Trace ideals, conductors, and ideals of finite (phantom) projective dimension

Kaito Kimura

REVIEW 2 major objections 5 minor 1 cited by

Trace ideals, conductors, and ideals of finite (phantom) projective dimension

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Trace ideals of big Cohen–Macaulay modules avoid ideals of finite projective dimension and parameter ideals in complete local rings; a two-dimensional Cohen–Macaulay domain refutes the conductor question.

desk verdict Strong Section 3, a fixable c/d swap in the counterexample, and an unpublished Hochster-Yao dependency—worth reviewing but not ready as is. read the letter →

arxiv 2501.03442 v2 pith:EIQ7EAOP submitted 2025-01-07 math.AC

Kaito Kimura This is my paper · ORCID

classification math.AC MSC 13A3513B2213D0213D05

keywords traceidealparametertestconductorF-idealfiniteprojectivedimensiondualizingcomplexbigCohen-Macaulaymodule

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 asks whether the conductor, the parameter test ideal, and trace ideals of big Cohen–Macaulay modules—modules on which some system of parameters acts as a regular sequence—can be contained in ideals of finite projective dimension or in ideals generated by a system of parameters. It proves that in complete local rings with a canonical module such containment never happens for trace ideals, and in quasi-Gorenstein complete local domains the conductor and parameter test ideal also avoid such ideals, even when the ring is not Cohen–Macaulay. It also constructs a two-dimensional Cohen–Macaulay local domain whose conductor is contained in the parameter ideal $(x,w)$, giving a negative answer to an open question about conductors and to its parameter-test-ideal analogue in positive characteristic. If the results are correct, non-containment theorems that were known for Gorenstein rings continue to hold more broadly, while the older conjecture is false in full generality.

What carries the argument

The proof replaces the Cohen–Macaulay hypotheses in earlier trace-ideal arguments with a dualizing complex $D$. The canonical module is $\omega \simeq H^0(D)$, and the key orthogonality statement, Proposition 3.6, is that $H_i(G\otimes^{\mathbf L}_R \operatorname{RHom}_R(N,D))=0$ for $i>0$ whenever $G$ is a bounded complex of free modules satisfying the standard conditions on rank and height (the rank of the $i$-th differential equals the alternating sum of the free ranks, and the ideal of maximal minors has height at least $i$); Lemma 3.8 adds the nonvanishing $H^0(\operatorname{RHom}_R(M,D))\neq 0$ for a big Cohen–Macaulay module $M$. These two facts turn an exactness argument into the desired non-containment. A separate mechanism powers the Frobenius-side result, Theorem 2.4: a cited embedding theorem embeds a module of finite projective dimension into a direct sum of quotients by an $R$-regular sequence, after which a tight-closure contradiction using Frobenius powers shows that a non-zero $F$-ideal cannot sit inside the ideal.

What would settle it

Compute the conductor of the localization at $(x,y,z,w)$ of $R=K[x,y,z,w]/(x^3z-y^2,\,x^3w^4-yz,\,w^4y-z^2)$; if any conductor element lies outside $(x,w)R$, the claimed counterexample to the conductor question fails.

Watch

Extended reading notes

Core claim

The paper's central claim is a non-containment principle for trace ideals: if $R$ is a complete local ring with a canonical module $\omega$, $M$ is a big Cohen–Macaulay module, and $I$ is a proper ideal such that $R/I$ is presented by a bounded complex of free modules satisfying the standard conditions on rank and height, then $\operatorname{tr}_\omega(M)\nsubseteq I\omega$; in particular this covers ideals of finite projective dimension and parameter ideals (Corollary 3.10). When $R$ is quasi-Gorenstein the conclusion upgrades to $\operatorname{tr}_R(M)\nsubseteq I$, and for a complete local domain the conductor is therefore never contained in such an ideal. In prime characteristic the parameter test submodule $\tau(\omega)$ is shown to equal $\operatorname{tr}_\omega(R^+)$, so $\tau(\omega)\nsubseteq I\omega$ whenever $R/I$ has finite phantom projective dimension, and the parameter test ideal itself avoids $I$ in the quasi-Gorenstein case (Corollary 3.11). The paper also presents two explicit rings: one shows that a parameter test ideal of a reduced, equidimensional, complete, two-dimensional Cohen–Macaulay ring need not be an $F$-ideal, and the other, a two-dimensional Cohen–Macaulay local domain, has its conductor contained in the parameter ideal $(x,w)R$, giving a negative answer to the conductor question.

