–Macaulay modules and Tate cohomology over Gorenstein rings,http://hdl.handle.net/1807/16682(1986), 155 pp

1986

[3] [3]

Xiao-Wu Chen and Zhengfang Wang,Differential graded enhancements of singularity categories, arXiv:2312.12138v1[math.RT]

work page arXiv

[4] [4]

Algebra272(2004), no

Vladimir Drinfeld,DG quotients of DG categories, J. Algebra272(2004), no. 2, 643–691

2004

[5] [5]

Algebra321(2009), no

Ching-Hwa Eu and Travis Schedler,Calabi-Yau Frobenius algebras, J. Algebra321(2009), no. 3, 774–815

2009

[6] [6]

Pure Appl

Bernhard Keller,Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123(1998), no. 1-3, 223–273

1998

[7] [7]

Pure Appl

,On the cyclic homology of exact categories, J. Pure Appl. Algebra136(1999), no. 1, 1–56

1999

[8] [8]

301, Springer-Verlag, Berlin, 1998, Appendix E by María O

Jean-LouisLoday,Cyclic homology, seconded., GrundlehrenderMathematischenWissenschaften[Fun- damental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili

1998

[9] [9]

Dmitri Orlov,Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova246(2004), 240–262

2004

[10] [10]

Noncommut

Manuel Rivera and Zhengfang Wang,Singular Hochschild cohomology and algebraic string operations, J. Noncommut. Geom.13(2019), no. 1, 297–361

2019

[11] [11]

Math.360(2022), 491-496

Yu Wang, Umamaheswaran Arunachalam, and Bernhard Keller,On the Hochschild homology of sin- gularity categories, C.R. Math.360(2022), 491-496. School of Mathematics and Statistics, Taiyuan Normal University, Shanxi 030619, P. R. China Email address:yu.wang@tynu.edu.cn School of Mathematics and Statistics, Taiyuan Normal University, Shanxi 030619, P. R. Chi...

2022