{"id":"9af23659-b46e-4aa1-a033-ff1cfa0eb4d6","arxiv_id":"2501.03442","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper unifies and extends several non-containment results for trace and test ideals, and answers an open question of Huneke-Swanson with an explicit counterexample.","lead":"This paper proves that in quasi-Gorenstein complete local domains, special ideals such as test ideals, conductors, and trace ideals cannot lie inside an ideal of finite projective dimension or a parameter ideal. It also constructs a two-dimensional Cohen-Macaulay local domain where the conductor is contained in a parameter ideal, giving a negative answer to a question of Huneke and Swanson.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example 2.9 Claim 3 swaps definitions of c and d: the claimed integral elements can have negative exponents, undermining the Huneke–Swanson counterexample as written.","rationale":"The reader's weakest assumption targeted the unpublished Hochster–Yao theorem used in Theorem 2.4. That is a real external dependency, but it does not affect the paper's central counterexample to Huneke–Swanson, which the reader believed to be self-contained. The more acute problem is internal to Example 2.9. In Claim 3 the integrality of [x^a w^d]/[z] requires 3d ≥ 2b, while the displayed definition of d gives only 3d ≥ 2a; symmetrically, [x^c w^b]/[y] requires 3c ≥ 2a, but c is defined via 2b. The two definitions are swapped. This is almost certainly a transposition typo rather than a fatal flaw, since with the corrected definitions the subsequent inequalities and coefficient argument appear to be what the author intended. However, as written the proof of the conductor containment is invalid, and this directly undermines the stated negative answer to Huneke–Swanson. Because the error is local and readily repairable, the appropriate verdict remains CONDITIONAL, matching the reader's verdict; the reason for conditionality should be updated accordingly.","tokens_in":21123,"tokens_out":27962,"duration_ms":245416,"concrete_test":"Correct Claim 3 by setting c = ⌈2a/3⌉ and d = ⌈2b/3⌉, and use the elements [x^a w^d]/[z] and [x^c w^b]/[y]. Re-run the coefficient argument (now using d < b and c < a) to verify that C ⊆ (x,w)R_m. Independently, for a concrete pair such as (a,b) = (3,4), compute the conductor in Macaulay2 and check whether it is contained in (x,w)R_m; also check that the uncorrected elements genuinely fail to be integral.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Example 2.9, Claim 3 sets c = min{n | 2b ≤ 3n} and d = min{n | 2a ≤ 3n}, then asserts that [x^a w^d]/[z] and [x^c w^b]/[y] are integral over R_m. But the displayed cube computation gives ([x^a w^d]/[z])^3 = [x^{2a} w^{3d-2b}]. This lies in R only if 3d ≥ 2b; the definition of d only guarantees 3d ≥ 2a. For example, when a=3, b=4, d=2 and 3d-2b = -2, so the cube is not in R. Similarly, ([x^c w^b]/[y])^3 = [x^{3c-2a} w^{2b}] requires 3c ≥ 2a, but c is defined via 2b. The definitions of c and d are swapped: one needs c = ⌈2a/3⌉ and d = ⌈2b/3⌉ for the two elements to be integral. The later inequality d < b, used for the coefficient argument, also holds only for the corrected d. This is an internal inconsistency in the proof of the conductor containment, which is the paper's negative answer to the Huneke–Swanson question.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21327,"tokens_out":16933,"duration_ms":142139,"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":[{"comment":"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.","section":"Theorem 2.4 proof, use of [15]"},{"comment":"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.","section":"Example 2.8, final paragraph; Remark 2.10(3)"}],"minor_comments":[{"comment":"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.","section":"Example 2.9, Claim 3"},{"comment":"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.","section":"Example 2.9, opening"},{"comment":"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.","section":"Theorem 2.4 proof"},{"comment":"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.","section":"Example 2.9, Claim 3"},{"comment":"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.","section":"Proposition 3.6(1), proof"}],"recommendation":"major_revision","confidential_remarks":"This is a serious contribution to commutative algebra. The main counterexample appears sound, and the derived-category framework is attractive. The two blocking issues are the reliance on an unpublished Hochster-Yao theorem for the central Section 2 result and the unverifiable Macaulay2 claims. If the author can make the Hochster-Yao material available and provide code or proofs for the computational assertions, I would support acceptance. The alleged swap of c and d in Example 2.9 is not present in the manuscript as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nSo: 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.","headline":"Strong