Load-bearing premise

A theorem quoted from a preliminary manuscript says that every module with a finite free resolution embeds, in a controlled way, into a direct sum of quotients by regular sequences; the paper's main F-ideal non-containment proof assumes that theorem is valid exactly as cited, while the counterexample is self-contained.

Editorial extensions

If this is right

In every complete local ring with a canonical module, the trace of any big Cohen–Macaulay module is not contained in $I\omega$ when $I$ has finite projective dimension, is a parameter ideal, or more generally $R/I$ has a free complex satisfying the standard conditions on rank and height.
In quasi-Gorenstein complete local rings, the trace ideal itself, and hence the conductor in the domain case, is not contained in any such $I$.
In a complete local domain of prime characteristic, the parameter test submodule $\tau(\omega)$ is not contained in $I\omega$ whenever $R/I$ has finite phantom projective dimension; in the quasi-Gorenstein case the parameter test ideal itself is not contained in $I$.
The conductor question is answered negatively: a two-dimensional analytically unramified Cohen–Macaulay local domain has its conductor contained in the parameter ideal $(x,w)$, and in positive characteristic its parameter test ideal is likewise contained in $(x,w)$.
The parameter test ideal of a reduced equidimensional complete two-dimensional Cohen–Macaulay ring need not be an $F$-ideal, so the earlier Gorenstein argument cannot extend to all Cohen–Macaulay rings without extra hypotheses.

Reading between the lines

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

One could test whether the $F$-ideal non-containment of Theorem 2.4 extends to every excellent reduced local ring of prime characteristic, since the proof's test-element step appears to work once the embedding theorem is granted.
The counterexample suggests that the real boundary for conductor non-containment may be quasi-Gorensteinness rather than Cohen–Macaulayness; searching among Cohen–Macaulay local rings of minimal multiplicity for which conductors are or are not contained in parameter ideals would sharpen that line.
Because $\tau(\omega)=\operatorname{tr}_\omega(R^+)$, any construction of other big Cohen–Macaulay algebras with controlled trace could transfer the non-containment theorem to mixed characteristic, where Frobenius and tight closure are unavailable.

Share X Bluesky LinkedIn Reddit HN

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

Fair warning: the paper's headline counterexample has a real bug. In Example 2.9, Claim 3 defines c by 2a ≤ 3n and d by 2b ≤ 3n, then asserts [x^a w^d]/[z] is integral using 3d ≥ 2b and [x^c w^b]/[y] using 3c ≥ 2a. The definitions are swapped. With a=3, b=4, d=2 and 3d−2b = −2, so the cube is not in R. The fix is mechanical—take c = ceil(2a/3) and d = ceil(2b/3)—and the rest of the example's coefficient argument then goes through, but as written the negative answer to Huneke-Swanson is unsupported. This is not a fatal blow, but it is a load-bearing proof flaw.

What the paper does well is separate. Section 3 is a clean generalization of Dey-Dutta to non-Cohen-Macaulay rings via the dualizing complex. The standard-rank-and-height hypothesis is a genuine idea, and the derived-category arguments look coherent. I checked the structure of Proposition 3.6 and Theorem 3.9; the claims are not just restatements, and the proof of the key exactness in Theorem 3.9 is plausible. This part deserves a serious referee. The same goes for Theorem 2.4, but it leans on Hochster-Yao's unpublished 'Preliminary Version' [15]. That is a real availability risk; the paper should either include a proof or cite a public version. Another soft spot: several Macaulay2 computations (Example 2.8, Remark 2.10(3)) are asserted without code or output. Minor, but annoying.

So: the core ideas are good, the Section 3 results look sound, and the counterexample is probably fixable. But the written counterexample is broken, and the unpublished dependency should be resolved. I would send it to peer review, with the complaint to the author explicit. The reader's conditional verdict is about right, though I'd say the c/d swap is more visible than the H-Y dependency.

Referee Report

2 major / 5 minor

Summary. The paper studies containment of parameter test ideals, conductors, F-ideals, and trace ideals in ideals whose quotient has finite (phantom) projective dimension. It proves several non-containment results, most notably Theorem 2.4 (non-zero F-ideals are not contained in ideals of finite projective dimension under Cohen-Macaulay or excellent equidimensional reduced hypotheses), Corollary 2.6, and the derived-category Theorem 3.9/Corollary 3.10 (trace ideals of big Cohen-Macaulay modules avoid such ideals in complete local rings, with consequences for conductors and parameter test ideals). It also constructs two families of examples: Example 2.8, where the parameter test ideal is not an F-ideal, and Example 2.9, a two-dimensional Cohen-Macaulay local domain whose conductor is contained in a parameter ideal, answering negatively a question of Huneke-Swanson.

Significance. If the results are correct, they substantially extend earlier non-containment theorems of Smith, Dey-Dutta, and Asgharzadeh from Gorenstein or Cohen-Macaulay rings to larger classes, and they give a negative answer to the Huneke-Swanson question with a self-contained two-dimensional example. The use of standard conditions on rank and height in place of finite projective dimension is a conceptually valuable generalization. The paper also contains a self-contained counterexample to [31, Proposition 4.5] (Example 2.8). However, the central Section 2 argument depends on an unpublished 'Preliminary Version' of Hochster-Yao, and several computational claims are delegated to unshown Macaulay2 code, which limits verifiability.

major comments (2)

[Theorem 2.4 proof, use of [15]] The proof of Theorem 2.4 depends essentially on [15, Theorem 2.3], which is cited only as a 'Preliminary Version.' This theorem supplies the embedding R/I into a direct sum of quotients by a regular sequence on which the entire contradiction argument rests, and Theorem 2.4 in turn feeds Corollary 2.6. Without access to the statement (or proof) of [15, Theorem 2.3], the main non-containment theorem of Section 2 is not checkable by readers. Please quote the theorem explicitly, provide a proof in an appendix, or update to a published reference.
[Example 2.8, final paragraph; Remark 2.10(3)] Several claims are assigned to Macaulay2 without accompanying code or a reproducible transcript: that the parameter test ideal of R is (x,y,z)R in Example 2.8, and that for a≤2 or b≤2 the conductor and parameter test ideal of S and R_m are not contained in a parameter ideal in Remark 2.10(3). These claims are not load-bearing for the Huneke-Swanson counterexample in Example 2.9, but they are presented as part of the evidence for the paper's main examples. Please supply the Macaulay2 code or replace these assertions with self-contained arguments.

minor comments (5)

[Example 2.9, Claim 3] The definitions c = min{n : 2a ≤ 3n} and d = min{n : 2b ≤ 3n} are correct as written: they give 3c ≥ 2a and 3d ≥ 2b, so the displayed integrality computations are consistent. A concern that these definitions are swapped does not appear to be supported by the manuscript text.
[Example 2.9, opening] The maximal ideal is written m = (x,y,z,v,w)R, but the ring involves only the variables x,y,z,w; this should read (x,y,z,w)R.
[Theorem 2.4 proof] The notation 'Jr^p_{ij}' is not defined; it denotes the ideal generated by elements z r_{ij}^p with z in J, and should be introduced for clarity.
[Example 2.9, Claim 3] The phrase 'Due to symmetry' in the argument that D = E = 0 is terse, since the defining equations are not literally symmetric under exchanging x and w along with a and b. Expanding the analogous coefficient argument would improve readability.
[Proposition 3.6(1), proof] The reduction to the finite-length case is compressed: after localizing at the chosen minimal prime p, the argument that the resulting homology modules have finite length should be stated more explicitly, as it is a key step in the general case.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the non-containment theorems are derived from independent external results, with no fitted parameters or self-referential definitions.