Section 3, a fixable c/d swap in the counterexample, and an unpublished Hochster-Yao dependency—worth reviewing but not ready as is.","tokens_in":21892,"tokens_out":3403,"would_cite":false,"duration_ms":29561,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A35","13B22","13D02","13D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["trace ideal","parameter test ideal","conductor","F-ideal","finite projective dimension","parameter ideal","dualizing complex","big Cohen-Macaulay module"],"falsifier":"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.","tokens_in":20864,"feed_emoji":"🎯","tokens_out":18836,"duration_ms":162426,"temperature":0.7,"pith_summary":"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.","feed_headline":"Conductor conjecture fails in a two-dimensional local domain","feed_subtitle":"Trace ideals still escape ideals of finite projective dimension in complete local rings; a two-dimensional domain refutes the conductor…","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the embedding theorem for modules of finite projective dimension that starts the contradiction proof of Theorem 2.4.","marker":"[15]"},{"why":"Defines F-ideals and parameter test ideals and supplies the Gorenstein non-containment statement that Theorem 2.4 and Remark 2.10(4) take as their starting point.","marker":"[31]"},{"why":"Gives the trace-ideal non-containment theorem for Cohen–Macaulay rings that Section 3 generalizes beyond Cohen–Macaulayness.","marker":"[9]"},{"why":"Equates the conductor question with the monomial conjecture, making the counterexample a negative answer to that question.","marker":"[18]"},{"why":"Provides the quasi-Gorenstein conductor non-containment result that Corollary 3.10 refines in the complete local case.","marker":"[4]"},{"why":"Supplies tight closure, test elements, and the phantom-acyclicity facts behind the standard conditions on rank and height.","marker":"[11]"},{"why":"Sets up tight closure of parameter ideals and identifies the parameter test submodule with the trace of $R^+$ used in Corollary 3.11.","marker":"[30]"},{"why":"Gives the big Cohen–Macaulay algebra construction whose trace lies inside the conductor, completing Corollary 3.10.","marker":"[12]"}],"fun_headline_variants":["Trace ideals evade finite projective dimension ideals in complete rings","No conductor containment in finite phantom projective dimension ideals","2D domain answers Huneke-Swanson: conductor containment possible","Quasi-Gorenstein complete local domains resist trace ideal containment"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Trace ideals evade finite projective dimension ideals in complete rings","No conductor containment in finite phantom projective dimension ideals","2D domain answers Huneke-Swanson: conductor containment possible","Quasi-Gorenstein complete local domains resist trace ideal containment"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001022,"raw_usage":{"total_tokens":4301,"prompt_tokens":927,"completion_tokens":3374,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":3307}},"tokens_in":543,"tokens_out":3374,"duration_ms":22303,"temperature":1.0,"reasoning_tokens":3307,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:53:45.349892+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Hochster; Y","cited_arxiv_id":null,"evidence_quote":"Supplies the embedding theorem for modules of finite projective dimension that starts the contradiction proof of Theorem 2.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines F-ideals and parameter test ideals and supplies the Gorenstein non-containment statement that Theorem 2.4 and Remark 2.10(4) take as their starting point."},{"cited_title":"On containment of trace ideals in ideals of finite homological dimension","cited_arxiv_id":"2312.01230","evidence_quote":"Gives the trace-ideal non-containment theorem for Cohen–Macaulay rings that Section 3 generalizes beyond Cohen–Macaulayness."},{"cited_title":"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","cited_arxiv_id":null,"evidence_quote":"Equates the conductor question with the monomial conjecture, making the counterexample a negative answer to that question."},{"cited_title":"Homology of conductor","cited_arxiv_id":"2404.02005","evidence_quote":"Provides the quasi-Gorenstein conductor non-containment result that Corollary 3.10 refines in the complete local case."},{"cited_title":"Hochster; C","cited_arxiv_id":null,"evidence_quote":"Supplies tight closure, test elements, and the phantom-acyclicity facts behind the standard conditions on rank and height."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sets up tight closure of parameter ideals and identifies the parameter test submodule with the trace of $R^+$ used in Corollary 3.11."},{"cited_title":"Hochster; C","cited_arxiv_id":null,"evidence_quote":"Gives the big Cohen–Macaulay algebra construction whose trace lies inside the conductor, completing Corollary 3.10."}],"review_version":1}