full rationale

The paper does not exhibit any of the enumerated circularity patterns. The main results are proved from external ingredients rather than from their own conclusions. Theorem 2.4 invokes the Hochster–Yao embedding theorem [15, Theorem 2.3] and standard tight-closure results; this is an independent structural theorem, not an assertion equivalent to the F-ideal non-containment being proved. Corollary 3.10 is derived through the dualizing-complex orthogonality of Proposition 3.6, Lemma 3.8, and the Dey–Dutta strategy; the trace ideal tr_ω(M) is defined as a sum of homomorphic images and Iω is an arbitrary proper module, so the non-containment conclusion is not built into the definition. There are no fitted parameters or data-driven 'predictions': no quantity is adjusted to a subset of data and then reported as a prediction. There is no load-bearing self-citation: the author cites no prior work of his own, and the only preliminary-version citation ([15]) is to Hochster–Yao, not to the present authors. The paper explicitly notes in Remark 3.2 that the standard conditions are treated as independent hypotheses and that some supporting facts are not even required, which further reduces any concern that the conclusion is being assumed. The caveats that exist are correctness or verification risks, not circularity: [15] is a Preliminary Version, Smith's Proposition 4.5 gap is explicitly acknowledged in Remark 2.10(4), and some claims in Remark 2.10(3) rely on Macaulay2 computations. In particular, the alleged swap of c and d in Example 2.9 is not present in the full text: the paper defines c by 2a ≤ 3n and d by 2b ≤ 3n, matching the displayed cube computations, so the example's internal consistency is a separate mathematical question. None of these issues makes a derivation reduce to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the external theorems listed above. There are no fitted parameters and no invented entities. The most fragile input is the unpublished Hochster-Yao theorem [15], used in Theorem 2.4. The Macaulay2 computations are empirical checks rather than proofs. All other axioms are standard published results in tight closure theory and commutative algebra.

assumptions (5)

domain assumption Hochster-Yao embedding theorem: for a local ring R and an R-module of finite projective dimension, there is a regular sequence x_1,...,x_s and an exact sequence 0 -> R/I -> direct sum of (R/(x_1,...,x_i))^{n_i} -> N -> 0.
Used at the start of Theorem 2.4; the cited reference [15] is an unpublished Preliminary Version, so the theorem cannot be independently checked by the reader. This is a load-bearing external input for the F-ideal result.
domain assumption Existence of big Cohen-Macaulay algebras: over a complete local domain, the absolute integral closure R+ is a big Cohen-Macaulay R-module (Andre and others).
Used in Corollaries 3.10 and 3.11 to translate trace-ideal non-containment of a big CM module into conductor and parameter test ideal statements.
standard math Stably phantom acyclic complexes satisfy standard conditions on rank and height [11, Theorem 9.8]; existence of test elements [13, Theorem 6.1(a)].
Used in Theorem 2.4 (test element) and Corollary 3.11 (phantom projective dimension implies standard conditions).
standard math Karpilovsky's irreducibility criterion for X^n - a over fraction fields [20, Chapter 8, Theorem 1.6].
Used in Remark 2.7 and Example 2.9 to show certain hypersurfaces are integral domains.
domain assumption Macaulay2 computations: the parameter test ideal of Example 2.8 equals (x,y,z)R; conductor and test ideal facts in Remark 2.10(3) are accepted as correct without shipped code.
Several statements rely on computer algebra results stated as 'according to Macaulay2'. No scripts or logs are provided, so these are empirical inputs rather than verified proofs.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Trace ideals, conductors, and ideals of finite (phantom) projective dimension." pith.science (2026). https://pith.science/paper/EIQ7EAOP

@misc{pith2026250103442,
  author       = {Pith},
  title        = {Pith review of: Trace ideals, conductors, and ideals of finite (phantom) projective dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EIQ7EAOP}},
  note         = {Machine review of arXiv:2501.03442}
}

abstract

In this paper, we consider whether parameter test ideals, conductors, $F$-ideals, and trace ideals are contained in an ideal whose quotient ring has finite phantom projective dimension (for example, ideals generated by a system of parameters or ideals with finite projective dimension). One of the main results asserts that such inclusions do not exist in quasi-Gorenstein complete local domains. We also provide examples of Cohen-Macaulay local rings with good properties where such inclusions occur, thus answering negatively a question of Huneke-Swanson.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

When do pseudo-Gorenstein rings become Gorenstein?
math.AC 2025-02 conditional novelty 6.0 of 10

A pseudo-Gorenstein graded ring becomes Gorenstein when the trace ideal of its canonical module contains a length-two regular sequence in the initial degree, with applications to nearly and almost Gorenstein rings.

Reference graph

Works this paper leans on

32 extracted references · 30 canonical work pages · cited by 1 Pith paper

[15]

Hochster; Y

M. Hochster; Y. Yao , An embedding theorem for modules of ﬁnite (G-)projective d imension, Preliminary Version

work page
[1]

I. M. Aberbach , Finite phantom projective dimension, Amer. J. Math. 116 (1994), no. 2, 447—477

work page 1994
[2]

Andr ´e, La conjecture du facteur direct, Publ

Y. Andr ´e, La conjecture du facteur direct, Publ. Math. Inst. Hautes ´Etudes Sci. 127 (2018), 71-–93

work page 2018
[3]

Andr ´e, W eak functoriality of Cohen–Macaulay algebras, J

Y. Andr ´e, W eak functoriality of Cohen–Macaulay algebras, J. Amer. Math. Soc. 33 (2020), no. 2, 363–380

work page 2020
[4]

Homology of conductor

M. Asgharzadeh , Homology of conductor, preprint (2024), arXiv:2404.02005

work page Pith review arXiv 2024
[5]

Bhatt , Cohen–Macaulayness of absolute integral closures, prepr int (2021), arXiv:2008.08070

B. Bhatt , Cohen–Macaulayness of absolute integral closures, prepr int (2021), arXiv:2008.08070

arXiv 2021
[6]

Bruns; J

W. Bruns; J. Herzog , Cohen–Macaulay rings, revised edition, Cambridge Studie s in Advanced Mathematics 39, Cambridge University Press, Cambridge , 1998

1998
[7]

D. A. Buchsbaum; D. Eisenbud , What makes a complex exact?, J. Algebra 25 (1973), 259—268

work page 1973

Show all 32 references

[8]

L. W. Christensen , Gorenstein dimensions, Lecture Notes in Math. 1747, Springer-Verlag, Berlin , 2000

2000
[9]

S. Dey; M. Dutta , On containment of trace ideals in ideals of ﬁnite homologic al dimension, preprint (2024), arXiv:2312.01230

2024 arXiv
[10]

Enescu; M

F. Enescu; M. Hochster , The Frobenius structure of local cohomology, Algebra Number Theory 2 (2008), no. 7, 721—754

2008
[11]

Hochster; C

M. Hochster; C. Huneke , Tight closure, invariant theory, and the Brian¸ con-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31-–116

1990
[12]

Hochster; C

M. Hochster; C. Huneke , Inﬁnite integral extensions and big Cohen–Macaulay algeb ras, Ann. of Math. (2) 135 (1992), no. 1, 53—89

1992
[13]

Hochster; C

M. Hochster; C. Huneke , F -regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1—62

1994
[14]

Hochster; C

M. Hochster; C. Huneke , Applications of the existence of big Cohen–Macaulay algeb ras, Adv. Math. 113 (1995), no. 1, 45–117

1995
[16]

Huneke , Tight closure, parameter ideals, and geometry, in Six Lectures on Commutative Algebra , 187—239

C. Huneke , Tight closure, parameter ideals, and geometry, in Six Lectures on Commutative Algebra , 187—239. Mod. Birkh¨ auser Class. Birkh¨ auser Verlag, Basel, 2010

2010
[17]

Huneke; I

C. Huneke; I. Sw anson , Integral closure of ideals, rings, and modules, London Math. Soc. Lecture Note Ser. , 336 Cambridge University Press, Cambridge , (2006)

2006
[18]

Ikeda , Conductor ideals of Gorenstein domains and local rings of m ultiplicity 2, The Annals of Gifu University for Education and Languages 33 (1997) 109–116

S. Ikeda , Conductor ideals of Gorenstein domains and local rings of m ultiplicity 2, The Annals of Gifu University for Education and Languages 33 (1997) 109–116. 14 KAITO KIMURA

1997
[19]

S. B. Iyengar; L. Ma; K. Schwede; M. E. W alker , Maximal Cohen–Macaulay complexes and their uses: A partia l survey, Commutative algebra , 475—500. Springer, Cham , (2021)

2021
[20]

Karpilovsky , Topics in ﬁeld theory, North-Holland Math

G. Karpilovsky , Topics in ﬁeld theory, North-Holland Math. Stud., 155 Notas Mat., 124[Mathematical Notes] North- Holland Publishing Co., Amsterdam, (1989)

1989
[21]

Katzman , Parameter-test-ideals of Cohen–Macaulay rings, Compos

M. Katzman , Parameter-test-ideals of Cohen–Macaulay rings, Compos. Math. 144 (2008), no. 4, 933—948

2008
[22]

Kaw asaki, On arithmetic Macaulayﬁcation of Noetherian rings, Trans

T. Kaw asaki, On arithmetic Macaulayﬁcation of Noetherian rings, Trans. Amer. Math. Soc. 354 (2002), no. 1, 123– 149

2002
[23]

Kumashiro; N

S. Kumashiro; N. Matsuoka; T. Nakashima , Nearly Gorenstein local rings deﬁned by maximal minors of a 2 × n matrix, preprint (2023), arXiv:2308.04234

2023 arXiv
[24]

L. Ma; K. Schwede , Singularities in mixed characteristic via perfectoid big Cohen–Macaulay algebras, Duke Math. J., 170 (2021), no. 13, 2815—2890

2021
[25]

Maitra , Partial trace ideals and Berger’s conjecture, J

S. Maitra , Partial trace ideals and Berger’s conjecture, J. Algebra 598 (2022), 1—23

2022
[26]

Murayama, Uniform bounds on symbolic powers in regular rings, To appe ar in J

T. Murayama, Uniform bounds on symbolic powers in regular rings, To appe ar in J. Reine Angew. Math
[27]

P ´erez; Rebecca R

F. P ´erez; Rebecca R. G. , Characteristic-free test ideals, Trans. Amer. Math. Soc. Ser. B 8 (2021), 754-–787

2021
[28]

Peskine; L

C. Peskine; L. Szpiro , Dimension projective ﬁnie et cohomologie locale, Inst. Hautes ´Etudes Sci. Publ. Math. (1973), no. 42, 47—119

1973
[29]

Shimomoto , Integral perfectoid big Cohen–Macaulay algebras via Andr ´ e’s theorem,Math

K. Shimomoto , Integral perfectoid big Cohen–Macaulay algebras via Andr ´ e’s theorem,Math. Ann. 372 (2018), no. 3-4, 1167–1188

2018
[30]

K. E. Smith , Tight closure of parameter ideals, Invent. Math. 115 (1994), no. 1, 41-–60

1994
[31]

K. E. Smith , Test ideals in local rings, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3453-–3472

1995
[32]

K. E. Smith , F -rational rings have rational singularities, Amer. J. Math. 119 (1997), no. 1, 159-–180. Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan Email address : m21018b@math.nagoya-u.ac.jp

1997

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [15]

Hochster; Y

M. Hochster; Y. Yao , An embedding theorem for modules of ﬁnite (G-)projective d imension, Preliminary Version

work page

[2] [1]

I. M. Aberbach , Finite phantom projective dimension, Amer. J. Math. 116 (1994), no. 2, 447—477

work page 1994

[3] [2]

Andr ´e, La conjecture du facteur direct, Publ

Y. Andr ´e, La conjecture du facteur direct, Publ. Math. Inst. Hautes ´Etudes Sci. 127 (2018), 71-–93

work page 2018

[4] [3]

Andr ´e, W eak functoriality of Cohen–Macaulay algebras, J

Y. Andr ´e, W eak functoriality of Cohen–Macaulay algebras, J. Amer. Math. Soc. 33 (2020), no. 2, 363–380

work page 2020

[5] [4]

Homology of conductor

M. Asgharzadeh , Homology of conductor, preprint (2024), arXiv:2404.02005

work page Pith review arXiv 2024

[6] [5]

Bhatt , Cohen–Macaulayness of absolute integral closures, prepr int (2021), arXiv:2008.08070

B. Bhatt , Cohen–Macaulayness of absolute integral closures, prepr int (2021), arXiv:2008.08070

arXiv 2021

[7] [6]

Bruns; J

W. Bruns; J. Herzog , Cohen–Macaulay rings, revised edition, Cambridge Studie s in Advanced Mathematics 39, Cambridge University Press, Cambridge , 1998

1998

[8] [7]

D. A. Buchsbaum; D. Eisenbud , What makes a complex exact?, J. Algebra 25 (1973), 259—268

work page 1973